Proceedings ISSAC (Beijing)

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Foreword ISSAC 2005 is a continuation of a well-established series of international conferences for the presentation of the latest advances in the field of Symbolic and Algebraic Computation. The first meeting of the series (1966) was held in Washington, DC, and sponsored by the Association for Computing Machinery (ACM). Since then, the abbreviated name of the meeting has evolved from SYMSAM, SYMSAC, EUROSAM, EUROCAL to finally settle on the present name ISSAC. This 30th meeting was hosted by the Key Laboratory of Mathematics Mechanization, Chinese Academy of Sciences, Beijing, China from July 24 to July 27. The topics of the conference include, but are not limited to: •

Algorithmic mathematics. Algebraic, symbolic and symbolic-numeric algorithms. Simplification, function manipulation, equations, summation, integration, ODE/PDE, linear algebra, number theory, group and geometric computing.

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Computer Science. Theoretical and practical problems in symbolic computation. Systems, problem solving environments, user interfaces, software, libraries, parallel/distributed computing and programming languages for symbolic computation, concrete analysis, benchmarking, theoretical and practical complexity of computer algebra algorithms, automatic differentiation, code generation, mathematical data structures and exchange protocols.

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Applications. Problem treatments using algebraic, symbolic or symbolic-numeric computation in an essential or a novel way. Engineering, economics and finance, physical and biological sciences, computer science, logic, mathematics, statistics, education.

Following tradition, ISSAC 2005 featured invited talks, contributed papers, tutorials, poster sessions, software exhibitions, and satellite workshops. This volume contains all the contributed papers which were presented at the meeting as well as the abstracts of the invited talks. The picture on the front cover shows a page from the classic Chinese math book bearing the title “Jade Mirrors of Four Elements” by Zhu Shijie, written in 1303 AD during the Yuan Dynasty. In this page, a system of equations of three unknowns and degree three is reduced to a univariate equation by eliminating variables. For ISSAC 2005, a total of 111 papers were submitted and each was distributed to members of the program committee and external reviewers. An average of 2.5 referee reports was obtained for each submission, and finally 48 papers were selected for presentation. We are particularly pleased that the papers in these proceedings represent such a broad spectrum of topics found in the field of computer algebra. Of course it is our pleasure to acknowledge the contributions of all the researchers and educators who submitted papers for consideration and all those who assisted in the selection process. We would also like to express our sincere gratitude to all the organizers listed in the front material of these proceedings. The success of ISSAC 2005 is in large part due to the efforts of these people. Finally, we thank the Association for Computing Machinery (ACM) and its Special Interest Group on Symbolic and Algebraic Computation (SIGSAM) for their sponsorship and for their assistance in the organization. We also would like to thank the following sponsors •

the National Natural Science Foundation of China (NSFC),

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the Chinese Academy of Mathematics and Systems Science (AMSS),

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the Institute of Systems Sciences (ISS),

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the Key Laboratory of Mathematics Mechanization (KLMM),

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Maplesoft Inc.

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and the Institut national de recherche en informatique et automatique (INRIA) for both their financial and logistic support.

We write this note in anticipation that the attendees of the ISSAC 2005 conference will find the experience both scientifically rewarding and personally satisfying.

Xiao-Shan Gao, General Co-Chair

George Labahn, General Co-Chair

Peter Paule, Program Chair

Manuel Kauers, Editor

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These proceedings are dedicated to the memory of our friend and colleague

Manuel Bronstein who died suddenly in June 2005. He will be missed by us all.

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Table of Contents Dedication to Manuel Bronstein ...............................................................................................................v ISSAC 2005 Conference Organization..................................................................................................xi ISSAC Steering Committee ........................................................................................................................xii Reviewers .............................................................................................................................................................xii Sponsor & Supporters .................................................................................................................................xiv Invited Talks •

A View on the Future of Symbolic Computation .......................................................................................1

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D-finiteness: Algorithms and Applications.................................................................................................2

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On a Finite Kernel Theorem for Polynomial-Type Optimization Problems and Some of its Applications .............................................................................................................................................4

B. Buchberger (Johannes Kepler University) B. Salvy (INRIA Rocquencourt)

W. Wen-tsun (Academia Sinica)

Contributed Talks •

Gosper’s Algorithm, Accurate Summation, and the Discrete Newton-Leibniz Formula.................5

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Signature of Symmetric Rational Matrices and the Unitary Dual of Lie Groups.............................13

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Sum of Roots with Positive Real Parts ......................................................................................................21

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Algebraic General Solutions of Algebraic Ordinary Differential Equations.....................................29

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Adherence is Better than Adjacency: Computing the Riemann Index Using CAD ........................37

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Fast Algorithms for Polynomial Solutions of Linear Differential Equations....................................45

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Non Complete Integrability of a Magnetic Satellite in Circular Orbit .................................................53

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Symmetric and Semisymmetric Graphs Construction Using G-graphs ...........................................61

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Picard–Vessiot Extensions for Linear Functional Systems .................................................................68

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On Using Bi-equational Constraints in CAD Construction...................................................................76

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Hybrid Symbolic-Numeric Integration in Multiple Dimensions via Tensor-Product Series .........84

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A BLAS Based C Library for Exact Linear Algebra on Integer Matrices ...........................................92

S. A. Abramov (Russian Academy of Sciences), M. Petkovšek (University of Ljubljana) J. Adams (University of Maryland), B. D. Saunders, Z. Wan (University of Delaware) H. Anai (Fujitsu Laboratories Ltd.), S. Hara (The University of Tokyo), K. Yokoyama (Rikkyo University) J. M. Aroca, J. Cano (Univ. de Valladolid), R. Feng, X. S. Gao (Academia Sinica) J. C. Beaumont, R. J. Bradford, J. H. Davenport, N. Phisanbut (University of Bath) A. Bostan (Algorithms Project), T. Cluzeau (Université de Limoges), B. Salvy (INRIA Rocquencourt) D. Boucher (Université de Rennes 1) A. Bretto, L. Gillibert (Université de Caen), B. Laget (Ecole Nationale d’Ingénieurs de Saint-Etienne) M. Bronstein (INRIA – CAFÉ), Z. Li (Acad. of Math. and Syst. Sci.), M. Wu (INRIA – CAFÉ, Acad. of Math. and Syst. Sci.) C. W. Brown (United State Naval Academy), S. McCallum (Macquarie University NSW) O. A. Carvajal, F. W. Chapman, K. O. Geddes (University of Waterloo) Z. Chen, A. Storjohann (University of Waterloo)

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Structure and Asymptotic Expansion of Multiple Harmonic Sums ..................................................100

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Lifting Techniques for Triangular Decompositions..............................................................................108

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Computing the Multiplicity Structure in Solving Polynomial Systems ...........................................116

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Algorithms for the Non-monic Case of the Sparse Modular GCD Algorithm ................................124

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Computing µ-Bases of Rational Curves and Surfaces Using Polynomial Matrix Factorization ......................................................................................................................................132

C. Costermans, J. Y. Enjalbert, H. Ngoc Minh, M. Petitot (Université Lille 2) X. Dahan (LIX, École polytechnique), M. Moreno Maza (University of Western Ontario), É. Schost (LIX, École polytechnique), W. Wu, Y. Xie (ORCCA, UWO) B. H. Dayton, Z. Zeng (Northeastern Illinois University) J. de Kleine, M. Monagan (Simon Fraser University), A. Wittkopf (Maplesoft)

J. Deng, F. Chen, L. Shen (University of Science and Technology of China) •

Efficient Computation of the Characteristic Polynomial.....................................................................140

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Selfintersections of a Bézier Bicubic Surface........................................................................................148

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A Procedure for Proving Special Function Inequalities Involving a Discrete Parameter ....................................................................................................................................156

J.-G. Dumas, C. Pernet (Université Joseph Fourier), Z. Wan (University of Delaware) A. Galligo, J. P. Pavone (Université de Nice Sophia-Antipolis)

S. Gerhold, M. Kauers (Johannes Kepler Universität) •

Generalized Loewy-Decomposition of D-Modules................................................................................163

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On Computing Nearest Singular Hankel Matrices ................................................................................171

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A Reliable Block Lanczos Algorithm over Small Finite Fields ..........................................................177

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Schur Partition for Symmetric Ternary Forms and Readable Proof to Inequalities.....................185

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Affine Transformations of Algebraic Numbers......................................................................................193

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Architecture-Aware Classical Taylor Shift by 1 .....................................................................................200

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On the Complexity of Factoring Bivariate Supersparse (Lacunary) Polynomials........................208

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Generic Matrix Multiplication and Memory Management in LinBox.................................................216

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Exact Analytical Solutions to the Nonlinear Schrödinger Equation Model....................................224

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Half-GCD and Fast Rational Recovery .....................................................................................................231

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Application of Wu’s Method to Symbolic Model Checking ................................................................237

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Probabilistic Algorithms for Computing Resultants ............................................................................245

D. Grigoriev (Université de Rennes, Beaulieu), F. Schwarz (FhG, Institut SCAI) M. A. Hitz (North Georgia College & State University) B. Hovinen (University of Toronto), W. Eberly (University of Calgary) F. Huang, S. Chen (Chinese Academy of Science) D. J. Jeffrey, Pratibha (The University of Western Ontario), K. B. Roach (The University of Waterloo) J. R. Johnson, W. Krandick, A. D. Ruslanov (Drexel University) E. Kaltofen (North Carolina State University), P. Koiran (École Normale Supérieure de Lyon) E. Kaltofen (North Carolina State University), D. Morozov (Duke University), G. Yuhasz (North Carolina State University) B. Li, Y. Chen (Ningbo University, Chinese Academy of Sciences), Q. Wang (Dalian University of Technology, Chinese Academy of Sciences) D. Lichtblau (Wolfram Research, Inc.) W. Mao (Chinese Academy of Sciences), J. Wu (Chinese Academy of Sciences, Lanzhou University, Universität Mannheim) M. Monagan (Simon Fraser University)

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Generalized Normal Forms and Polynomial System Solving ............................................................253

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Domains and Expressions: An Interface Between Two Approaches to Computer Algebra ....................................................................................................................................261

B. Mourrain (INRIA)

C. E. Oancea, S. M. Watt (The University of Western Ontario) •

Symbolic-Numeric Completion of Differential Systems by Homotopy Continuation ..................269

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Algorithms for Symbolic/Numeric Control of Affine Dynamical Systems ......................................277

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Finding Telescopers with Minimal Depth for Indefinite Nested Sum and Product Expressions ............................................................................................................................285

G. Reid (University of Western Ontario), J. Verschelde (University of Illinois at Chicago), A. Wittkopf (Simon Fraser University), W. Wu (The University of Western Ontario) A. Rondepierre, J.-G. Dumas (Université Joseph Fourier)

C. Schneider (J. Kepler University) •

Multivariate Power Series Multiplication .................................................................................................293

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Partial Degree Formulae for Rational Algebraic Surfaces ..................................................................301

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Computing the Rank and a Small Nullspace Basis of a Polynomial Matrix ...................................309

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Approximation of Dynamical Systems using S-Systems Theory: Application to Biological Systems............................................................................................................317

É. Schost (LIX, École Polytechnique) S. Pérez-Díaz, J. R. Sendra (Universidad de Alcalá) A. Storjohann (University of Waterloo), G. Villard (École Normale Supérieure de Lyon)

L. Tournier (Laboratoire de Modélisation et Calcul) •

Generalized Laplace Transformations and Integration of Hyperbolic Systems of Linear Partial Differential Equations....................................................................................................325 S. P. Tsarev (Krasnoyarsk State Pedagogical University)

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Preconditioners for Singular Black Box Matrices.................................................................................332

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Solving Second Order Linear Differential Equations with Klein’s Theorem ..................................340

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Deterministic Equation Solving over Finite Fields ...............................................................................348

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Stability Analysis of Biological Systems with Real Solution Classification ..................................354

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An Open Problem on Metric Invariants of Tetrahedra .........................................................................362

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Admissible Orderings and Finiteness Criteria for Differential Standard Bases ...........................365

W. J. Turner (Wabash College) M. van Hoeij (Florida State University), J.-A. Weil (Université de Limoges) C. van de Woestijne (Universiteit Leiden) D. Wang (Beihang University), B. Xia (Peking University) L. Yang, Z. Zeng (East China Normal University) A. Zobnin (Moscow State University)

Author Index......................................................................................................................................................373

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ISSAC 2005 Conference Organization General Chairs:

SIGSAM Chair: Program Committee:

Poster Committee:

Tutorials: Software Exhibits: Proceedings: Local Arrangements:

Treasurer: Publicity Committee:

Xiao-Shan Gao, MMRC, Chinese Academy of Sciences (China) George Labahn, University of Waterloo, Ontario (Canada) Emil Volcheck, National Security Agency (USA) Peter Paule (Chair), RISC-Linz (Austria) Ron Boisvert, NIST (USA) John Cannon, University of Sydney (Australia) Howard Cheng, University of Lethbridge (Canada) Frédéric Chyzak, INRIA-Rocquencourt (France) Robert Corless, University of Western Ontario (Canada) Mark Giesbrecht, University of Waterloo, Ontario (Canada) Andreas Griewank, Humboldt-Universität zu Berlin (Germany) Tudor Jebelean, RISC-Linz (Austria) Hongbo Li, MMRC, Chinese Academy of Sciences (China) Daniel Lichtblau, Wolfram Research (USA) Michael Monagan, Simon Fraser University (Canada) Teo Mora, Università di Genova (Italy) Marko Petkovšek, University of Ljubljana (Slovenia) Tomás Recio, University of Cantabria (Spain) Felix Ulmer, Université de Rennes 1 (France) Paul Wang, Kent State University (USA) Kazuhiro Yokoyama, Kyushu University (Japan) Austin Lobo (Chair), Washington College (USA) Alin Bostan, INRIA-Rocquencourt (France) Ha Le, Simon Fraser University (Canada) William Turner, Wabash College (USA) Claude-Pierre Jeannerod, INRIA-Lyon (France) Dongming Wang, Université Pierre et Marie Curie (France) Manuel Kauers, RISC-Linz (Austria) Ziming Li (Chair), MMRC, Chinese Academy of Sciences (China) Dongdai Lin, MMRC, Chinese Academy of Sciences (China) Huilin Liu, MMRC, Chinese Academy of Sciences (China) Yujie Ma, MMRC, Chinese Academy of Sciences (China) Lihong Zhi, MMRC, Chinese Academy of Sciences (China) Zhuojun Liu, MMRC, Chinese Academy of Sciences (China) Ilias Kotsireas, Wilfried Laurier University (Canada) xi

Web/Registration:

ISSAC Steering Committee:

Dingkang Wang (Chair), MMRC, Chinese Academy of Sciences (China) Zhuosheng Lu, MMRC, Chinese Academy of Sciences (China) Erich Kaltofen (Chair), North Carolina State University (USA) Mark Giesbrecht, University of Waterloo (Canada) Wolfram Koepf, Universität Kassel (Germany) Gilles Villard, INRIA-Lyon (France) Emil Volcheck, National Security Agency (USA) Kazuhiro Yokoyama, Kyushu University (Japan)

Reviewers:

P. Abbott S.A. Abramov I. Ajwa M. Angeles Gomez-Molleda A. Antonov D. Aruliah G. Ateniese M. Audin E. Bach M. Barkatou D.A. Bini U. Bodenhofer P. Borwein A. Bostan F. Boulier D. Bradley M. Bronstein J. Carette H. Cheng B. Chen W.Y.C. Chen E.-W. Chionh R. Churchill F. Chyzak G.E. Collins D. Coombs G. Cooperman G. Craciun J.-G. Dumas W. Eberly M. El Kahoui I.Z. Emiris T. Erlebach J. Farr G. Fee R. Feng

C. Fieker S. Fortune A. Galligo X.-S. Gao J. Gerhard P. Giorgi M. Giusti L. Gonzalez-Vega T. Hagerup M. Harrison W. Harris G. Havas T.F. Havel F. Hess Q.-H. Hou E. Hubert R. Israel T. Jebelean R. Joan J. Johnson B. Juettler D. Kapadia D. Kapur M. Kauers M. Kida E.P. Klement J. Knoop I.S. Kotsireas W. Krandick G. Labelle G. Landsmann J. Lauri D. Lazard G. Lecerf H. Le D. Lichtblau xii

P. Lisonek Z. Li A. Lobo A.J. Maciejewski E. Mansfield M.G. Marinari A. Martens R. Martin D. Masulovic J. May J.M. McNamee E. Melis M. Monagan T. Mora M. Moreno-Maza J. Moulin Ollagnier B. Mourrain J.-M. Muller M. Noro F. Ollivier A. Orlando P. Orponen V. Pan H. Park C. Pernet M. Petkovsek G. Pfister R. Pozo A. Quadrat A. Quiros S. Ratschan D. Richardson A. Riese E. Rodriguez Carbonell C.M. Roney-Dougal F. Rouillier

Reviewers (continued)

S.M. Rump B. Salvy S. Saminger F. Santos T. Sasaki D. Saunders J. Schicho C. Schneider W. Schreiner F. Schwarz J. Segura A. Seidl R. Sendra T. Shimoyama I. Shparlinski

M. Singer A. Solomon G. Sommer A. Steel H.J. Stetter A. Storjohann A. Szanto N. Temme V. Timofte S. Tsarev B. Unger C. van de Woestijne J. van der Hoeven M. van der Put M. van Hoeij

In addition, 52 reviewers have requested that their name not appear.

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J. Verschelde G. Villard K. Weber J.A. Weil V. Weispfenning F. Winkler W. Wu Y. Wu L. Yang P. Yu M. Zhang L. Zhi B. Zimmermann E. Zuazua W. Zudilin

Sponsor & Supporters ISSAC 2005 is sponsored by

Association for Computing Machinery SIGSAM — Special Interest Group on Symbolic and Algebraic Manipulation with financial support from

National Natural Science Foundation of China

Maplesoft

Academy of Mathematics and Systems Science

Institute of Systems Science

Key Laboratory of Mathematics Mechanization

Institut national de recherche en informatique et en automatique

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A View on the Future of Symbolic Computation Invited Talk Abstract Bruno Buchberger Research Institute for Symbolic Computation Johannes Kepler University Linz, Austria

[email protected] Since approximately 1960, symbolic computation added algebraic algorithms (polynomial algorithms, simplification algorithms for expressions, algorithms for integration, algorithms for the analysis of algebraic structures like groups etc.) to numerics and provided both numerical and algebraic algorithms in the frame of powerful integrated mathematical software systems like Macsyma, Reduce, . . . , Mathematica, Maple, . . . Various wonderful tools like graphics, notebook facilities, extensible two-dimensional syntax etc. greatly enhanced the attractivity of these systems for mathematicians, scientists, and engineers. Over the recent decades, sometimes based on very early work in the 19th century, new and deep research results in various branches of mathematics have been developed by the symbolic computation research community which led to an impressive variety of new algebraic algorithms. In parallel, in a different community, based on new and deep results in mathematical logic, algorithms and systems for automated theorem proving were developed. In the editorial for the Journal of Symbolic Computation (1985), I tried to offer this journal as a common forum for both the computer algebra and the computational logic community and for the interaction and merge of the two fields. In fact,

in some specific theorem proving methods (as, for example, decision methods for the first order theory of real closed fields and decision methods for geometry), algebraic techniques play an important role. However, we are not yet at a stage where both worlds, the world of computational algebra (the algorithmization of the object level of mathematics) and the world of computational logic (the algorithmization of the meta-level of mathematics) would find there common frame in terms of integrated mathematical software systems. In the talk, I will sketch a view on future symbolic computation that hopefully will integrate numerics, computer algebra, and computational logic in a unified frame and will offer software systems for supporting the entire process of what could be called “mathematical theory exploration” or “mathematical knowledge management”. In this view, symbolic computation is not only a specific part of mathematics but, rather, will be specific way of doing mathematics. This will have drastic effects on the way how research, education, and application in mathematics will be possible and the publication, accumulation, and use of mathematical knowledge will be organized. We envisage a kind of “Bourbakism of the 21st century”, which will be very different — and partly in opposition to — the Bourbakism of the 20th century.

Copyright is held by author/owner ISSAC’05, July 24–27, 2005, Beijing, China. ACM 1-59593-095-7/05/0007.

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D-finiteness: Algorithms and Applications Invited Talk Abstract Bruno Salvy Algorithms Project Inria Rocquencourt 78153 Le Chesnay (France)

[email protected] Differentially finite series are solutions of linear differential equations with polynomial coefficients. P-recursive sequences are solutions of linear recurrences with polynomial coefficients. Corresponding notions are obtained by replacing classical differentiation or difference operators by their qanalogues. All these objects share numerous properties that are described in the framework of “D-finiteness”. Our aim in this area is to enable computer algebra systems to deal in an algorithmic way with a large number of special functions and sequences. Indeed, it can be estimated that approximately 60% of the functions described in Abramowitz & Stegun’s handbook [1] fall into this category, as well as 25% of the sequences in Sloane’s encyclopedia [20, 21]. In a way, D-finite sequences or series are non-commutative analogues of algebraic numbers: the role of the minimal polynomial is played by a linear operator. Ore [14] described a non-commutative version of Euclidean division and extended Euclid algorithm for these linear operators (known as Ore polynomials). In the same way as in the commutative case, these algorithms make several closure properties effective (see [22]). It follows that identities between these functions or sequences can be proved or computed automatically. Part of the success of the gfun package [17] comes from an implementation of these operations. Another part comes from the possibility of discovering such identities empirically, with Pad´e-Hermite approximants on power series [2] taking the place of the LLL algorithm on floating-point numbers. The discovery that a series is D-finite is also important from the complexity point of view: several operations can be performed on D-finite series at a lower cost than on arbitrary power series. This includes multiplication, but also evaluation at rational points by binary splitting [4]. A typical application is the numerical evaluation of π in computer algebra systems; we give another one in these proceedings [3]. Also, the local behaviour of solutions of linear differential equations in the neighbourhood of their singularities is well understood [9] and implementations of algorithms comput-

ing the corresponding expansions are available [24, 13]. This gives access to the asymptotics of numerous sequences or to analytic proofs that sequences or functions cannot satisfy such equations [10]. Results of a more algebraic nature are obtained by differential Galois theory [18, 19], which naturally shares many subroutines with algorithms for D-finite series. The truly spectacular applications of D-finiteness come from the multivariate case: instead of series or sequences, one works with multivariate series or sequences, or with sequences of series or polynomials,. . . . They obey systems of linear operators that may be of differential, difference, qdifference or mixed types, with the extra constraint that a finite number of initial conditions are sufficient to specify the solution. This is a non-commutative analogue of polynomial systems with a finite number of solutions. It turns out that, as in the polynomial case, Gr¨ obner bases give algorithmic answers to many decision questions, by providing normal forms in a finite dimensional vector space. This has been observed first in the differential case [11, 23] and then extended to the more general multivariate Ore case [8]. A crucial insight of Zeilberger [27, 15] is that elimination in this non-commutative setting computes definite integrals or sums. This is known as creative telescoping. In the hypergeometric setting (when the quotient is a vector space of dimension 1), a fast algorithm for this operation is known as Zeilberger’s fast algorithm [26]. In the more general case, Gr¨ obner bases are of help in this elimination. This is true in the differential case [16, 25] and to a large extent in the more general multivariate case [8]. Also, Zeilberger’s fast algorithm has been generalized to the multivariate Ore case by Chyzak [5, 6]. Still, various efficiency issues remain and phenomena of non-minimality of the eliminated operators are not completely understood. A further generalization of D-finite series is due to Gessel [12] who developed a theory of symmetric series. These series are such than when all but a finite number of their variables (in a certain basis) are specialized to 0, the resulting series is D-finite in the previous sense. Closure properties under scalar product lead to proofs of D-finiteness (in the classical sense) for various combinatorial sequences. Again, algorithms based on Gr¨ obner bases make these operations effective [7]. The talk will survey the nicest of these algorithms and their applications. I will also indicate where current work is in progress, or where more work is needed.

Copyright is held by author/owner ISSAC’05, July 24–27, 2005, Beijing, China. ACM 1-59593-095-7/05/0007.

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Categories and Subject Descriptors

[11] A. Galligo. Some algorithmic questions on ideals of differential operators. In B. F. Caviness, editor, Proceedings EUROCAL’85, volume 204 of Lecture Notes in Computer Science, pages 413–421. Springer-Verlag, 1985. [12] I. M. Gessel. Symmetric functions and P -recursiveness. Journal of Combinatorial Theory, Series A, 53:257–285, 1990. [13] M. van Hoeij. Formal solutions and factorization of differential operators with power series coefficients. Journal of Symbolic Computation, 24(1):1–30, 1997. [14] O. Ore. Theory of non-commutative polynomials. Annals of Mathematics, 34:480–508, 1933. [15] M. Petkovˇsek, H. S. Wilf, and D. Zeilberger. A = B. A. K. Peters, Wellesley, MA, 1996. [16] M. Saito, B. Sturmfels, and N. Takayama. Gr¨ obner deformations of hypergeometric differential equations. Springer-Verlag, Berlin, 2000. [17] B. Salvy and P. Zimmermann. Gfun: a Maple package for the manipulation of generating and holonomic functions in one variable. ACM Transactions on Mathematical Software, 20(2):163–177, 1994. [18] M. F. Singer. Liouvillian solutions of n-th order homogeneous linear differential equations. American Journal of Mathematics, 103(4):661–682, 1981. [19] M. F. Singer. Algebraic relations among solutions of linear differential equations. Transactions of the American Mathematical Society, 295(2):753–763, 1986. [20] N. J. A. Sloane. The On-Line Encyclopedia of Integer Sequences. 2005. Published electronically at http://www.research.att.com/~njas/sequences/. [21] N. J. A. Sloane and S. Plouffe. The Encyclopedia of Integer Sequences. Academic Press, 1995. [22] R. P. Stanley. Enumerative combinatorics, volume 2. Cambridge University Press, 1999. [23] N. Takayama. Gr¨ obner basis and the problem of contiguous relations. Japan Journal of Applied Mathematics, 6(1):147–160, 1989. ´ Tournier. Solutions formelles d’´equations [24] E. diff´erentielles. Doctorat d’´etat, Universit´e scientifique, technologique et m´edicale de Grenoble, 1987. [25] H. Tsai. Algorithms for algebraic analysis. PhD thesis, University of California at Berkeley, Spring 2000. [26] H. S. Wilf and D. Zeilberger. Rational function certification of multisum/integral/“q” identities. Bulletin of the American Mathematical Society, 27(1):148–153, July 1992. [27] D. Zeilberger. A holonomic systems approach to special functions identities. Journal of Computational and Applied Mathematics, 32(3):321–368, 1990.

I.1.2 [Symbolic and Algebraic Manipulation]: Algorithms

General Terms Algorithms

Keywords Computer algebra, Linear differential equations, Linear recurrences, Creative telescoping, Elimination.

1.

REFERENCES

[1] M. Abramowitz and I. A. Stegun, editors. Handbook of mathematical functions with formulas, graphs, and mathematical tables. Dover Publications Inc., New York, 1992. Reprint of the 1972 edition. [2] B. Beckermann and G. Labahn. A uniform approach for the fast computation of matrix-type Pad´e approximants. SIAM Journal on Matrix Analysis and Applications, 15(3):804–823, July 1994. [3] A. Bostan, T. Cluzeau, and B. Salvy. Fast algorithms for polynomial solutions of linear differential equations. In M. Kauers, editor, Symbolic and Algebraic Computation, New York, 2005. ACM Press. Proceedings of ISSAC’05, July 2005, Beijing, China. [4] D. V. Chudnovsky and G. V. Chudnovsky. Approximations and complex multiplication according to Ramanujan. In Ramanujan revisited, pages 375–472. Academic Press, Boston, MA, 1988. [5] F. Chyzak. Fonctions holonomes en calcul formel. Phd ´ thesis, Ecole polytechnique, 1998. [6] F. Chyzak. An extension of Zeilberger’s fast algorithm to general holonomic functions. Discrete Mathematics, 217(1-3):115–134, 2000. [7] F. Chyzak, M. Mishna, and B. Salvy. Effective scalar products of D-finite symmetric functions. Journal of Combinatorial Theory, Series A, 2005. 51 pages. To appear. [8] F. Chyzak and B. Salvy. Non-commutative elimination in Ore algebras proves multivariate holonomic identities. Journal of Symbolic Computation, 26(2):187–227, Aug. 1998. [9] E. Fabry. Sur les int´egrales des ´equations diff´erentielles lin´eaires ` a coefficients rationnels. Th`ese de doctorat `es sciences math´ematiques, Facult´e des Sciences de Paris, July 1885. [10] P. Flajolet, S. Gerhold, and B. Salvy. On the non-holonomic character of logarithms, powers, and the nth prime function. The Electronic Journal of Combinatorics, 11(2), Apr. 2005. A2, 16 pages.

3

On a Finite Kernel Theorem for Polynomial-Type Optimization Problems and Some of its Applications Invited Talk Abstract Wu Wen-tsun MMKL, Academy of Mathematics and System Sciences, Academia Sinica Beijing, 100080, P.R. China

[email protected] Extremalization and optimization problems are considered as of utmost importance both in the past and in the present days. Thus, in the early beginning of infinitesimal calculus the determination of maxima and minima is one of the stimulating problems causing the creation of calculus and one of the successful applications which stimulates its rapid further development. However, the maxima and minima involved are all of local character leading to equations difficult to be solved, not to say the inherent logical difficulties involving necessary and/or sufficient conditions to be satisfied. In recent years owing to creation of computers various kinds of numerical methods have been developed which involve usually some converging processes. These methods, besides such problems as stability or error-control, can hardly give the greatest or least value, or global-optimal value for short over the whole domain, supposed to exist in advance. How-

ever, the problem becomes very agreeable if we limit ourselves to the polynomial-type case. In fact, based on the classical treatment of polynomial equations-solving in ancient China and its modernization due to J.F. Ritt, we have discovered a Finite Kernel Theorem to the effect that a finite set of real values, to be called the finite kernel set of the given problem, may be so determined that all possible extremal values will be found among this finite set and the corresponding extremal zeros are then trivially determined. Clearly it will give the global optimal value over the whole domain in consideration, if it is already known to exist in some way. Associated packages wsolve and e val have been given by D.K. Wang and had been applied with success in various kinds of problems, polynomial-definiteness, non-linear programming, etc., particularly problems involving inequalities.

Copyright is held by author/owner ISSAC’05, July 24–27, 2005, Beijing, China. ACM 1-59593-095-7/05/0007.

4

Gosper’s Algorithm, Accurate Summation, and the Discrete Newton-Leibniz Formula ∗

S. A. Abramov†

M. Petkovˇsek‡

Russian Academy of Sciences Dorodnicyn Computing Centre Vavilova 40, 119991, Moscow GSP-1, Russia

Department of Mathematics University of Ljubljana Jadranska 19, SI-1000 Ljubljana, Slovenia

[email protected]

[email protected]

ABSTRACT

hypergeometric term u(n) which satisfies the key equation

Sufficient conditions are given for validity of the discrete Newton-Leibniz formula when the indefinite sum is obtained either by Gosper’s algorithm or by Accurate Summation algorithm. It is shown that sometimes a polynomial can be factored from the summand in such a way that the safe summation range is increased.

u(n + 1) − u(n) = t(n)

for all n ∈ I \ S where S is a finite set. Summing this equation on n from v to w we get the discrete analog of the Newton-Leibniz formula w


t(n) = u(w + 1) − u(v)

(2)

n=v

Categories and Subject Descriptors: G.2.1 [Combinatorics]: Counting problems; I.1.2 [Algorithms]: Algebraic algorithms

provided that [v, w] ∩ Z ⊆ I \ S. In many existing implementations of Gosper’s algorithm, however, indiscriminate use of (2) sometimes results in wrong answers. Here is a case in point.

General Terms: Algorithms Keywords: symbolic summation, Gosper’s algorithm, Accurate Summation algorithm, Newton-Leibniz formula

1.

(1)

Example 1. Consider the sequence 2n−3

INTRODUCTION

t(n) =

Let K be a field of characteristic zero. A function t : I
→ K defined on an interval of integers I ⊆ Z is a

n

4n

,

(3)

which is defined for all n ∈ Z. This is a hypergeometric term which satisfies

• hypergeometric term if there are nonzero polynomials a0 , a1 ∈ K[n] such that a1 (n)t(n + 1) + a0 (n)t(n) = 0 for all n ∈ Z such that n, n + 1 ∈ I;

2(n + 1)(n − 2)t(n + 1) = (2n − 1)(n − 1)t(n)

(4)

for all n ∈ Z. Gosper’s algorithm succeeds with input t(n) and returns   2n(n + 1) 2n−3 n u(n) = . (n − 2)4n

• P -recursive sequence if there are polynomials a0 , a1 , . . . , aρ ∈ K[n] such that a0 aρ = 0 and aρ (n)t(n + ρ) + · · · + a1 (n)t(n + 1) + a0 (n)t(n) = 0 for all n ∈ Z such that n, n + 1, . . . , n + ρ ∈ I.

Summing equation (1) on n from 0 to m the left-hand side telescopes, and we obtain   m
(m + 1)(m + 2) 2m−1 m+1 t(n) = (?) u(m + 1) − u(0) = . 2(m − 1)4m n=0 (5) But the expression on the right gives the true value of the sum only at m = 0. At m = 1 it is undefined, while at each m ≥ 2 its value is 3/8 less than the actual value of the sum. The problem here is that u(n) is undefined at n = 2, hence equation (1) does not hold for n ∈ {1, 2}, and summing it over a range including 1 or 2 may give a wrong answer. This is not an isolated example: a similar phenomenon seems to occur with the sum m 2n−p


Each hypergeometric term is, of course, a P -recursive sequence. If t(n) is a hypergeometric term, one can use the wellknown Gosper’s algorithm [6] to find (if it exists) another ∗ The work is partially supported by the ECO-NET program of the French Foreign Affairs Ministry. † Partially supported by RFBR under grant 04-01-00757. ‡ Partially supported by ARRS under grant P1-0294.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. ISSAC’05, July 24–27, 2005, Beijing, China. Copyright 2005 ACM 1-59593-095-7/05/0007 ...$5.00.

n

n=0

for each positive integer p.

5

4n

If t is a P -recursive sequence, then one can use Accurate Summation algorithm from [3], or its generalization in [5], to solve equation (1) (we discuss this algorithm in Section 5). Problems similar to those arising in Example 1 are possible when one uses the resulting Newton-Leibniz formula. Notice that one can apply Accurate Summation algorithm in the case ρ = 1 as an alternative to Gosper’s algorithm; then the incorrect formula (5) will appear again. This common error is the discrete analogon of a wellknown error in definite integration committed by some of the early symbolic integrators: when attempting to evalub ate I = a f (x)dx by computing first an antiderivative F (x) such that F  (x) = f (x), and then using the Newton-Leibniz formula I = F (b)−F (a), we may obtain an incorrect answer unless F (x) is continuous on [a, b]. For example, the actual value of  1 x2 + 1 dx 4 2 −1 x − x + 1

1 (β)n

Note that if (α)n resp. 1/(β)n is defined for some n ∈ Z, then (α)n+1 resp. 1/(β)n−1 is defined for that n as well. More precisely, if α ∈ Z and α ≥ 1 then (α)n is defined on [−α + 1, ∞) ∩ Z, otherwise it is defined on all Z. Similarly, if β ∈ Z and β ≤ 0 then 1/(β)n is defined on (−∞, −β] ∩ Z, otherwise it is defined on all Z. Thus (α)n and 1/(β)n are hypergeometric terms which satisfy (α)n+1 = (α + n)(α)n ,

(β + n)/(β)n+1 = 1/(β)n (6)

whenever (α)n and 1/(β)n+1 are defined. If I ⊆ Z is an infinite interval of integers we denote  (−∞, a + 1] ∩ Z, if I = (−∞, a] ∩ Z, I+ = I, otherwise;

is π, but using the antiderivative arctan(x − 1/x) in the Newton-Leibniz formula gives 0. The obvious solution is to split the summation interval into several subintervals that do not contain the exceptional points from S. In this paper we analyze the exceptional set S that appears in Gosper’s algorithm when summing hypergeometric terms, and more generally, in the Accurate Summation algorithm [3] when summing P -recursive sequences. Section 3 provides sufficient conditions for the NewtonLeibniz formula (2) to hold when the indefinite sum u(n) is obtained by Gosper’s algorithm, and Section 5 does the same for Accurate Summation. These conditions provide a bounding interval for the exceptional set S, and are of two kinds: a priori, which are weaker but readily available even before running the algorithms, as they are based on the singularities of the operator annihilating the summand; and a posteriori, which are stronger but available only after running the algorithms, as they are based on their output. On the other hand, in Section 4 we prove that for proper hypergeometric terms the discrete Newton-Leibniz formula is valid without restrictions. For general P -recursive sequences Section 6 shows that sometimes a polynomial can be factored from the summand in such a way that the size of the bounding interval in the a priori condition is decreased. A thorough analysis of the relationship between hypergeometric terms as syntactic objects and their analytic meaning in the context of summation has been provided by M. Schorn in [8]. The solution proposed there for evaluation of sums such as the one in Example 1 is by means of suitably chosen limiting processes.

2.

 n−1

1    , n ≥ 0, β = 0, −1, . . . , 1 − n,     k=0 β + k |n| =

  (β − k), n < 0,      k=1 undefined , otherwise.

I− =



(−∞, a − 1] ∩ Z, if I = (−∞, a] ∩ Z, I, otherwise.

We use E to denote the shift operator w.r.t. n, so that E t(n) = t(n + 1). Since juxtaposition can mean not only operator application but also composition of operators, we use ◦ to denote the latter in case of ambiguity, so that, e.g., E ◦ t(n) = t(n + 1) ◦ E = t(n + 1)E. Sometimes we use parentheses to denote operator application, writing, e.g., E(1) = E 1 = 1. Definition 2. For a linear difference operator L = aρ E ρ + aρ−1 E ρ−1 + · · · + a0

(7)

where ρ ≥ 1, aρ , . . . , a0 ∈ K[n], aρ a0 = 0 and gcd(a0 , . . . , aρ ) = 1, we define the sets SLl of leading and SLt of trailing integer singularities by SLl SLt

= {x ∈ Z; aρ (x − ρ) = 0}, = {x ∈ Z; a0 (x) = 0}.

We call • mLl = min(SLl ∪ {+∞}) the minimal leading singularity of L, • MLl = max(SLl ∪ {−∞}) the maximal leading singularity of L, • mLt = min(SLt ∪ {+∞}) the minimal trailing singularity of L,

PRELIMINARIES

• MLt = max(SLt ∪ {−∞}) the maximal trailing singularity of L.

Definition 1. Following conventional notation, the rising factorial power (α)n and its reciprocal 1/(β)n are defined for α, β ∈ K and n ∈ Z by  n−1

   (α + k), n ≥ 0,     k=0 |n| (α)n =

1   , n < 0, α = 1, 2, . . . , |n|,   α − k    k=1 undefined , otherwise;

Proposition 1. Let L be as in (7) and b ∈ K[n]. If a rational function y ∈ K(n) satisfies aρ (n)y(n + ρ) + · · · + a0 (n)y(n) = b(n),

(8)

then y(n) has no integer poles outside the interval (possibly empty) [mLl , MLt ]. For a proof, see [1].

6

3.

WHEN CAN GOSPER’S ALGORITHM BE USED TO SUM HYPERGEOMETRIC TERMS?

Proof: By assumption, t(n), t(n + 1), r(n), r(n + 1), u(n), u(n + 1) are defined for all n ∈ I1 , and (9), (10) are valid on I1 . Therefore, for all n ∈ I1 ,

We denote Gosper’s algorithm hereafter by GA. Consider the case when (8) has the form a1 (n)t(n + 1) + a0 (n)t(n) = 0

a0 (n)u(n + 1)

(9)

and set L = a1 (n)E + a0 (n). Let a hypergeometric term t(n) satisfy equation (9). Given a0 (n), a1 (n) as input, GA tries to construct r ∈ K(n) such that a0 (n)r(n + 1) + a1 (n)r(n) = −a1 (n)

= = =

a0 (n)r(n + 1)t(n + 1) −a1 (n)(1 + r(n))t(n + 1) (by (10)) a0 (n)(1 + r(n))t(n) (by (9))

=

a0 (n)u(n) + a0 (n)t(n),

or equivalently, a0 (n)(u(n + 1) − u(n)) = a0 (n)t(n).

(10)

(12)

Pick an n ∈ I1 . If a0 (n) = 0 then (12) implies (1). If a0 (n) = 0 then, since gcd(a0 , a1 ) = 1, we have a1 (n) = 0. Hence (9) implies t(n + 1) = 0 and (10) implies r(n) + 1 = 0. Therefore u(n + 1) = r(n + 1)t(n + 1) = 0 and u(n) + t(n) = (r(n) + 1)t(n) = 0, so (1) holds for all n ∈ I1 . Summing (1) over I1 yields (11).

(this can also be done by the algorithms from [1] or [2]). If such r(n) exists then u(n) = r(n)t(n) satisfies the key equation (1), possibly with finitely many exceptions. We now give two kinds of sufficient conditions for this u(n) to satisfy equation (1) and for the discrete Newton-Leibniz formula in the form w
t(k) = u(w) − u(v) + t(w) (11)

Theorem 1. (a priori condition for GA) If [v, w] ∩ [mLt + 1, MLl −1] = ∅, then the key equation (1) holds for all n ∈ I1 , and the discrete Newton-Leibniz formula (11) is valid.

k=v

to be valid: Proof: Since r(n) satisfies (10), Proposition 1 implies that r(n) has no integer poles outside the interval [α, β] where

1. an a posteriori condition, depending on the poles of r(n) (Proposition 2),

α

2. an a priori condition, depending only on the integer singularities of L (Theorem 1).

β

In both, we make the following assumptions: • L = a1 (n)E + a0 (n) is an operator of type (7) with ρ = 1,

= = = =

min({x ∈ Z; a0 (x − 1) = 0} ∪ {+∞}) mLt + 1, max({x ∈ Z; a1 (x) = 0} ∪ {−∞}) MLl − 1.

By assumption, the interval [v, w] is disjoint from [α, β]. Hence r(n) has no poles in [v, w], and the assertion follows from Proposition 2.

• r ∈ K(n) is a rational function which satisfies (10) as an equation in K(n),

In practice, one would run GA and then check if the a posteriori condition of Proposition 2 is satisfied, i.e., if r(n) has any integer poles in the summation interval. If yes, this interval would be split into several subintervals in order to guarantee correct evaluation. But it may be useful to check the a priori condition of Theorem 1 first, because this will, in general, restrict the relevant domain to check for poles of r(n).

• v, w are integers such that v ≤ w, • I1 := [v, w − 1] ∩ Z, • t(n) is a K-valued sequence which is defined for all n ∈ [v, w] ∩ Z and satisfies (9) for all n ∈ I1 , • u(n) is a K-valued sequence such that u(n) = r(n)t(n) whenever both r(n) and t(n) are defined.

Example 2. For the hypergeometric term t(n) = 2n−3  n of Example 1, we have L = 2(n + 1)(n − 2)E − /4 n (2n − 1)(n − 1),  r(n) = n2n(n + 1)/(n − 2), and u(n) =  /((n − 2)4 ). Thus SLt = {1}, SLl = {0, 3}, 2n(n + 1) 2n−3 n mLt = 1, MLl = 3, [mLt + 1, MLl − 1] = {2}, and the only integer pole of r(n) is n = 2. In this case both the a priori and the a posteriori conditions give the same point n = 2 to be avoided by the summation interval. As predicted by either condition, the key equation (1) fails at n = 1 and n = 2 because u(n) or u(n + 1) are undefined there. One is tempted to absorb the denominator factor n − 2 into the binomial coefficient, by, say, the sequence  replace u(n)  and n ) which is defined ev/((2n − 1)4 u ¯(n) = n(n + 1) 2n−1 n erywhere, and agrees with u(n) for all n = 1, 2. But then equation

Remark 1. Since u(n) = r(n)t(n) it is clear that, in general, formula (11) should be used instead of (2), because the latter formula needs values of the summand lying outside the summation interval which however may be undefined. A nice example is provided by the sum 2n
2n k 4n (−1) / 2k k k=0 whose evaluation was posed as Problem 10494 in the Amer. Math. Monthly in 1996. Here GA succeeds (see [7]), but the summand is undefined everywhere outside the summation interval. Proposition 2. (a posteriori condition for GA) If r(n) has no integer poles in [v, w], then the key equation (1) holds for all n ∈ I1 , and the discrete Newton-Leibniz formula (11) is valid.

u ¯(n + 1) − u ¯(n) = t(n) fails at n = 0 and n = 1.

7

(13)

Example 3. Let  (n − 2)(n − 3)(n − 5)(n − 1)!, n t(n) = (n − 2)(n − 3)(n − 5) (−1) , (−n)!

The hypergeometric term t(n) from Example 4 is an instance of a proper term which we are going to define now. Then we show in Theorem 2 that there are no restrictions on the validity of the discrete Newton-Leibniz formula for proper terms.

n≥2 n≤1

where we define as usual 1/k! = 0 when k is a negative integer. This is a hypergeometric term which satisfies

Definition 3. A hypergeometric term t(n) defined on an interval I of integers is proper if there are

(n − 5)(n − 3) t(n + 1) = (n − 4)(n − 1)n t(n) for all n ∈ Z. Here we have a0 (n) = −(n − 4)(n − 1)n, a1 (n) = (n − 5)(n − 3), r(n) = (n − 6)/((n − 2)(n − 3)), and u(n) = r(n)t(n). Thus SLt = {0, 1, 4}, SLl = {4, 6}, mLt = 0, MLl = 6, [mLt + 1, MLl − 1] = [1, 5], and the set of integer poles of r(n) is {2, 3}. The set to be avoided by the summation interval given by the a priori condition is {1, 2, 3, 4, 5}, while the analogous set given by the a posteriori condition is {2, 3}. The key equation (1) fails at n = 1, 2, 3 as predicted by the a posteriori condition, because u(n) or u(n + 1) are undefined there. One can try cancelling the factor (n − 2)(n − 3) and replace u(n) by the sequence  (n − 5)(n − 6)(n − 1)!, n ≥ 2 n u ¯(n) = , n≤1 (n − 5)(n − 6) (−1) (−n)!

• a polynomial p ∈ K[n], • a constant z ∈ K, • nonnegative integers q, r, • constants α1 , . . . , αq , β1 , . . . , βr ∈ K such that

Theorem 2. Let t(n) be a proper hypergeometric term defined on an  interval I of integers and  given by (15). Denote a(n) = z qi=1 (n + αi ) and b(n) = rj=1 (n + βj ). If a polynomial y ∈ K[n] satisfies a(n)y(n + 1) − b(n − 1)y(n) = p(n)

SUMMATION OF PROPER HYPERGEOMETRIC TERMS

and if

It is clear that the a priori condition given in Theorem 1 is, in general, too cautious: e.g., if the summand is a polynomial sequence then the integer singularities of the corresponding recurrence present no obstacles to validity of the discrete Newton-Leibniz formula (11). The following example shows that even the a posteriori condition given in Proposition 2 can sometimes be too pessimistic.

=

t(n).

(17)

Since (αi )n and 1/(βj )n are defined for all n ∈ I, (αi )n+1 n−1 are defined there too.  and 1/(βj )  By (6), a(n) qi=1 (αi )n = z qi=1 (αi )n+1 and b(n−1)/ rj=1 (βj )n =  1/ rj=1 (βj )n−1 . Hence (17) is the same as (13), and (14) follows by summing it over [v, w] ∩ Z.

(−1/2)n , 2n!

t(n) = u ¯(w + 1) − u ¯(v)

q (αi )n u ¯(n) = y(n)z n r i=1 (β j )n−1 j=1

Proof: By  (16) holds for all n ∈ I. Multiplying  assumption, it by z n qi=1 (αi )n / rj=1 (βj )n yields q (αi )n z n  ri=1 a(n)y(n + 1) j=1 (βj )n q (αi )n − z n  ri=1 b(n − 1)y(n) j=1 (βj )n

then equation (13) holds for all n ∈ Z, and the discrete Newton-Leibniz formula w


(16)

for all n ∈ I + (see Section 2 for notation), then equation (13) holds for all n ∈ I, and the discrete Newton-Leibniz formula (14) is valid whenever [v, w] ∩ Z ⊆ I.

Example 4. Let t(n) = (2 − n)(−1/2)n /(4n!). This hypergeometric term is defined for all n ∈ Z (note that t(n) = 0 for n < 0) and satisfies Lt(n) = 0 for all n ∈ Z where L is the same operator as in Example 2. Thus both Theorem 1 and Proposition 2 require the point n = 2 to be excluded from the summation interval. Equation (1) indeed fails at n = 1 and n = 2 because u(n) = r(n)t(n) is undefined at n = 2. But if we cancel the factor n − 2 in the product r(n)t(n), where r(n) = 2n(n + 1)/(n − 2), and replace u(n) by the resulting sequence u ¯(n) = −n(n + 1)

(15)

for all n ∈ I.

which is defined everywhere, and agrees with u(n) for all n = 2, 3. But equation (13) still fails at n = 1.

4.

q (αi )n t(n) = p(n)z n ri=1 j=1 (βj )n

Example 5. Even though the hypergeometric term (3) from Example 1 defined on I = Z can be written in terms of rising factorials as

(14)

n=v

is valid for all v ≤ w. This example also shows that, thanks to possible singularities, a hypergeometric term (or a P -recursive sequence) is, in general, not uniquely defined by its annihilating operator and an appropriate number of initial values. In fact, it is shown in [4] that every positive integer is the dimension of the kernel of some operator of type (7) with ρ = 1 in the space of sequences t : Z → K.

t(n) =

(n − 2)n , 4n (1)n

one can show that it is not a proper term on Z. However, it can also be written as  2 t∗ (n), n < 2, t(n) = n ≥ 2, t∗ (n),

8

and q = P ∗ (1). Then

where t∗ (n) = (2 − n)

(−1/2)n 4(1)n

(E − 1) ◦ Q + q =

is a proper term (namely the one discussed in Example 4). So to evaluate w n=v t(n) one can first split the summation range at n = 2, then use Theorem 2 on both subranges.

ρ−1 ρ−k



bk+j (n − j + 1)E k+1 − Q + P ∗ (1)

k=0 j=1

=

ρ ρ−k+1

k=1

j=1

ρ−1 ρ−k+1

−



k=0

5.

WHEN CAN ACCURATE SUMMATION BE USED TO SUM P-RECURSIVE SEQUENCES?

=

By Accurate Summation algorithm (hereafter denoted by AS) we mean a specific version of the general Accurate Integration algorithm given in [3] for integration/summation of solutions of Ore equations. This version, which is adapted for sequences that satisfy equations of the form (8) with b(n) = 0, solves the following problem: Let a minimal annihilator L of the form (7) be known for a K-valued sequence t(n). Determine if there exists a sequence u(n) which satis˜ of order ρ. fies (1), and has a minimal annihilator L It is shown in [3] that if such a u exists then it can be expressed as R t where R is an operator of order ρ − 1 with rational-function coefficients. AS constructs R if it exists. (GA solves this problem when ρ = 1.) In order to analyze the validity of the discrete Newton-Leibniz formula (11) in this case, we need to express explicitly the quotient and the remainder of a linear difference operator when divided by the first-order operator E − 1 from the left. The notion of the adjoint difference operator is useful here.

−

bk+j−1 (n − j + 1)E k

j=2 ρ−1




bk (n)E k = P.

k=0

As the quotient and remainder in operator division are unique, it follows that R = Q and p = q. Note that just to find the remainder p = P ∗ (1), it suffices to take adjoints on both sides of equation P = (E −1)◦R+p which results in P ∗ = R∗ ◦ (E − 1)∗ + p = R∗ ◦ (E −1 − 1) + p, and apply this to 1. Remark 2. Let r ∈ K(n) be a rational function, and L a difference operator as in (7). By Lemma 1, the remainder of 1 − rL when divided by E − 1 from the left is equal to (1 − rL)∗ (1) = (1 − L∗ ◦ r)(1) = 1 − (L∗ ◦ r)(1) = 1 − L∗ r.

ρ


bk (n)E k

Algorithm AS

k=0

 Input: L = ρk=0 ak (n)E k ∈ K[n, E]. Output: r ∈ K(n) and R ∈ K(n)[E] such that 1 − rL = (E − 1) ◦ R, if they exist.

be an operator in K(n)[E]. Its adjoint L∗ ∈ K(n)[E −1 ] is defined as E −k ◦ bk (n) =

k=0

ρ


if there exists r ∈ K(n) such that L∗ r = 1 then for k := 0, 1, . . . , ρ − 1 do  ck (n) := − ρ−k j=1 r(n − j)ak+j (n − j); ρ−1 k R := k=0 ck (n)E ; return (r(n), R) else such r(n) and R do not exist.

bk (n − k)E −k .

k=0

It is straightforward to verify that (L1 ◦ L2 )∗ = L∗2 ◦ L∗1 .   k Lemma 1. Let P = ρk=0 bk (n)E k , R = ρ−1 k=0 ck (n)E be operators from K(n)[E], and p ∈ K(n) a rational function  such that P = (E−1)◦R+p. Then ck (n) = ρ−k j=1 bk+j (n−j) ∗ and p = P (1).

We can find a rational-function solution r(n) of L∗ r = 1 using, e.g., the algorithm from [1] or the algorithm from [2]. A generalization of [3] was given in [5]; however, the approach taken in [3] has the advantage of simplicity, as it only uses the adjoint operator and algorithms for finding rational solutions. This simplifies the investigation of solutions that are obtained by AS, and enables us to formulate a priori conditions for AS, similar to Theorem 1 (see Theorems 3 and 5 below).

Proof: Let

Q =

bk+j−1 (n − j + 1)E k

Hence an operator R such that 1 − rL = (E − 1) ◦ R exists if and only if L∗ r = 1. This observation forms the basis of Accurate Summation.

L =

ρ


bj (n − j)

j=0

= bρ (n)E ρ +

Definition 4. Let

L∗ =

ρ


j=1

ρ−1 ρ−k+1

k=0

bk+j−1 (n − j + 1)E k +

j=2

ρ ρ−k+1

k=0

bk+j−1 (n − j + 1)E k

ρ−1 ρ−k



bk+j (n − j)E k

k=0 j=1

9

Assume that AS succeeds with L, returning r and R. It is shown in [3] that

Proof: Write R =

ρ−1

k=0 ck (n)E

(i) By assumption, r(n − j) has no integer poles in [α +  j, β + j], hence ck (n) has no integer poles in = [max1≤j≤ρ−k (α + 1≤j≤ρ−k [α + j, β + j] j), min1≤j≤ρ−k (β +j)] = [α+ρ−k, β +1],  and the coefficients of R have no integer poles in 0≤k≤ρ−1 [α + ρ − k, β+1] = [max0≤k≤ρ−1 (α+ρ−k), β+1] = [α+ρ, β+1]. (ii) To prove the second assertion, we need to express the coefficients ck (n) in a different way. Since 1 − rL = (E − 1) ◦ R, it follows from Remark 2 that L∗ r =  ρ j=0 r(n − j)aj (n − j) = 1. Shifting this k times we   find that ρj=0 r(n+k−j)aj (n+k−j) = ρ−k j=−k r(n− j)ak+j (n − j) = 1. Therefore

1. an a posteriori condition, depending on the poles of r(n) and of the coefficients of R (Proposition 3), 2. an a priori condition, depending only on the integer singularities of L (Theorem 3). In either case, we make the following assumptions:

ρ−k

• L ∈ K[n, E] is an operator of type (7),

ck (n)

= −

• r ∈ K(n) is a rational function which satisfies L r = 1 as an equation in K(n),




r(n − j)ak+j (n − j) − 1

j=−k

=

• v, w are integers such that v ≤ w − ρ,

k


r(n + j)ak−j (n + j) − 1

j=0

• Iρ := [v, w − ρ] ∩ Z,

for 0 ≤ k ≤ ρ − 1. By assumption, r(n + j) has no integer poles in [α − j, β − j], hence ck (n) has no integer poles in 0≤j≤k [α − j, β − j] = [max0≤j≤k (α − j), min0≤j≤k (β − j)] = [α, β − k],  and the coefficients of R have no integer poles in 0≤k≤ρ−1 [α, β − k] = [α, min0≤k≤ρ−1 (β − k)] = [α, β − ρ + 1].

• t(n) is a K-valued sequence which is defined for all n ∈ [v, w] ∩ Z and satisfies L t(n) = 0 for all n ∈ Iρ , • u(n) is a K-valued sequence such that u(n) = R t(n) whenever R t(n) is defined. Proposition 3. (a posteriori condition for AS) If r(n) has no poles in Iρ and the coefficients of R have no integer poles in [v, w − ρ + 1], then equation (1) holds for all n ∈ Iρ , and the discrete Newton-Leibniz formula

Theorem 3. (a priori condition for AS) If [v, w − ρ] ∩ [mLt , MLl − ρ] = ∅, then equation (1) holds for all n ∈ Iρ , and the discrete Newton-Leibniz formula (18) is valid. Proof: Rewrite L∗ r =1 in the equivalent form L r = 1 where L = E ρ ◦ L∗ = ρk=0 aρ−k (n + k)E k ∈ K[n, E]. By Lemma 1, r(n) has no integer poles outside [mL
l , ML
t ]. But SL
l = SLt and SL
t = SLl − ρ, therefore mL
l = mLt and ML
t = MLl −ρ, hence r(n) has no integer poles outside [mLt , MLl − ρ]. If mLt ≤ MLl − ρ, then both intervals [v, w − ρ] and [mLt , MLl − ρ] are nonempty, hence either w − ρ < mLt or MLl − ρ < v. In the former case, r(n) has no integer poles in (−∞, w − ρ], so by Lemma 2(i), the coefficients of R have no integer poles in (−∞, w−ρ+1]. In the latter case, r(n) has no integer poles in [v, ∞), so by Lemma 2(ii), the coefficients of R have no integer poles in [v, ∞). In either case, the result follows from Proposition 3. If mLt > MLl − ρ then r(n) has no integer poles at all. By Lemma 2, the coefficients of R also have no integer poles, and the result again follows from Proposition 3.

t(w − ρ + k) (18)

k=1

is valid. Proof: By assumptions on t and R, u(n) and u(n + 1) are defined for all n ∈ Iρ . As r(n) has no poles in Iρ , = = =

r(n − j)ak+j (n − j)

ρ−k

+

• R ∈ K(n)[E] is an operator of order ρ − 1 which satisfies 1 − rL = (E − 1) ◦ R in K(n)[E],

u(n + 1) − u(n)




j=1

∗

t(k) = u(w − ρ + 1) − u(v) +

r(n − j)ak+j (n − j)

for 0 ≤ k ≤ ρ − 1.

We now give two sufficient conditions for this u(n) to satisfy equation (1) and for the discrete Newton-Leibniz formula (11) to be valid:

k=v




j=1

• the sequence u(n) = R t(n) satisfies (1), possibly with finitely many exceptions, for any sequence t such that Lt = 0 (not only for those t’s whose minimal annihilator is L).

ρ


. By Lemma 1,

ρ−k

ck (n) = −

• if L is a minimal annihilator for t, then a minimal ˜ = 1 − R ◦ (E − 1) (note annihilator for u = R t is L ˜ has the same order ρ as L); that L

w


k

(E ◦ R) t(n) − R t(n) ((E − 1) ◦ R) t(n) = (1 − rL) t(n) t(n) − r(n) L t(n) = t(n)

for every n ∈ Iρ . Thus (1) holds for all n ∈ Iρ , and summing it over Iρ yields (18). Lemma 2. Let α, β ∈ Z ∪ {−∞, ∞}. If r(n) has no integer poles in [α, β] then the coefficients of R have

A similar remark to the one stated immediately after the proof of Theorem 1 about the use of the a priori and a posteriori conditions in practice applies here as well.

(i) no integer poles in [α + ρ, β + 1], and also (ii) no integer poles in [α, β − ρ + 1].

10

Example 6. Let L = (n−3)(n−2)(n+1)E 2 −(n−3)(n2 − 2n−1)E −(n−2)2 . Define t(n) by the initial values t(2) = a, t(3) = 0, t(4) = b, t(5) = c where a, b, c are arbitrary fixed complex numbers, and by the recurrence L t(n) = 0 when n ≤ 1 or n ≥ 6. Then it can be checked that L t(n) = 0 for all n ∈ Z. Algorithm AS succeeds with input L and returns r(n) = −1/((n−2)(n−3)), R = nE +1/(n−3). In this case SLt = {2}, mLt = 2, SLl = {1, 4, 5}, MLl − ρ = 5 − 2 = 3. So both the a posteriori and the a priori conditions reduce to 3 ∈ / [v, w − 1]. This is the best possible, as the sequence u(n) = R t(n) is undefined at n = 3, and equation (1) does not hold for n = 2, 3. It can be verified that except in the special case b + 4c = 0, there is no way to define u(3) so that (1) would hold for all n ∈ Z.

If I ∩ [mLt ¯ + 1, MLl ¯ − 1] = ∅ then equation (13) holds for all n ∈ I − , and the discrete Newton-Leibniz formula w


(19)

is valid. Proof: By assumption, we have for all n ∈ I − , ¯0 (n)t¯(n) a ¯1 (n)t¯(n + 1) + a ¯ a1 (n)p(n + 1)t(n + 1) + a0 (n)p(n)t¯(n)

= =

0, 0.

(20) (21)

¯1 (n) Multiplying (20) by a1 (n)p(n + 1), (21) by a and subtracting, we find a ¯0 (n)t¯(n)a1 (n)p(n + 1) = a1 (n). Since t¯(n) has infinitely many nonzero a0 (n)p(n)t¯(n)¯ − values on I , this implies that

Remark 3. When ρ = 1 and v ≤ w − 1, Theorem 3 implies Theorem 1 in the following way. If r(n) satisfies (10) then it is easy to verify that r¯(n) := −r(n + 1)/a1 (n) satisfies L∗ r¯ = 1, and 1− r¯ L = (E −1)◦R where R = r(n) is an operator of order 0. Thus u(n) = r(n)t(n) of Theorem 1 agrees with u(n) = R t(n) of Theorem 3. By the assumption of Theorem 1, [v, w] ∩ [mLt + 1, MLl − 1] = ∅. If mLt + 1 ≤ MLl − 1 then either w ≤ mLt or MLl − 1 ≤ v − 1, hence [v, w − 1] ∩ [mLt , MLl − 1] = ∅, and the conclusion follows by Theorem 3. If mLt + 1 > MLl − 1 then mLt ≥ MLl − 1. Again we distinguish two cases: If mLt > MLl − 1, the conclusion follows by Theorem 3. If mLt = MLl − 1, then a0 (n) and a1 (n) have a common zero since a0 (mLt ) = a1 (MLl − 1) = 0. But this contradicts the assumption of relative primality of the coefficients of L. Note that when ρ ≥ 2, polynomials a0 (n) and aρ (n) need not be relatively prime.

6.

t(k) = u ¯(w) − u ¯(v) + t(w)

k=v

a1 (n) = a ¯0 (n)a1 (n)p(n + 1) a0 (n)p(n)¯

(22)

holds infinitely often, hence also as an equation in K[n]. Multiplying (10) by a ¯1 (n)p(n), (22) by r(n+1), subtracting, and cancelling a1 (n) in K(n), we obtain r (n + 1) + a ¯1 (n)¯ r (n) = −¯ a1 (n)p(n) a ¯0 (n)¯

(23)

as an equation in K(n). It follows from Proposition 1 that r¯ ∈ K(n) has no integer poles outside the interval [mLt ¯ + ¯(n) and u ¯(n) are defined on I, r¯(n+1) 1, MLl ¯ −1]. Therefore r and u ¯(n + 1) are defined on I − , and (23) is valid for all n ∈ I −. Pick an n ∈ I − . Multiplying (20) by r¯(n + 1), (23) by t¯(n) and subtracting, we obtain r (n + 1)t¯(n + 1) − a ¯1 (n)¯ r(n)t¯(n) = a ¯1 (n)p(n)t¯(n). a ¯1 (n)¯ ¯1 (n) = 0 then, by If a ¯1 (n) = 0 this reduces to (13). If a assumption, a ¯0 (n) = 0, so (20) implies t¯(n) = t(n) = u ¯(n) = 0 and (23) implies r¯(n + 1) = u ¯(n + 1) = 0, hence (13) holds in this case as well. So (13) holds for all n ∈ I − , and the second assertion follows by summing (13) on n from v to w − 1.

EXPLOITING POLYNOMIAL FACTORS

In this section we show how polynomial factors of hypergeometric terms (even non-proper ones, such as the one in Example 3) and of P -recursive sequences can be used to strengthen the statements of Theorems 1 and 3 (i.e., to weaken the a priori conditions for validity of the discrete Newton-Leibniz formula).

Example 7. Let t(n) be the hypergeometric term from Example 3 which satisfies L t(n) = 0 for all n ∈ Z where L = (n − 5)(n − 3) E − (n − 4)(n − 1)n. Define p(n) = (n − 5)(n − 3)(n − 2) and  (n − 1)!, n ≥ 2, t¯(n) = (−1)n , n ≤ 1. (−n)!

Theorem 4. Assume that ¯ = a • L = a1 (n)E + a0 (n) and L ¯1 (n)E + a ¯0 (n) are operators of type (7) with ρ = 1,

¯ t¯(n) = 0 for all n = 1 where L ¯ = E − n. So, in Then L the notation of Theorem 4, the maximal possible interval I ¯ has no leading is either (−∞, 1] ∩ Z or [2, ∞) ∩ Z. As L singularities, MLl ¯ = −∞ and [mLt ¯ + 1, MLl ¯ − 1] = ∅. With r(n) = (n − 6)/((n − 2)(n − 3)), r¯(n) = (n − 5)(n − 6) and u ¯(n) = r¯(n)t¯(n), all the assumptions of Theorem 4 are satisfied, and it follows that formula (19) is valid provided that w ≤ 1 or v ≥ 2.

• t(n), t¯(n) are K-valued sequences with infinitely many nonzero values, defined on an infinite interval of inte¯ t¯(n) = 0 on I − (see gers I and satisfying L t(n) = L Section 2 for notation), • p ∈ K[n] is a polynomial such that t(n) = p(n)t¯(n) for all n ∈ I, • r ∈ K(n) is a rational function which satisfies (10) as an equation in K(n),

Now we consider the general case with ρ ≥ 1. Proposition 4. Let (r(n), R) be the result of applying AS to an operator L ∈ K[n, E] of type (7), and let r = s/q ¯ ∈ where s, q ∈ K[n]. Then there exist p ∈ K[n] and L K[n, E] such that ¯ L ◦ p = q L, (24)

• r¯ = p r ∈ K(n), • u ¯(n) is a K-sequence such that u ¯(n) = r¯(n)t¯(n) whenever both r¯(n) and t¯(n) are defined, • v, w are integers such that v ≤ w and [v, w] ∩ Z ⊆ I.

¯ and R ◦ p ∈ K[n, E]. (E − 1) ◦ R ◦ p = p − s L,

11

Proof: Let d ∈ K[n] be a polynomial and B ∈ K[n, E] an operator such that E ρ ◦ L∗ ◦

in (25). Then by Theorem 5, the sequence t(n) = (n−2)t¯(n) satisfies Lt(n) = 0 for all n ∈ [v, w − 1] ∩ Z , and the formula w−1


1 1 = B. q d

t(n) = 2w(w + 1)t¯(w) − 2v(v + 1)t¯(v)

(26)

n=v

Then

¯ =0 is valid whenever v ≤ w − 1. The general solution of Ly (−1/2)n ¯ is t(n) = c (1)n where c is an arbitrary constant. Thus, by taking c = −1/4, we see that (26) can be used to sum n the term t(n) = (n − 2)t¯(n) = (2 − n) (−1/2) considered in 4(1)n Example 4.

1 1 L ◦ E −ρ = B ∗ ◦ . q d Therefore L ◦ E −ρ ◦ d = qB ∗ . Multiplying this by E ρ on the right gives L ◦ E −ρ ◦ d ◦ E ρ = L ◦ d(n − ρ) = qB ∗ ◦ E ρ . Take ¯ = B ∗ ◦ E ρ . Then (24) is satisfied and p(n) = d(n − ρ) and L ¯ = p − rq L ¯ = p − rL ◦ p = (1 − rL) ◦ p = (E − 1) ◦ R ◦ p. p−s L ¯ by Hence the operator R ◦ p is the left quotient of p − s L E − 1 and, consequently, has polynomial coefficients.

Acknowledgements. The authors wish to express their thanks to J¨ urgen Gerhard for providing (a close variant of) the hypergeometric term in Example 1, and to anonymous referees for their helpful remarks.

Theorem 5. Let ¯ R, r, p, q be such as in Proposition 4, • L, L,

7. REFERENCES [1] S. A. Abramov, Rational solutions of linear difference and differential equations with polynomial coefficients, USSR Comput. Math. Phys. 29 (1989), 7–12. Transl. from Zh. vychisl. mat. mat. fyz. 29 (1989), 1611–1620. [2] S. A. Abramov, Rational solutions of linear difference and q-difference equations with polynomial coefficients, Programming and Comput. Software 21 (1995), 273–278. Transl. from Programmirovanie 21 (1995), 3–11. [3] S. A. Abramov, M. van Hoeij, Integration of solutions of linear functional equations, Integral Transforms and Special Functions 8 (1999), 3–12. [4] S. A. Abramov, M. Petkovˇsek, Solution spaces of H-systems and the Ore-Sato theorem, Proceedings FPSAC ’05, to appear. [5] F. Chyzak, An extension of Zeilberger’s fast algorithm to general holonomic functions, Discrete Math. 217 (2000), 115–134. [6] R. W. Gosper, Jr., Decision procedure for indefinite hypergeometric summation, Proc. Natl. Acad. Sci. USA 75 (1978), 40–42. [7] I. Nemes, M. Petkovˇsek, H. S. Wilf, D. Zeilberger, How to do Monthly problems with your computer, Amer. Math. Monthly 104 (1997), 505–519. [8] M. Schorn, Contributions to Symbolic Summation, RISC, J. Kepler University, Diploma Thesis, December 1995.

• v, w ∈ Z be such that v ≤ w − ρ, • Iρ = [v, w − ρ] ∩ Z, • t¯(n) be a K-valued sequence defined for all n ∈ [v, w] ∩ ¯ t¯(n) = 0 for all n ∈ Iρ . Z such that L Then the K-valued sequence t(n) = p(n)t¯(n) satisfies Lt(n) = 0 for all n ∈ Iρ , and the discrete Newton-Leibniz formula (18) can be applied to t(n) with u(n) = (R ◦ p)t¯(n). Proof: By (24), ¯ t¯(n) = 0 Lt(n) = (L ◦ p)t¯(n) = q L for all n ∈ Iρ . Also, u(n) = (R ◦ p)t¯(n) and u(n + 1) = (E ◦ R ◦ p)t¯(n) are defined for all n ∈ Iρ . Therefore, by Proposition 4, u(n + 1) − u(n)

¯ t¯(n) = ((E − 1) ◦ R ◦ p)t¯(n) = (p − sL) ¯ ¯ ¯ = p(n)t(n) − s(n)Lt(n) = t(n)

for all n ∈ Iρ , and (18) follows by summing this over Iρ . Example 8. Consider again the operator L = 2(n + 1)(n − 2)E − (2n − 1)(n − 1) from Example 2. Here mLt = 1 and MLl = 3, so, following Theorem 3, we can apply formula (18) if [v, w−1]∩[1, 2] = ∅. Using the algorithm from [1] or the algorithm from [2] we compute the solution r(n) = −(n + 2)/((n − 1)(n − 2)) of L∗ r = 1, and set q(n) = (n − 1)(n − 2). Then E ◦ L∗ ◦

1 1 = (−(2n + 1)E + 2(n + 1)), q(n) n−1

therefore we have d(n) B = ¯ = L p(n)

= n − 1, −(2n + 1)E + 2(n + 1), B ∗ ◦ E = 2(n + 1)E − (2n − 1), = n − 2,

(25)

R = 2n(n + 1)/(n − 2), u(n) = 2n(n + 1)t¯(n). Let t¯(n) be a sequence defined for all n ∈ [v, w] ∩ Z and ¯ t¯(n) = 0 for all n ∈ [v, w−1]∩ Z where L ¯ is given satisfying L

12

Signature of Symmetric Rational Matrices and the Unitary Dual of Lie Groups [Extended Abstract] ∗

†

B. David Saunders

Jeffrey Adams

Department of Mathematics University of Maryland College Park, MD, 20742 USA

[email protected]

Department of Computer and Information Sciences University of Delaware Newark, DE, 19716 USA

Department of Computer and Information Sciences University of Delaware Newark, DE, 19716 USA

[email protected]

[email protected]

ABSTRACT

definiteness properties of symmetric integer matrices. The latter is sometimes easier than the signature. Also there is often a considerable difference in the cost of verifying (certifying) a property and of certifying it’s negation. This is primarily an experimental paper whose purpose is to assess the feasibility of exact integer linear algebra methods for signature and definiteness determinations that are useful in the study of Lie group representations. The combination of measurements and analysis of asymptotic growth rates of time and memory use that we provide is for the purpose of predicting the cost of the larger computations of interest, so as to determine the most promising algorithms and the hardware resource needs. The algorithmic asymptotic complexities reported here are largely straightforward applications of known results. We have an interesting observation which allows both efficient binary tree structured Chinese remaindering to be used to construct numbers moduli a bunch of distinct primes and an early termination strategy at the same time. Practical and asymptotic speed up is given when using early termination. We have measured the time and memory costs of three algorithms on some matrices arising in group representations, studied their relative merits, and developed formulas to predict the costs for large matrices. We remark that the largest computation we’ve succeeded in doing with a standard package such as Mathematica is for a matrix of dimension 400, whereas the largest needed is for dimension 7168 Our measurements indicate it will be feasible. In section 2, the motivation for this work in the study of Lie group representations is presented. This application creates large problems straining our limits of time and memory. In section 3 we present the proposed algorithms and discuss their mathematical basis and complexity. Also included is some discussion of unimplemented alternatives and of the space issues for large instances of the problems. In section 4, we apply these algorithms to the matrices from Lie group representations. Finally, in section 5, experimental results are reported and, in section 6, conclusions are drawn on the state of these problems.

A key step in the computation of the unitary dual of a Lie group is the determination if certain rational symmetric matrices are positive semi-definite. The size of some of the computations dictates that high performance integer matrix computations be used. We explore the feasibility of this approach by developing three algorithms for integer symmetric matrix signature and studying their performance both asymptotically and experimentally on a particular matrix family constructed from the exceptional Weyl group E8 . We conclude that the computation is doable, with a parallel implementation needed for the largest representations.

Categories and Subject Descriptors G.4 [Mathematical Software]: Algorithm Design and Analysis; F.2.2 [Analysis of Algorithms and Problem Complexity]: Nonnumerical Algorithms and Problems—Computations on discrete structures

General Terms Algorithms, Performance

Keywords matrix signature, symmetric matrix, Lie group

1.

†

Zhendong Wan

INTRODUCTION

We propose, analyze, and test several algorithms for computing the signature and for verifying or disproving specific ∗Supported by NSF grant DMS 0200851. †Supported by NSF grants CCR 0098284 and 0112807.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. ISSAC ’05, KLMM, Chinese Academy of Sciences, Beijing, China Copyright 2005 ACM 1-59593-095-7/05/0007 ...$5.00.

2. LIE GROUP REPRESENTATIONS AND UNITARY DUALITY We assume some familiarity with root systems and Weyl

13

Each facet contains an element ν with rational coordinates. It is well known we may choose a basis of Vρ so that ρ(w) is a rational, or even an integral, matrix for all w ∈ W . For n ≤ 8 the former has been carried out explicitly in [16], and the latter in [1] except for E8 . Then the matrices Aρ (ν) and Jρ (ν) will have rational entries. We may clear denominators when testing for positive semi–definiteness. Now let G be a split semi-simple group over R or a p–adic field F, with root system R. Associated to ν is an irreducible spherical representation πν of G.

groups, for example see [11]. We begin with a formal construction. Let R be a root system with Weyl group W . Thus R is a finite subset of V = Rn satisfying certain properties; in particular for each α ∈ R the reflection sα ∈ Hom(V, V ) takes R to itself. By definition W acts on V . We may choose simple roots S = {α1 , . . . , αn } so that W is generated by {si = sαi | i = 1, . . . , n}, with relations order(si sj ) = mi,j for certain mi,j ∈ {2, 3, 4, 6}. For ν ∈ V define ν, α∨  = 2(ν, α)/(α, α). Let V + be the set of ν ∈ V which are dominant (i.e., v, α∨ i  ≥ 0 (1 ≤ i ≤ n)). Fix a finite dimensional representation (ρ, Vρ ) of W . Thus ρ : W → GL(Vρ ) is a group homomorphism. Fix an invariant Hermitian form (, )ρ on Vρ (i.e., satisfying (v1 , v2 )ρ = (ρ(w)v1 , ρ(w)v2 )ρ for all v1 , v2 ∈ Vρ , w ∈ W ). Choose a matrix Jρ so that (v1 , v2 )ρ = v1 Jρ v2t . For α ∈ R and ν ∈ V + we define Aρ (α, ν) =

1 + ν, α∨ ρ(sα ) ∈ Hom(Vρ , Vρ ). 1 + ν, α∨ 

Lemma 3. If F is p–adic πν is unitary if and only if τν is unitary. Conjecture 4. If F = R then πν is unitary if and only if τν is unitary.

(1)

Let w be the longest element of W and choose a reduced expression w = siN siN −1 . . . si1 (1 ≤ j ≤ N, 1 ≤ ij ≤ n). Set w0 = 1 and define wj = sαij sαij−1 . . . sαi1 (1 ≤ j ≤ N ). For ν ∈ V define Aρ (ν) =

n−1 Y

Aρ (αij , wj−1 ν) ∈ Hom(Vρ , Vρ )

(2)

j=1

and Jρ (ν) = Aρ (ν)Jρ .

(3)

Lemma 1. 1. Aρ (ν) is independent of the choice of reduced expression for w , 2. Aρ (0) = Id and Aρ (δ) = 0 (unless ρ is trivial) where δ is one-half the sum of the positive roots, 3. limν→∞ Aρ (ν) = ρ(w ), 4. Aρ (ν) is invertible unless ν, α∨ i  = 1 for some i, 5. Assume w ν = −ν. Then (Aρ (ν)v1 , v2 )ρ = (v1 , Aρ (ν)v2 )ρ for all v1 , v2 ∈ V , and Jρ (ν) is symmetric.

Thus computing the unitary representations τν of Lemma 2 tells us about a subset of the unitary dual of Lie groups (the “spherical” unitary dual). We may as well assume that G is simple, or equivalently that R is irreducible. The irreducible root systems are of type An , Bn , Cn , Dn (classical) or E6 , E7 , E8 , F4 or G2 (exceptional). The most interesting case is that of E8 . The Weyl group has order 696, 729, 600; it has 112 representations, the largest of which has dimension 7, 168. There are 1, 070, 716 facets. In the classical case the classification of the unitary representations τν , and Conjecture 4, are known [2]. Dan Barbasch and Dan Ciubotaru have also computed the unitary τν in the exceptional cases. Thus the problem which the calculation of Jρ (ν) solves is already known. However this calculation is the prototype of a much more general calculation which will be needed to calculate the unitary dual of Hecke algebras, real and p–adic Lie groups. The Atlas of Lie Groups and Representations is a project to compute the unitary dual by theoretical and computational means, see http://atlas.math.umd.edu. Information about what computations are feasible is of great importance in the continuation of this project.

3. SIGNATURE ALGORITHMS For the rest of this paper we will consider the question of signatures and sign patterns of symmetric matrices in general, but constantly keeping in mind the operators generated from Lie group matrix representations. The signature of a real symmetric matrix A is generally defined as σ = p−n, the number by which positive eigenvalues outnumber negative ones. Define the signature triple to be the σ ∗ (A) = (p, z, n), where p is the number of positive eigenvalues, z is the multiplicity of zero as an eigenvalue, and n is the number of negative eigenvalues. A real symmetric matrix is positive definite if z and n are zero and positive semi-definite if n is zero. For a polynomial with only real roots, Descartes’ rule of signs can be used to determine the number of positive roots and the number of negative roots. Thus the signature triple of a matrix can be determined from the signs in the vector of coefficients of the characteristic polynomial. It is computationally useful that, when only the zero eigenvalue has multiplicity greater than 1, the characteristic polynomial is just a shift of the minimal polynomial. If A has minimal polynomial m(x) of degree d, we define the shifted minimal

For proofs of this and other statements in this section see [2] and http://atlas.math.umd.edu/papers. Let H be the affine Hecke algebra of W [11, Chapter 7]. Associated to ν is an irreducible spherical representation τν . Lemma 2. The representation τν of H is unitary if and only if for every irreducible representation ρ of W , the operator Jρ (ν) is positive semi–definite. It is also of interest to determine the signature of Jρ (ν). The denominator in (1) is a convenient normalization, which makes Lemma 1(2) and (3) hold. Since ν ∈ V + it is positive and does not affect whether Jρ (ν) is positive semi– definite. The question of whether Jρ (ν) is positive semi–definite only depends on whether 1 − ν, α∨  is positive, negative, or 0, for each α ∈ R. Therefore the set of dominant parameters ν is decomposed into a finite number of facets, each one determined by an element of {+, 0, −}n (not every such n-tuple arises). The classification of the unitary representations τν of H is therefore reduced to a finite calculation.

14

polynomial to be xn−d m(x). Less well known is that signature can also be determined in certain cases, in a similar way, from the vector consisting of the leading principal minors of A. In view of these connections, for a vector the signature triple is σ ∗ (v) = (p, z, n), where p is the number of alternating successive nonzero pairs, and n is the number of constant successive nonzero pairs, z is the number of zeros at the end of the vector (which corresponds to the multiplicity of zero as an eigenvalue in our case). To be precise, a pair of entries, (vi , vj ) of (v0 , v1 , . . . vn ) is successive if i < j and the entries between them are zero. A successive pair of nonzero entries is alternating(constant) if their signs are opposite(same). The vector in question will be either the coefficient vector of a polynomial or w = (w0 = 1, w1 , . . . , wn ), where wi is (−1)i mi , with mi being the i × i leading principal minor of the given matrix. Following [12], say a matrix of rank r has generic rank profile if the leading principal minors of dimensions 1 through r are nonzero.

prime may divide many leading principal minors of A (and the rank may be lower mod p than the integer rank). Note that in both of these cases the segment of non-zero values computed mod p is shorter than it should be. In the interest of presenting the Chinese Remaindering issues in a generic way, we refer to a likely image function, which given A and p, returns a vector which is likely to be the image mod p of the desired integer vector but may not be, and if not, will be shorter. We call a prime p good if likely image(A, p) returns the correct image, otherwise p is bad. The key feature of this generic Chinese remainder algorithm for such vectors is its early termination technique. Algorithm: GenCRA [Chinese Remainder Algorithm with early termination] Input: A, a symmetric integer matrix. vp = likely image(A, p), a function as described above. P, a set of primes. Ω, the random sample size for the certificate. Output: The revealed vector v with integer coefficients.

Theorem 1 (Signature theorem). Let A be a real symmetric n × n matrix. The following hold: 1. Signature is invariant under congruence, that is, if Q is nonsingular then σ ∗ (A) = σ ∗ (QAQT ).

1. Set list L := ∅. This will be a list of pairs: (good prime, imaged answer mod that prime). Set l = 0. 2. Choose a random vector x of length n with entries independently and uniformly chosen from [0, Ω − 1]. 3. Uniformly and randomly remove a prime p from P. 4. Call vp = likely image(A, p). If the length of vp is less than l , [reject] goto statement 3. If the length of vp is greater than l , [restart] reset L := {(p, vp )}. Otherwise, [continue] append pair (p, vp ) to L. 5. Use Chinese remainder algorithm to construct the certificate c(i) where i the size of L, such that c(i) = x · vq (mod q), for each pair (q, vq ) in L. This construction is done incrementally, combining the previous result c(i−1) with the current residue and modulus. 6. Otherwise the termination condition, c(i) = c(i−1) , is met. Return the vector v, which is constructed from pairs in L by the Chinese remainder algorithm, such that v = vq (mod q), for every pair (q, vq ) in L. This remaindering is done using a divide-and-conquer method. Notes:

2. σ ∗ (A) = σ ∗ (charpoly(A)). Also, if the nonzero eigenvalues of A are distinct, σ ∗ (A) = σ ∗ (shiftminpoly(A)). 3. If A is in generic rank profile with rank r, σ ∗ (A) = σ ∗ (1, −m1 , . . . , (−1)r mr , 0, . . . , 0), where mi is the ith leading principal minor of A, and the last (n − r) entries are zero. 4. A matrix in generic rank profile has a unique A = LDLT decomposition with unit lower triangular L and diagonal D = diag(d1 , . . . , dn ). If A has generic rank profile and rank r, then (ending with Q (n − r) zeroes) σ ∗ (A) = σ ∗ (1, −d1 , d1 d2 , . . . , (−1)r 1≤i≤r di , 0, . . . , 0). Proof. A good source for these fundamental facts is [9]. In particular the third statement is a theorem of Jacobi, [9, Chapter X, §3, theorem 2]. The fourth item follows since di = mi /mi−1 . See [10, Chapter 4] for a good discussion of LDLT decomposition. ✷ The generic rank profile condition assures that the mi consist of nonzero entries followed by zeros with no intermingling of zero and nonzero values, and the same applies to the diagonal D of the LDLT decomposition. Interestingly, again see [9], the signature can be recovered even when there are some scattered zeroes among the nonzero leading minors, hence something less than generic rank profile is needed. We will not pursue this point further. Call a vector v a σ ∗ -revealing vector for A if σ ∗ (A) = ∗ σ (v). The algorithms we propose all work by computing images of a signature revealing vector mod a series of primes and constructing the integer vector via the Chinese Remainder Algorithm. The vector will be the coefficients of the characteristic polynomial, of the shifted minimal polynomial, or the vector of leading principal minors. In most of these cases there can be bad primes. For instance there are primes p for which the minimal polynomial of image mod p of A is not the image mod p of the integer minimal polynomial of A. This leads to a shifted minpoly mod p that is not a true image of the integer shifted minpoly. Similarly a

1. In order to capture negative numbers, we normalize the final number a to lie in [−(m − 1)/2, (m − 1)/2], Q where m= 1≤i≤n pi . 2. The early termination technique which may be used to reduce the practical run time has been studied before -see e.g. [8, 14, 7]. Here, we use a different and more efficient termination technique. At each step, only one integer called a “certificate” is constructed at each prime, instead of the entire vector answer. This method has almost the same probability as the probability when the whole answer is constructed at each prime. It allows the more efficient divide and conquer remaindering to be done for the n values in the answer while using the incremental remaindering only for the certificate. This technique can be easily adopted to other cases, such as solving non-singular integer systems over the rationals.

15

Proof: These preconditionings are discussed in detail in [4] ✷ For the generic rank profile condition, The butterfly is chosen as preconditioner because the specified matrix QAQT can be computed in O˜(n2 ) time. For our purposes here we can afford preconditioning complexity up to O˜(n3 ) time. A general random matrix could be used for the preconditioner, or Toeplitz [13] or sparse preconditioners [4, Section 6]. It is of interest to keep the size of the entries of the resulting matrix as small as possible. We build three algorithms on the theory described above, two using the minimal polynomial (for blackbox and for dense matrices) and one using LDLT decomposition.

3. The pre-selected primes idea in [14] may be use here also. It works with preselected prime stream so that one can pre-compute the constants which are independent of the actual answer, for instance (p1 . . . pi )−1 mod pi+1 (The precomputed constant will help if the Newton interpolation algorithm is applied to construct the final answer). Such moduli are not random, additional post-check of the result at one additional random moduli will guarantee the correctness of the final answer with very high probability. Please see [14] for more details. Theorem 2. The algorithm above computes the revealed vector with error probability at most

Algorithm: BBSM [BlackBox Signature by Minpoly] Input: A, a symmetric matrix in blackbox form, S, a set of integers from which to make random selections. Output: The signature σ ∗ (A) = (p, z, n). 1. [ Preconditioning may be necessary ] Let q be a random prime. Let r := rank(A, q) and mq = minpoly(A, q). If deg(mq ) < n and deg(mq ) ≤ r, let B := DAD [A blackbox], for a random diagonal matrix D with entries chosen from S, Otherwise, let B := A. [ Now B has the same signature as A and its charpoly is its shifted minpoly, with high probability. ] 2. Choose a set P of primes, and the sample size Ω, such that the error probability is as small as desired. Return σ ∗ (xmax(0,n−r−1) GenCRA(B, minpoly(), P, Ω)).

log2β (2l ΩB) 2 +  , Ω M − logβ (2l ΩB) where B is the infinity norm of the correct answer, M is the number of good primes in P, and β is the minimal prime in P. Note: For a typical problem, such as characteristic polynomial, or minimal polynomial, it is easy to choose a reasonable set of primes and random sample size such that the error probability is tiny. Runs are independent. If you repeat the algorithm, you square the error probability. PROOF: Let vector α denote the correct answer and c denote x · α. We assume the early termination condition is met when c(n) is equal to c(n−1) , for some number n. If both |c| ≥ α∞ and c = c(n) are true, then the algorithm returns the correct answer. This is true since the modulus which is the product of the primes in L is at least 2α∞ under these hypotheses. If x is a random vector with entries independently and uniformly chosen from the integer set [0, Ω−1], then Prob(|x· α|) < α∞ ) ≤ Ω2 . This is true since there is at least one entry of α, whose absolute value is equal to α∞ . Without loss of generosity, we assume |α0 | = α∞ . Then for any x1 , . . . , xl−1 , then there are at most two integers x0 , such that |x · α| < α∞ . The probability analysis of c = c(n) on the condition that the early termination condition c(n) = c(n−1) is met, is straightforward - see e.g. [14, Theorem 1.] for details. So

The r = rank(A, p) and v = minpoly(A, p) algorithms used to compute the rank and the minimal polynomial of A mod p respectively are as in [17, 13, 4], for example. Here minpoly() and rank() run in time O˜(nE) and are probabilistic with probability of error no more than 1/p, where E is the cost of a single matrix-by-vector product. But rank() will never return a value greater than the true rank and minpoly() always computes at least a factor of the true minimal polynomial of A mod p. DAD is the blackbox whose matrix-vector product is formed as y = D(A(Dx)). Algorithm: DSM [Dense Signature by Minpoly] Input: Matrix A, in dense form. Output: σ ∗ (A). Method: Apply algorithm BBSM, except use r = rank(A, p) [6] and v = minpoly(B, p), [5], algorithms which are available for the explicit (dense) matrix representation. Then rank() and minpoly() are deterministic eliminations running in time O˜(n3 ) and using O(n2 d) memory Of course, in this case the DAD preconditioning is done explicitly (and cheaply). Again, of course, p may be a bad prime (minpoly of A mod p not equal mod p image of integer minpoly of A..

log2 (2lΩB)

β . ✷ the total error probability is at most Ω2 + M −log β (2lΩB) The following theorem gives additional flexibility in using the strategies for signature computation made available by GenCRA.

Theorem 3. Let A be an n × n real symmetric matrix and let S be a set of nonzero integers of sufficient size that , = n2 log(n)2 /|S| is as small as desired. 1. Let D be a diagonal matrix whose n diagonal entries are chosen uniformly at random from S, and let B = DADT . Then

Especially for blackbox matrices, it is useful that minpoly computation can suffice, because we have faster algorithms for minpoly than for charpoly in that case. But for the DSM algorithm above, an alternative is to use the charpoly() function as the likely image function, see [5] for example. Also note that the minpoly suffices for determining the sign pattern (not full signature) even without preconditioning. When the minpoly is of low degree or has small coefficients, this is a great savings. In general, though, BBSM is not a fully memory efficient algorithm because of the size of the

Prob(shiftminpoly(B) = charpoly(B) ≥ 1 − ,. 2. Let Q be a butterfly matrix whose n log(n) defining entries are chosen uniformly at random from S, and let B = QAQT . Then Prob(B is in generic rank profile) ≥ 1 − ,.

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σ ∗ -revealing vector. It is possible that the technique of [3] could be used to determine the signs using less memory and perhaps less time. This approach deserves further examination. Alternatively, combining Theorem 1, part 3, 4, and theorem 3, part 2, we may work from the LDLT decomposition of a matrix in generic rank profile. If the matrix should fail to have generic rank profile, this will be detected during the eliminations because of the need for pivoting. We use preconditioners Q to assure that QAQT has generic rank profile. In some cases symmetric pivoting could be used instead to avoid the increased entry size caused by preconditioning. The LDLT decomposition based signature algorithm assumes a procedure lpm(v, A, p) which computes the vector of leading principal minors with alternating signs as described above, up to but not including the first which is zero mod p. The matrix mod p can fail to have generic rank profile. We could modify the LDLT to successfully handle such cases, but it is a remote possibility and not worth the overhead. We simply reject such primes as bad primes. The procedure lpm meets the requirements of a σ ∗ -revealing function for GenCRA.

Also, with any of the σ ∗ -revealing vectors, the entries tend to grow in proportion to their index. In particular the ith entry is either an i × i minor or sum of i × i minors, so is bounded by the Hadamard bound which is in O˜(id). A heuristic to determine indefiniteness computes the first few vector entries, using many fewer remaindering steps than are required for the later entries. If the sign pattern fails to be constant or strictly alternating the matrix is indefinite. It is an open question whether a conjugacy class preconditioning, A → QAQT , could make probable that early entries indicate definiteness (cheaply).

4. APPLICATION TO LIE MATRICES The matrices from Lie group representation are rational matrices. Our algorithms in the previous section focus on integer matrices. There are at least two ways to adapt. One is to compute the minimal polynomial, or the leading principal minors over the rationals. Rational numbers must be reconstructed at the GenCRA, and it is easy to adopt our GenCRA, including the early termination technique, to this case. The other way is to multiply the matrices by the lcm of the denominators of all entries. For these specially constructed matrices from Lie group representation, the lcm of the denominators of all entries is just a little larger than each individual one, so this latter way is a better. We found for some models that the gcd of all the numerators in the dense representation is not trivial and can be removed. Next we present blackbox algorithms to compute the lcm of denominators of all entries, and gcd of numerators of all entries, respectively.

Algorithm: DSLU [Dense matrix Signature by LU-decomposition] Input: A, an integer symmetric k × k matrix. Output: Signature σ ∗ (A) = (p, z, n). 1. [ Preconditioning may be necessary ] Let q be a random prime. Let lpm(vq , A, q), r := rank(A, q). If length of vq less than r + 1, let B := QAQT , for a random integer matrix Q with entries chosen from [1, · · · , s] (or any set of s nonzero integers). [Q may be a Toeplitz or Butterfly matrix for speed] Otherwise, let B := A. [ Now B has the same signature as A and has generic rank profile. ] Choose a set P of primes, and the sample size Ω, such that the error probability is as small as desired. 2. Return σ ∗ (xn−r GenCRA(B, lpm(), P, Ω)).

Algorithm: LD [LCM of Denominators] Input: A, a rational matrix M , sample size n, number of trials Output: d, the lcm of denominators of all entries of dense representation of A. 1. for i from 1 to n do Choose a random vector x(i) with entries independently and uniformly chosen from [0, M − 1] y (i) = Ax(i) , apply x(i) to A d(i) = the lcm of denominators of every entry of y (i) . 2. d := lcm(d(1) , · · · , d(i) ) 3. return d.

Theorem 4. Let A be an integer blackbox n × n symmetric matrix, whose matrix-vector product costs e operations and whose entries would be of bit length at most d if they were explicitly constructed. Algorithms BBSM, DSM, and DSLU are Las Vegas probabilistic algorithm if the Hadamard bound is used and are Monte Carlo if early termination is used.Even with early termination, DSM is Las Vegas if the computed integer minimal polynomial is checked by evaluation at A over the integers. Also a minimal polynomial verification by application to the identity could be done after BBSM making it Las Vegas for the sign pattern of the signature. Let h be a bound for the bit lengths of the values constructed using GenCRA, (the length of the largest characteristic polynomial coefficient or of the largest leading principal minor). Then the expected run times are in O˜(neh) for BBSM and in O˜(n3 h) for DSM and DSLU. By the Hadamard bound, h is in O˜(nd), so the expected run times are also in O˜(n2 ed) for BBSM and in O˜(n4 d) for DSM and DSLU. In particular if e ∈ O˜(n), and d ∈ O(log(n)), then the BBSM expected run time is O˜(n3 ) and the DSM, DSLU expected times are in O˜(n4 ). ✷

Algorithm: GN [Gcd of Numerators] Apply algorithm LD by replacing lcm with gcd and denominators with numerators. The algorithm LD always returns a factor of the true answer. The algorithm GN always returns a multiple of the true answer. For a rational matrix A, if d and g are the lcm of all denominators and gcd all numerators of entries of A, respectively, then If A = dg A is an integer matrix. For each

individual trial, if all entries of A x(i) are coprime, then d(i) is correct in both algorithms. Please see [15] for probability analysis. Roughly speaking, the error probability is 2−n+1 . Each matrix Jρ (ν) in equation 3 can be represented as a product of many sums of sparse matrices and scalar matrices. The algorithm BBSM is very suitable for this case. We also reduce the product of 121 sums of sparse matrices and

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5.

EXPERIMENTS

Our goal is to know which algorithm to use and how much resources it will take for computing the signature of the Jρ (ν) in equation 3, the matrix generated by the facet ν for the model of dimension n of the group E8 . In particular, we focus on the case when the facet is the challenge facet ν0 = 1/40(0, 1, 2, 3, 4, 5, 6, 23), and the goal is to verify (or disprove) that Jn (ν0 ) is positive definite. To emphasize the model dimension n in what follows, let Jn = Jρ (ν0 ). The three signature algorithms we have discussed are BBSM, DSM, DSLU. We have computed full solutions by each model for the models Q2, . . ., Q12, Q22, Q32, Q42, Q52(dimension 1008), and we have computed single modular step costs for every 10th model on up to the largest, Q112(dimension 7168). The experiments involved individual run times up to about one day and verified, where the full computation was done, that Jn is indeed positive definite. All computations reported here were performed on a 3.2GHz Xeon processor with 6GB of main memory. Our purpose here is to determine the algorithm to use and resources needed for the full signature computation on examples of all dimensions (up to 7168) and to have tools to estimate the cost for matrices defined by other facets. The blackbox representation of Jn is a product of very sparse factors. Since the blackbox method cost is sensitive to e, the cost of the single matrix-by-vector product, we examined the number of nonzero elements in total in these factors. To guess the expected growth of e with dimension, we measured the total number of nonzero entries in the factors of every 10th model up to Q112. Since the number of factors is fixed and they are extremely sparse we expect a linear or n log(n) term to dominate. The formula e = 132.5900691n + 13.12471811n ln(n), a least squares fit to the data, is plotted in figure 1 along with the data values. The fit is extremely good and we see that e is only slightly super-linear. Therefore we expect the runtime of characteristic polynomial computation mod a single prime to grow at a rate only slightly above quadratic in n.

Each algorithm involves a series of steps mod primes pi , each step being a computation of σ ∗ -revealing vector (a minpoly or diagonal of LDLT -decomposition) mod pi . The computation is finished when the CRA remainderingQhas given an image of the vector with modulus, M = 2 pi , sufficiently large to recover the actual (signed) vector entries. If we know a bound d for the length of the entries of Jn , we have by the Hadamard bound a maximal length b = n(log(n)/2 + d) + 1 for M . The dense matrices are computed by applying the black box to the columns of the identity matrix (over Z, not modularly), and so the computation is sensitive to the length of the actual integer entries. Let d be the bit length of the largest entry. We do not have a theory to predict how matrix entry bit length d may depend on the model dimension for a fixed facet. We have plotted the bit length d of the largest entry of Jn along with the fitted curve d = 67.7 + 33.2 log(n) in figure 2. One sees the log(n) playing a stronger role than that for the non-zeroes in the blackbox, but the fit to the data is poorer and likely to have less predictive value. We noted that the n2 entries of A all have about the same length, so n2 d accurately describes the storage required for A. Figure 2: Bit length of entries 350

300

bit length

scalar matrices to a product of 94 sparse matrices for the E8 case. We explicitly compute each sum of a sparse matrix and a scalar matrix and store them as a single sparse matrix. Also, if the result is a diagonal matrix, explicitly multiply it into the previous factor. By reducing the number of sparse matrices in this way, 20% run time was saved for the BBSM algorithm. After multiplying these collapsed matrices by the lcm of denominators of all entries and dividing by gcd of numerators, these matrices can be handled by the algorithms discussed in the previous section.

200

150 actual bit length estimated by: 176.5334824+18.42786376ln(n)

100 10

100

1000

10000

matrix order

It will also be useful to use d in estimating the number of remaindering steps. The needed bit length b for the modulus M is at most 1 bit (for sign) more than the length of the p Hadamard bound. The rows norms are no more than n(ed )2 , so their product has length bounded by n(log(n)/2 + d). Reasonably consistent with this prediction is the number of bits used when we ran the full algorithms on Jn for n up to 1008 (model Q52). The curve in figure 3 is a least squares fit, b = 150.1n + 17.30nln(n), The number of bits for the charpoly coefficients (sums of minors) are expected to be slightly larger than for the entries of the leading principal minors vector, but most likely, the final term, determinant of Jn , is dominant. The number of bits needed appears to be slightly super-linear, not quite as large as the worst case Hadamard bound level. Figure 3: Bit length of final modulus

Figure 1: Blackbox number of non-zeroes 1e+07

1e+06

1e+06

100000

bits

number of non-zero entries

250

100000

10000

10000

1000

estimated by: 87.7n + 20nlogn Actual data

1000 10

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Actual required Estimated by:150.1n + 17.3nln(n)

100 10000

1

matrix order

18

10

100 matrix order

1000

10000

Next we consider the costs of the modular steps. Let step(s) denote the cost per step which varies with length, s, of the prime, but is essentially constant among primes of the same length. The total time is b/s×step(s) when using s bit primes. Thus we see the advantage of minimizing step(s)/s, the cost per bit. After discussing step(s)/s and determining it’s value, for each algorithm, we will know how to estimate the full runtime of the algorithms. We expect to be able to reduce the LU step cost by half, as an elimination better tailored to symmetric matrices can be done, while still taking advantage of block operations using BLAS. For the blackbox algorithm, block methods may be able to reduce the costs somewhat, and the DSM may perform relatively better than here if the dense charpoly() algorithm [5] is used rather than preconditioning. At any rate, in figure 4 we see the performance of the modular steps of the three algorithms, BBSM, DAME, DSLU as currently implemented. Asymptotically, the BBSM step runtime, s, is expected to grow in proportion to ne. There was a good fit to s = 61n2 log(n) nanoseconds per bit (recall that e has a n log(n) dominant term). The algorithms DSM and DSLU depend on the construction of the full integer matrix of Jn , which is done once. The step times consist of computing a modular image of that and proceeding with an elimination on the image. The implementation of the Krylov matrix construction in the minpoly() algorithm of DSM , uses a technique of recursive doubling to better exploit fast matrix multiplication, and gets a step time s which fits the formula t = 1.8n3 + 0.16n3 log(n) nanoseconds per bit. t = 0.37n3 log(n) nanoseconds per bit. quite well. The DSLU step fits t = 0.3838888362n3 nanosec/bit. But these formulas have to be worked to the bit cost level!!! The log factor for DSM step can be removed, so it remains possible that it can become competitive or participate with BBSM in a heterogeneous distributed computation, running on the machines with larger memory. The time formula for the dense matrices should remain valid as long as the modular step fits in main memory. After than swapping would dramatically increase times. Assuming a memory capable of holding a 2GB virtual memory, this would allow for n2 words, so n < 215 . For all practical purposes, the blackbox step has no memory limitation.

the expanded integer matrix initially. In figure 5 we show the overall run times (where the computations have been done so far) together with their least squares fit formulas. The formulas for time t in nanoseconds are BBSM: t = 10050n3 log(n), DSM: t = 6.08n4 log(n), and DSLU: t = 1.45n4 log(n).Theory predicts a second log factor in the BBSM time, coming from the the single matrix-by-vector product cost, e. However that effect is weak and the fit was poorer. Figure 5: Total run time 1e+09 1e+08 1e+07 1e+06

time (s)

100000 10000 1000 100 BB time Estimated BB MINBLAS timeS Estimated MINBLAS LUBLAS time Estimated LUBLAS

10 1 0.1 0.01 10

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The memory needed to store the σ ∗ -revealing vector mutes the memory advantage of BBSM. However the modular images of the vector are easily stored on hard disk and manipulated from there, so the memory advantage remains real. The crossover point of the runtime formulas for BBSM and DSLU is around n = 6931. However the formula t = 10.02n4 fits the DSLU data about as well as our formula above. Using that form, the crossover point is at n = 9150. We have also completed a study of the operators for the facet 1/12(0, 2, 2, 3, 4, 5, 7, 23). In this case the ranks of the operators in each model are very low. For the largest model of dimension 7168 the rank is 448, and for the other 111 models the rank seldom exceeds a tenth of the dimension. We were able to verify the positive semi-definiteness in all 112 models, using a total of about 125 cpu hours. The LU method was most often the fastest, and it’s time was dominated by the cost to expand the product of 121 sparse matrices into a single dense matrix. The blackbox minpoly method was faster for the models with lowest ranks (and thus smallest degree minimal polynomial). Also it was the only method to work for the largest model because of address space limitations.

Figure 4: Time per bit of modulus 100 10

6. CONCLUSION

time (s)

1

We have demonstrated that we can compute (on current hardware) the signature of a dense n × n integer matrix having entries of bit length around log(n) in a minute if n ≤ 200, in three hours if n = 1000 and (projected) in a CPU year for J7168 . Beyond that size, using algorithm BBSM, the time grows at a rate slightly higher than n3 and memory is not a constraint (except for storage of the sparse factors of the blackbox). However, we conclude that algorithm DSLU serves best for matrices of dimension n < 7000. It is tossup between DSLU and BBSM for dimensions 7000 ≤ n ≤ 9000 and BBSM is superior beyond that. DSLU time grows slightly above n4 . For J7168 , DSLU requires explicit use of file storage of the expanded matrix and all algorithms should do this for intermediate results (modular images of σ ∗ -revealing vectors) because of the large size. We have not measured the cost

0.1 0.01 0.001 0.0001 1e-05 100

BB per bit Estimated BB MINBLAS per bit Estimated BL LUBLAS per bit Estimated LU 1000 matrix order

10000

The total runtime then involves the time per bit in the modular steps times the number of bits in the σ ∗ -revealing vector. The Chinese remaindering adds a cost that is similar for each but large enough to mute somewhat the effect of the differing step costs. However with the early termination technique of the previous section this remaindering is a smaller factor than in earlier timings. The two dense algorithms also incur a lower order cost for the creation of

19

[8] I. Z. Emiris. A complete implementation for computing general dimensional convex hulls. Inter. J. Comput. Geom. Appl., 8:223–253, 1998. [9] F. R. Gantmacher. The Theory of Matrices. Chelsea, New York, NY, 1959. [10] G. H. Golub and C. F. Van Loan. Matrix Computations. Johns Hopkins University Press, Baltimore, Maryland, third edition, 1996. [11] James E. Humphreys. Reflection groups and Coxeter groups, volume 29 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1990. [12] E. Kaltofen and A. Lobo. On rank properties of Toeplitz matrices over finite fields. In ISSAC 96 Proc. 1996 Internat. Symp. Symbolic Algebraic Comput., pages 241–249. [13] E. Kaltofen and B. D. Saunders. On Wiedemann’s method of solving sparse linear systems. In H. F. Mattson, T. Mora, and T. R. N. Rao, editors, Proc. AAECC-9, volume 539 of Lect. Notes Comput. Sci., pages 29–38, Heidelberg, Germany, 1991. Springer Verlag. [14] Erich Kaltofen. An output-sensitive variant of the baby steps/giant steps determinant algorithm. In Proc. ISSAC’02, pages 138–144. ACM Press, 2002. [15] B. David Saunders and Zhendong Wan. Smith normal form of dense integer matrices fast algorithms into practice. In Proc. ISSAC’04, pages 274–281. ACM Press, 2004. [16] John R. Stembridge. Explicit matrices for irreducible representations of Weyl groups. Represent. Theory (electronic), 8:267–289, 2004. [17] D. Wiedemann. Solving sparse linear equations over finite fields. IEEE Trans. Inf. Theory, it-32:54–62, 1986.

of this file manipulation. At the crossover about one CPU year is required and DSLU needs a large memory. The run time is expected to be about a CPU year, so parallel computation is desirable (and is quite straightforward for either algorithm) on distributed or shared memory hardware. It is an open question whether definiteness can be determined fundamentally faster than signature. There is a fast Monte Carlo algorithm for rank, hence for distinguishing semi-definite from definite matrices. We have sketched a heuristic that sometimes can determine indefiniteness much faster than the signature computation. To provide for the needs of Lie group representation studies, both BBSM and DSLU will be further refined and their parallel implementation developed. Also the judicious incorporation of numeric computation is a possibility.

7.

REFERENCES

[1] J Adams. Integral models of representations of weyl groups. http://atlas.math.umd.edu/weyl/integral. [2] D. Barbasch. Unitary spherical spectrum for split classical groups, preprint. http://www.math.cornell.edu/~barbasch/nsph.ps. [3] H. Bronnimann, I. Z. Emiris, V.Y. Pan, and S. Pion. Sign determination in residue number systems. Theoret. Comput. Sci., 210:173197, 1999. [4] L. Chen, W. Eberly, E. Kaltofen, B. D. Saunders, W. J. Turner, and G. Villard. Efficient matrix preconditioners for black box linear algebra. Linear Algebra and Applications, 343-344:119–146, 2002. [5] C.Pernet and Z. Wan. LU based algorithms for the characteristic polynomial over a finite field. In Poster, ISSAC’03. ACM Press, 2003. [6] Jean-Guillaume Dumas, Pascal Giorgi, and Cl´ement Pernet. FFPACK: Finite field linear algebra package. [7] Wayne Eberly. Early termination over small fields. In Proc. ISSAC’03, pages 80–87. ACM Press, 2003.

20

Sum of Roots with Positive Real Parts

∗

Hirokazu Anai

Shinji Hara

Kazuhiro Yokoyama

Fujitsu Laboratories Ltd 4-1-1 Kamikodanaka, Nakahara-ku, Kawasaki 211-8588, Japan

The University of Tokyo 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-8656, Japan

Rikkyo University 3-34-1 Nishi Ikebukuro, Toshima-ku, Tokyo, 171-8501, Japan

[email protected]

shinji [email protected]

ABSTRACT

[email protected]

system with a plant P (s) = nc (s) dc (s)

In this paper we present a method to compute or estimate the sum of roots with positive real parts (SORPRP) of a polynomial, which is related to a certain index of “average” stability in optimal control, without computing explicit numerical values of the roots. The method is based on symbolic and algebraic computations and enables us to deal with polynomials with parametric coefficients for their SORPRP. This leads to provide a novel systematic method to achieve optimal regulator design in control by combining the method with quantifier elimination. We also report some experimental result for a typical class of plants and confirm the effectiveness of the proposed method.

np (s) dp (s)

controlled by a controller

C(s) = where np (s), dp (s), nc (s), dc (s) ∈ Q[s], the stability of the system is described as follows: The feedback system is stable if and only if all of the roots of the closed-loop characteristic polynomial g(s) = np nc + dp dp locate within the left-half plane of the Gaussian plane. This is called Hurwitz stability. We may consider more general notion of stability, called D-stability, which implies that all of the roots locate inside a restricted region D within the left-half plane of the Gaussian plane. Control design problem is to find a controller C(s) so that the system satisfies given specifications. As the controller C(s, q) has fixed-structure with some parameters q, what we have to do is to seek feasible controller parameters q which satisfy the controller design problem. For such problems, techniques in computer algebra have been successfully applied [9, 14, 1, 2]. Stability is the first necessary requirement for control system design, and assigning roots of a certain polynomial within a desired region is an essential problem for stability study. Root assignment problem for Hurwitz stability is to find controller parameters q so that the system is Hurwitz stable. This is easily verified by the well-known Routh-Hurwitz criterion. In the case of D-stability, a wedge shape region or a circle is usually used as stability region D. For root assignment problems with such stability regions, controller design problem is reduced to check a sign definite condition (SDC) ∀z > 0, f (z) > 0 where f (z) ∈ Q(q)[z], see [15, 13]. Applying real quantifier elimination (QE) to the sign definite condition, we can obtain possible regions of controller parameters q to meet D-stability. In particular, we can utilize an efficient quantifier elimination algorithm specialized to SDC [1, 10]. These controller synthesis methods with respect to stability are implemented as functions in a MATLAB toolbox for robust parametric control [3] In this paper we focus on the sum of roots with positive real parts (SORPRP) of a given even polynomial, and provide another successful application of computer algebra to control design problem, where the SORPRP is related to certain index of “average” stability in optimal control. Here we compute or estimate the SORPRP without computing explicit numerical values of roots. Hence, we can handle polynomials with parametric coefficients for their SORPRP. The key point of the method is that computing SORPRP is reduced to computation of the maximal real root of another univariate polynomial. Subsequently this enables us to achieve control system design with respect to SORPRP systematically. In fact, since the actual control design prob-

Categories and Subject Descriptors I.1.2 [Computing Methodologies]: Symbolic and Algebraic Manipulation – Algebraic Algorithms

General Terms Algorithm, Experimentation

Keywords Sum of roots with positive real parts, Gr¨ obner basis, resultant, quantifier elimination, optimal regulator control

1. INTRODUCTION In control and system theory, investigating location of roots of the characteristic polynomial is one of important and fundamental topics related to the stability of feedback control systems. For example, in case of a typical feedback ∗

This work has been supported in part by CREST of Japan Science and Technology Agency, the 21st Century COE: Information Science and Technology Strategic Core “Superrobust computation project” (The University of Tokyo) and “Development of dynamic mathematics with high functionality” (Kyushu University).

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. ISSAC’05, July 24–27, 2005, Beijing, China. Copyright 2005 ACM 1-59593-095-7/05/0007 ...$5.00.

21

¯ m,f (z) and Rm,f (z), since W is a real number. To compute R we use the following triangular set related to Cauchy moduli [5] defined by f (x).

lems treated are recast as simple conditions on an univariate polynomial with parametric coefficients (one of them is a sign definite condition), we can utilize an efficient quantifier elimination algorithm using Sturm-Habicht sequence [1, 10]. The proposed method is applied to an even polynomial derived from “linear quadratic regulator (LQR) problem” which is one of the main concerns in control theory. The rest of this paper is organized as follows: First we show algorithms to compute other polynomials whose maximal real root coincides with the SORPRP of a given even polynomial. Then, using such polynomials, we state our main problem that comes from control design problem in §3. In §4, we present how we formulate the basic problem as a first-order formula and solve it by using quantifier elimination. In §5 we demonstrate some examples of LQR problems in order to confirm the effectiveness of the proposed method. Finally this paper ends with conclusion in §6.

2.

Definition 2. Let D be an arbitrary computable integral domain and K its quotient field. For a polynomial g(x) of degree n in D[x], we define the following polynomials: {g1 (x1 ), g2 (x1 , x2 ), . . . , gn (x1 , . . . , xn )}, where g1 (x1 ) = g(x1 ) and gi (x1 , . . . , xi ) is the quotient of g(xi ) divided by (xi − x1 ) · · · (xi − xi−1 ) for each i > 1. We note that gi (x1 , . . . , xi ) ∈ D[x1 , . . . , xi ] and gi (x1 , . . . , xi ) coincides with the quotient of gi−1 (x1 , . . . , xi−2 , xi ) divided by xi − xi−1 . Here we call {g1 , . . . , gn } the standard triangular set defined by g(x), and also call {g1 , . . . , gk } the k-th standard triangular set defined by g(x). It is well-known that {g1 , . . . , gk } forms a Gr¨ obner basis of the ideal hg1 , . . . , gk i generated by itself with respect to the lexicographic order x1 < . . . < xk in K[x1 , . . . , xk ] and the set of all its zeros with multiplicities counted coincides with the set {(βi1 , . . . , βik ) | i1 , . . . , ik ∈ {1, . . . , n} are distinct to each other }, where β1 , . . . , βn are all roots of g(x) in the algebraic closure of K. Thus, when g(x) is squarefree, hg1 , . . . , gk i is a radical ideal. We note that for each gi its leading coefficient lc(gi ) with respect to the order < coincides with the leading coefficient lc(g) of g(x). Now let F = {f1 (x1 ), . . . , fm (x1 , . . . , xm )} be the m-th standard triangular set defined by f (x) in Q[x1 , . . . , xm ]. Rm,f (z) can be computed by successive resultant computa¯ m,f (z) can be computed as the minimal polynotion and R mial of z = x1 + · · · + xm modulo the ideal I = hFi (with square-free computation if necessary). As Rm,f (z) has huge degree, it is difficult to compute it as it is. But Rm,f (z) is very useful for estimating coefficients and so we explain its computation below.

SORPRP OF EVEN POLYNOMIALS

First we consider an even polynomial f (x) of degree 2m in Q[x] with non zero constant. Then for any root α of f (x), −α is also a root of f (x). Thus, there are m roots, say α1 , . . . , αm , which have positive real parts, and also m roots, say αm+1 , . . . , α2m , which have negative real parts. So, f (x)

= =

a2m x2m + a2m−2 x2m−2 + · · · + a2 x2 + a0 Q a2m 2m i=1 (x − αi ),

(1)

where a2k ∈ Q for 0 ≤ k ≤ m, a2m 6= 0 and a0 6= 0. Our first target is to compute W = α1 + . . . + αm , which is the sum of all roots with positive real parts, without computing all αi ’s. For simplicity, we call W the SORPRP of f . Since, for each non real root of f (x), its complex conjugate has the same real part, we have the following: Lemma 1. W is a real number.

2.1

Polynomial having SORPRP as its root

Computation of Rm,f (z) via resultant

Let Bi1 ,...,im = αi1 + · · · + αim for 1 ≤ i1 , . . . , im ≤ 2m, and B the set of all distinct values of Bi1 ,...,im .

Let Tm (z) = z − (x1 + · · · + xm ) and for each k ≤ m, we define Tk successively as follows:

Definition 1. Gathering all sums Bi1 ,...,im , we can con¯ m,f (z), struct a polynomial Rm,f (z) and its square-free part R where z is a new variable: Y Rm,f (z) = (z − Bi1 ,...,im ), (2)

Tk−1 (z, x1 , . . . , xk−1 ) = resxk (fk (x1 , . . . , xk ), Tk (z, x1 , . . . , xk )),

where resy means the resultant with respect to a variable y. Then T0 (z) ∈ Q[z] and T0 (z) coincides with ad2m Rm,f (z) for some positive integer d. This can be shown as follows: By construction of Sylvester matrices in the resultant computation, it follows that the leading coefficient of Ti with respect to xj , where Ti is considered as a univariate polynomial in xj , is some power of a2m for each j < i, and the same for the leading coefficient of Ti with respect to z. Then, by the property of resultant, we have

i1 ,...,im

¯ m,f (z) R

=

Y

(z − B).

(3)

B∈B

As there are cases where Bi1 ,...,im coincides with Bj1 ,...,jm for distinct (i1 , . . . , im ) and (j1 , . . . , jm ), the square-free part ¯ m,f (z) is much smaller than Rm,f (z). Since all Bi1 ,...,im R are algebraic numbers and the coefficients of Rm,f (z) are symmetric under all permutations of roots, it follows that ¯ m,f (z) ∈ Q[z]. We may call Rm,f (z) Rm,f (z) ∈ Q[z] and so R ¯ m,f (z) the characteristic polynomial of sums of m and R roots, and the minimal polynomial of sums of m roots, respectively. ¯ m,f (z)) ≤ We note that deg(Rm,f (z)) = (2m)! and deg(R m! (2m)! . C = 2m m (m!)2

T0 (z)

=

1 ad2m

2m Y

T1 (z, αi ),

i=1

T1 (z, αi1 )

=

2 ad2m

Y

T2 (z, αi1 , αi2 ),

i2 6=i1

.. . Tm−1 (z, αi1 , . . . , αim−1 )

=

m ad2m

Y

Tm (z, αi1 , . . . , αim ),

im 6=i1 ,...,im−1

Tm (z, αi1 , . . . , αim )

It is obvious that the SORPRP W = α1 + · · · + αm of f (x) ¯ m,f (z) (Rm,f (z)), coincides with the maximal real root of R

22

=

z − (αi1 + · · · + αim ),

where i1 , . . . , im are distinct to each other and each di is a positive integer. (See [8].) When f (x) ∈ Z[x], that is, all a2k are integers, T0 (z) belongs to Z[z]. In order to avoid “coefficient growth and degree growth” in resultant computation, we may apply factorization technique to each Tk or its factors for computing smaller factors of T0 . (See §4.2 for usage of factors.) We note that multi-polynomial resultants can be also applied for computing T0 (z) by considering the system {f1 = f2 = · · · = fm = z − (x1 + · · · + xm ) = 0} in the variable {x1 , · · · , xm }.

tational efficiency, computing M (z, p) is much better than computing T0 (z, p) in many cases.

2.3

¯ m,f (z) via minimal polynomial Computation of R

Theorem 2. (Theorem 3.5 in [18], see also [12]) For a polynomial h(x) of degree n in C[x], let ϕ(x), ψ(x) ∈ R[x] be its real part and its imaginary part, respectively, that is, h(x) = ϕ(x) + iψ(x). We assume that h(x) has no real root. (If h(x) has a real root, we can eliminate it by GCD computation of ϕ(x) and ψ(x). ) Moreover, let N, N 0 be the number of roots with positive imaginary parts, and the number of roots with negative imaginary parts, respectively. Moreover, if deg(ϕ) ≥ deg(ψ), let V (a) be the number of variations in the sign of the standard Sturm sequence of ϕ(x), ψ(x) evaluated at the point a, and otherwise, let V (a) be that of ψ(x), ϕ(x) evaluated at the point a. Then,

Let z = x1 + · · · + xm and I = hFi in Q[x1 , . . . , xm ]. Then, we consider a minimal polynomial M (z) of z modulo I, that is, M (z) has the smallest degree among all polynomials h(z) in Q[z] such that h(x1 + · · · + xm ) belongs to the ideal I. Since the set of all zeros of I with multiplicities counted is {(αi1 , . . . , αim ) | i1 , . . . , im ∈ {1, . . . , 2m} are distinct to each other }, it can be shown easily that M (z) ¯ m,f (z) as its factor. (We is a factor of Rm,f (z) and has R ¯ m,f (z). may say that M (z) stands between Rm,f (z) and R ) Especially, when f (x) is square-free, then M (z)/lc(M (z)) ¯ m,f (z). When f (x) ∈ Z[x], that is, all a2k coincides with R are integers, by removing denominators of coefficients appearing in M (z), we may assume that M (z) belongs to Z[z]. Then the leading coefficient lc(M ) divides some power of a2m , as M (z) divides T0 (z). As we already know the Gr¨ obner basis {f1 , . . . , fm } of I, M (z) can be computed rather easily. See §4 in [17].

2.2

Note on non-even polynomial case

For an even polynomial of degree 2m, we know the number m of roots with positive real parts from their degree 2m. However, for a non-even polynomial, we do not know the number m of roots with positive real parts in advance. We can compute the number m (without computing each roots) by utilizing the following theorem for h(x) = f (ix), where i is an imaginary unit. (For Sturm sequence, see [16].)

N=

1 1 {n+V (∞)−V (−∞)}, N 0 = {n−V (∞)+V (−∞)}. 2 2

Since the number N 0 of roots of h(x) with negative imaginary parts coincides with the number m of roots of f (x) with positive real parts, we can compute m from N 0 . Then, same as in the case of even polynomials, the SORPRP W = α1 + · · · + αm of f (x) coincides with the maximal real root of Rf,m (z). For parametric case, m ranges according to the values of parameters, and m can be computed by precise analysis on Sturm sequence of ϕ, ψ using quantifier elimination technique. However, the problem becomes much complicated and harder to solve.

Parametric case

Now we consider the case where each coefficient a2k is some polynomial in parameters p = {p1 , . . . , pt }. Thus, the even polynomial f (x) is considered as a multivariate polynomial f (x, p) in Q[x, p]. Setting D = Q[p] and K = Q(p), we can compute the m-th standard triangular set F = {f1 (x1 , p), . . . , fm (x1 , . . . , xm , p)} in D[x1 , . . . , xm ]. Then, as lc(fi ) = a2m (p) for each i, F¯ = {f1 /a2m , . . . , fm /a2m } is the reduced Gr¨ obner basis of hFi in K[x1 , . . . , xm ]. By F, we can compute T0 (z, p) by successive resultant computation and M (z, p) as a minimal polynomial of z modulo the ideal hFi in K[x1 , . . . , xm ]. We note that using a block order {xm > . . . > x1 } >> z, M (z, p) is found in a Gr¨ obner basis of hF ∪ {z − (x1 + · · · + xm )}i in K[x1 , . . . , xm , z]. Then T0 (z, p) belongs to Q[z, p], and by removing denominators, we may assume that M (z, p) also belongs to Q[z, p]. As F¯ is the reduced Gr¨ obner basis of hFi and the denominator coincides with a2m (p), the following holds. (See Exercises of Chapter 6.3 in [7].)

3.

FORMULATION OF BASIC PROBLEM

Here we explain the fundamental problem in this paper. We denote the polynomial obtained above (T0 (z, p) or M (z, p)) by R(z). What we do after obtaining R(z) is the following: Problem 1. Given a polynomial R(z) involving parameters p in coefficients, R(z) ∈ Q(p)[z] and M1 , M2 ∈ Q (M1 > M2 ). Then find feasible ranges of parameters p so that the maximal real root W of R(z) satisfies the following each requirement: (a) W < M1 , (b) W > M2 , (c) M2 < W < M1 .

Theorem 1. For each (c1 , . . . , ct ) ∈ Qt , consider the polynomial fc (z) obtained from f (x, p) by substituting the parameters (p1 , . . . , pt ) with (c1 , . . . , ct ). If the leading coefficient a2m (c1 , . . . , ct ) does not vanish, then T0 (z, c1 , . . . , ct ) coincides with cRm,fc (z) for some non-zero constant c in Q, ¯ m,fc (z) and M (z, c1 , . . . , ct ) is a factor of Rm,fc (z) and has R as its factor in Q[z].

Here we exclude ranges where the leading coefficient of R(z) or its constant term vanishes. In view of control theory the parameters p usually comes from controller or plant parameters of the control system to be designed, and the above three requirements are originated from control design specifications in terms of SORPRP.

By Theorem 1, we can handle the SORPRPs for polynomials with parametric coefficients. For the total compu-

23

4. 4.1

SOLVING THE BASIC PROBLEM

5.

We here consider a typical optimal control problem named linear quadratic regulator (LQR) problem. We will first briefly explain the problem in §5.1 and show some computational examples, by which we can confirm the effectiveness of our proposed method. All computations except quantifier elimination are done by using a computer algebra system Risa/Asir1 . All QE computations in this paper were carried out by QEPCAD2 since QEPCAD succeeded in achieving all of QE computations for our examples in a very small amount of time. For the larger sized problems, we may use an efficient QE algorithm based on Sturm-Habicht sequence [1, 10]. Some types of QE methods using Sturm-Habicht sequence are available in a maple package SyNRAC [4, 19].

Outline of Algorithm

Problem 1 is resolved by using quantifier elimination over the real closed field. Actually all of the requirements are reduced to simple first-order formulas for R(z) ∈ Q(p)[z] as follows: (a) W < M1 : This requirement is equivalent to the first-order formula: ∀z > M1 , R(z) 6= 0.

(4)

This is so called a sign definite condition [1], hence we can solve it by an efficient quantifier elimination algorithm using Sturm-Habicht sequence [11, 6].

5.1

LQR problem

Here we briefly explain about linear quadratic regulator (LQR) problem (see [20] for more details) and introduce our target polynomial of which we want to estimate the SORPRP. Let us consider a linear time-invariant SISO (singleinput single-output) system represented by

(b) W > M2 : This requirement is equivalent to the first-order formula: ∃z > M2 , R(z) = 0.

CONTROL APPLICATION

(5)

We can also solve it by an efficient quantifier elimination algorithm using Sturm-Habicht sequence, see [10].

x(t) ˙ y(t)

(6)

Hence, this is achieved by superposing both quantifier-free formulas obtained by performing quantifier elimination for (a) and (b).

4.2

(7)

where x ∈ Rm is the state variable, u ∈ R is the control input, y ∈ R is the output, A ∈ Rm×m is the system matrix, b ∈ Rm is the input matrix, and cT ∈ Rm is the output matrix. Then the LQR problem is to find a control input u which minimizes the cost function Z ∞ (qy 2 (t) + ru2 (t))dt, (8) J=

(c) M2 < W < M1 : This requirement is equivalent to the conjunction of (a) and (b), that is, (∀z > M1 , R(z) 6= 0) ∧ (∃z > M2 , R(z) = 0).

= Ax(t) + bu(t), = cx(t),

0

where q > 0 and r > 0 are called weights. If we take the larger value of q, we can get the faster response in general. On the other hands, the lager value of r is required when we have a severe restriction on the value of u, since r reflects the penalty on u(t). Note that the ratio q/r plays an essential role for finding the optimal control input and determines the closed-loop poles. Actually, it is well-known that the optimal closed-loop poles are determined by the corresponding polynomial given by

Strategy against large size problems

In many examples, the polynomial R(z) is frequently factorized into small factors over Q(p) like R(z) = R1 · · · Rs due to a certain symmetry on αi1 + · · · + αim (see Remark 1 for example). Then the formula (4) is equivalent to ^ (∀z > M1 , Ri (z) 6= 0).

ϕ(s) = r · d(s)d(−s) + q · n(s)n(−s),

i

(9)

where d(s) and n(s) are the denominator and numerator of the transfer function of the plant (7) represented by

This means that the result obtained performing quantifier elimination to (4) is equivalent to conjunction of all results obtained by applying quantifier elimination to each factor Ri respectively. Moreover, the formula (5) is equivalent to _ (∃z > M2 , Ri (z) = 0).

P (s) = c(sI − A)−1 b. In other words, P (s) =

i

This implies that the result obtained performing quantifier elimination to (5) is equivalent to disjunction of all results obtained by applying quantifier elimination to each factor Ri respectively. Dividing the original large problem into several smaller problems via factorization shall reduce the total cost of quantifier elimination much. Of course, we have to consider the cost for factorization. We can expect the performance of this approach is superior to applying quantifier elimination to original formula (4) and (5).

n(s) , d(n)

where d(s) := det(sI − A), n(s) := c adj(sI − A) b. Note that deg(d(s)) = m, deg(n(s)) < m hold. 1 2

24

See http://www.math.kobe-u.ac.jp/Asir/asir.html See http://www.cs.usna.edu/ qepcad/B/QEPCAD.html

The polynomial ϕ(s) is our target polynomial with deg(ϕ(s)) = 2m and it is an even polynomial. It is strongly desired to establish a guiding principle to choose appropriate values of r and q or the ratio q/r, since the closed-loop poles are all the poles of ϕ(s) which have negative real parts. In the sequel we carry out an investigation of the weights r and q in terms of average stability, that is, the sum of roots with negative real parts (SORNRP) of ϕ(s). We can attain this by just applying our method for SORPRP shown in the previous sections to R(−z), where the polynomial R(z) has SORPRP of ϕ(s) as its root. Because, as ψ(s) is even, the value of SORPRP coincides with the absolute value of SORPNP, and R(−z) also has −1×SORPNP as its maximal real root. Particularly we study some behaviors of a parameter involving in the plant P (s) and feasible bounds for SORPRP W versus the ratio of weights q/r or q with r = 1 under the specifications in §4. This kind of investigations is important in practice to see control performance limitations, since the average stability is one of appropriate measures for the quickness of feedback control systems.

R(z; q, L) = −9765625L10 z 14

5.2

Remark 1. R(z; q, L) is factorized as follows instantly:

A sample plant: 2nd-order system with time delay

R(z; q, L) = R1 R2 R3 R4 R5

Here we study the LQR problem for a class of typical second-order systems with time delay represented by P (s) =

where

1 − 21 Ls kωn2 ωn2 ke−Ls ' , s2 + 2ζωn s + ωn2 s2 + 2ζωn s + ωn2 1 + 12 Ls

R1 R2 R3

where the exponential e−Ls is transformed to a rational function by the Pad´e approximation. We consider the case where k = 1, ζ = 0.1, ωn = 30 (kHz), and r = 1. Here, initially we assume that L > 0, r, q > 0. Then the target even polynomial is expressed as ϕ(s; q, L) = =

+ (−95703125L10 + 195312500L8 )z 12 +((68359375q−306796875)L10 + 995312500L8 −1562500000L6 )z 10 +((382812500q− 352493750)L10 + (1757812500q + 3258437500)L8 − 3062500000L6 + 6250000000L4 )z 8 + ((−78125000q 2 + 519031250q − 123443875)L10 + 8 (3675000000q +4263245000)L +(−7812500000q − 9013000000)L6 + 2450000000L4 − 12500000000L2 )z 6 + ((294122500q + 11647251)L10 + (2187500000q 2 + 3474625000q + 8 1863605100)L + (−17150000000q − 14797020000)L6 + (1250000000q − 3552000000)L4 − 4900000000L2 + 10000000000)z 4 +((−156250000q 3 −168625000q 2 − 10 2 12620025q − 245025)L + (1225000000q + 97020000q + 1920996)L8 + (−6250000000q 2 − 23304500000q − 7830818400)L6 + (−19600000000q + 8635760000)L4 + 2 (−12500000000q + 25916000000)L + 19600000000)z 2 + (625000000q 3 + 674500000q 2 + 50480100q + 980100)L8 + (−4900000000q 2 − 5094040000q − 194040000)L6 + (−5000000000q 2 + 4406000000q + 9406000000)L4 + (19600000000q + 19600000000)L2 + 10000000000q + 10000000000.

R4 R5

5.3

d(s)d(−s) + q · n(s)n(−s) −25L2 s6 + (−49L2 + 100)s4 . +((−25q − 25)L2 + 196)s2 + 100q + 100

= Lz + 2, = Lz − 2, = 625L4 z 4 − 5000L3 z 3 + (2450L4 + 15000L2 )z 2 + (−9800L3 −20000L)z + (−2500rqk2 − 99)L4 + 9800L2 + 10000, = 625L4 z 4 + 5000L3 z 3 + (2450L4 + 15000L2 )z 2 + (9800L3 +20000L)z + (−2500rqk2 − 99)L4 + 9800L2 + 10000, = −25z 4 − 49z 2 − 25rqk2 − 25.

Relationship between L and q

Here we consider the case where the bounds for the SORPRP are given, that is, M1 and M2 are fixed. Then we check the behavior of the plant parameter L versus a change of q. The possible regions of (L, q) to meet the specifications in the L − q parameter space is obtained by applying quantifier elimination to R(z; q, L) as explained in §4.

We remark that the leading coefficient −25L2 of ϕ(s) never vanish as L > 0, and the constant term 100q +100 also never vanish as q > 0. Let I3 be the ideal generated by the 3rd standard triangular set of ϕ(s; q, L);

5.3.1

(a) W < M1

Let M1 = 500, then the specification (a) is equivalent to the following first-order formula:

{ϕ(x1 ; q, L), ϕ1 (x1 , x2 ; q, L), ϕ2 (x1 , x2 , x3 ; q, L)},

∀z > 500, R(z; q, L) 6= 0.

where ϕ1 (x1 , x2 ; q, L) is the quotient of ϕ(x2 , q, L) divided by x2 − x1 and ϕ2 (x1 , x2 ; x3 ; q, L) is the quotient of ϕ1 (x3 , q, L) divided by (x3 − x1 )(x3 − x2 ). Then we obtained the following minimal polynomial R(z; q, L) in z of x1 + x2 + x3 with respect to I3 immediately. The maximal real root of R(z) coincides with the SORPRP W of ϕ(s; q, L). Since we need to compute sum of roots with negative real parts in a sense of stability, we apply our method computing SORPRP to R(−z; q, L). But as ϕ is even it follows that R(−z; q, L) = R(z; q, L).

(10)

After performing quantifier elimination to this, we can obtain the following equivalent quantifier-free formula in (L, q) which describe feasible regions of (L, q) for (a): (q + 62500490001 >= 0 ∧ 250L − 1 ≥ 0 ∧ 2500L4 q − 39063112499901L4 + 625004900000L3 − 3750009800L2 + 10000000L − 10000 ≤ 0 ∧ 2500L4 q − 39063112499901L4 − 625004900000L3 − 3750009800L2 − 10000000L − 10000 ≤ 0

This is illustrated as a shaded region in Fig.1.

(b) W > M2 Let M2 = 300, then the specification (b) is equivalent to the following first-order formula: 5.3.2

∃z > 300, R(z; q, L) = 0.

25

(11)

Figure 1: Feasible region of L − q for (a)

Figure 3: Feasible region of L − q for (a) ∧ (b) shown in Figs. 1, 2 and 3 meets the above requirements in terms of the magnitude of SORPRP. We can obtain the following knowledge from Fig. 3. The plant parameter L is restricted within an interval for a fixed value of q under the specification of 300 < W < 500. The maximum and minimum edges of the feasible interval of L are monotonically increasing. Thus, for instance for the value of L around 0.01, q must be taken from the region which is larger than a certain value. We can obtain the exact threshold value easily since we have the feasible region as a semi-algebraic set by virtue of quantifier elimination. These greatly help control designers to choose appropriate value of the ratio of weights q/r for their control system more systematically.

5.4

Relationship between M1 /M2 and q Next we investigate the case where the controller is given, that is, L is fixed. We set L = 0.02(msec). Then we estimate the behavior of the possible bounds for M1 and M2 versus a change of q. The possible regions of M1 and M2 according to a change of q is obtained by performing quantifier elimination to R(z; 1/50) as explained in §4.

Figure 2: Feasible region of L − q for (b) After performing quantifier elimination to this, we can obtain a following equivalent quantifier-free formula in (L, q) which describe feasible regions of (L, q) for (b):

5.4.1

The specification (a) is equivalent to the following firstorder formula:

(q + 8100176401 < 0) ∨ (L > 0∧150L − 1 < 0) ∨ (2500L4 q − 5062720499901L4 − 135002940000L3 − 1350009800L2 − 6000000L − 10000 > 0) ∨ (2500L4 q − 5062720499901L4 + 135002940000L3 − 1350009800L2 + 6000000L − 10000 > 0)

∀z > M1 , R(z; q, 1/50) 6= 0.

(12)

After performing quantifier elimination to this, we can obtain the following equivalent quantifier-free formula in (M1 , q) which describe feasible regions of (M1 , q) for (a):

This is illustrated as a shaded region in Fig.2.

5.3.3

(a) W < M1

(c) M2 < W < M1

If M2 = 300, M1 = 500 for the requirement (c), the problem is recast as the following first-order formula:

M1 − 100 ≥ 0 ∧ 25q + 25M14 + 49M12 + 25 ≥ 0 ∧ 2500q − 625M14 + 250000M13 − 37502450M12 2500490000M1 − 62524499901 ≤ 0

(∀z > 500, R(z; q, L) 6= 0) ∧ (∃z > 300, R(z; q, L) = 0). A formula describing feasible regions of (L, q) for the requirement (c) can be obtained by superposing above two results for (a) and (b) in the parameter space L − q as shown in Fig.3.

+

This is illustrated in Fig.3 as a shaded region.

5.4.2 (b) W > M2 The specification (b) is equivalent to the following firstorder formula:

Control theoretical significance : Any system with parameter values of L and q within the feasible regions

∃z > M2 , R(z; q, 1/50) = 0.

26

(13)

cases certainly, and choose appropriate value of the weight q for their requirement level (i.e., M1 , M2 ) systematically.

6.

Figure 4: Feasible region of M1 − q

7.

REFERENCES

[1] H. Anai and S. Hara. Fixed-structure robust controller synthesis based on sign definite condition by a special quantifier elimination. In Proceedings of American Control Conference 2000, pp.1312–1316, 2000. [2] H. Anai and S. Hara. Linear programming approach to robust controller design by a quantifier elimination. In Proceedings of SICE Annual Conference 2002 (Osaka, Japan), pp.863–869, 2002. [3] H. Anai, H. Yanami, K. Sakabe, and S. Hara. Fixed-structure robust controller synthesis based on symbolic-numeric computation: design algorithms with a CACSD toolbox (invited paper). In Proceedings of CCA/ISIC/CACSD 2004 (Taipei, Taiwan), pp.1540–1545, 2004. [4] H. Anai and H. Yanami. SyNRAC: A maple-package for solving real algebraic constraints. In Proceedings of International Workshop on Computer Algebra Systems and their Applications (CASA) 2003 (Saint Petersburg, Russian Federation), P.M.A. Sloot et al. (Eds.): ICCS 2003, LNCS 2657, pp.828–837. Springer, 2003. [5] P. Aubry and A. Valibouze. Using galois ideals for computing relative resolvents. Journal of Symbolic Computation, 30: pp.635–651, 2000. [6] B. Caviness and J. Johnson, editors. Quantifier Elimination and Cylindrical Algebraic Decomposition. Texts and Monographs in Symbolic Computation. Springer, Berlin, Heidelberg, New York, 1998. [7] D. Cox, J. Little, and D. O’Shea. Ideals, Varieties and Algorithms. Undergraduate Texts in Mathematics. Springer-Verlag, New York, Berlin, Heidelberg, 1992. [8] D. Cox, J. Little, and D. O’Shea. Using Algebraic Geometry. Graduate Texts in Mathematics 185. Springer-Verlag, New York, Berlin, Heidelberg, 1998. [9] P. Dorato, W. Yang, and C. Abdallah. Robust multi-objective feedback design by quantifier elimination. J. Symb. Comp. 24, pp.153–159, 1997. [10] L. Gonz´ alez-Vega. A combinatorial algorithm solving some quantifier elimination problems. In B.F.

Figure 5: Feasible region of M2 − q After performing quantifier elimination to this, we can obtain the following equivalent quantifier-free formula in (M2 , q) which describe feasible regions of (M2 , q) for (b): M2 − 100 < 0 ∨ 25M24 + 49M22 + 25 < 0 ∨ 2500q − 625M24 + 250000M23 − 37502450M22 2500490000M2 − 62524499901 > 0

CONCLUSION

In this paper we have presented a method to compute or estimate the sum of roots with positive real parts (SORPRP) of a polynomial with parametric coefficients based on symbolic and algebraic computations. Since the method does not compute explicit numerical values of the roots, we can treat polynomials with parametric coefficients for their SORPRP. Combining the method with quantifier elimination, we succeeded in giving a novel systematic method for achieving optimal regulator design in control. In order to see its effectiveness and practicality, we made some experiments for a concrete example from optimal regulator control. The method proposed here shall provide one of promising direction for an ad hoc part (i.e., choice of weights) of optimal regulator design that is one of the main concerns in control and gives another successful application of computer algebra to control design problem.

+

This is illustrated in Fig.4 as a shaded region, which is the exact complementary set of the shaded region in Fig.3. Control theoretical significance : Fig.4 illustrates the behavior of lower bound of the possible magnitude of M1 for q so that the given plant P (s) (where L is fixed) satisfies the requirement (a). Fig.5 shows the behavior of upper bound of the possible magnitude of M2 for q so that the given plant P (s) satisfies the requirement (b). We can see from Fig.4 that the lower bound of M1 monotonically increasing. Thus, if we need to satisfy (a) for the smaller M1 , we must choose enough smaller q. We can easily obtain the feasible range of q for a given value of M1 by using the semi-algebraic set obtained by quantifier elimination which describes the feasible region in Fig.4. These information becomes significant supports for control designers to avoid infeasible requirement

27

[11]

[12] [13]

[14]

Caviness and J.R. Johnson, editors, Quantifier Elimination and Cylindrical Algebraic Decomposition, Texts and Monographs in Symbolic Computation, pp.365–375. Springer, Wien, New York, 1998. L. Gonz´ alez-Vega, T. Recio, H. Lombardi, and M.-F. Roy. Sturm-habicht sequences determinants and real roots of univariate polynomials. In B.F. Caviness and J.R. Johnson, editors, Quantifier Elimination and Cylindrical Algebraic Decomposition, Texts and Monographs in Symbolic Computation, pp.300–316. Springer, Wien, New York, 1998. H. Weber. Lehrbuch der Algebra, Volume I: Third Edition AMS, 2002. S. Hara, T. Kimura, and R. Kondo. H ∞ control system design by a parameter space approach. In Proceedings of MTNS-91, pages 287–292, Kobe, Japan, 1991. M. Jirstrand. Nonlinear control system design by quantifier elimination. Journal of Symbolic Computation, 24(2): pp.137–152, August 1997. Applications of quantifier elimination (Albuquerque, NM, 1995).

[15] T. Kimura and S. Hara. A robust control system design by a parameter space approach based on sign definition condition. In Proceedings of KACC-91, pp.1533–1538, Soul, Korea, 1991. [16] B. Mishra. Algorithmic Algebra Springer Verlag, 1993. [17] M. Noro and K. Yokoyama. Implementation of prime decomposition of polynomial ideals over small finite fields Journal of Symbolic Computation, 38 (2004) pp.1227-1246. [18] T. Takagi. Lecture of Algebra (‘Daisugaku Kougi’ in Japanese ). Kyoritsu Pub. Japan, 1930. [19] H. Yanami and H. Anai. Development of SyNRAC – formula description and new functions. In Proceedings of International Workshop on Computer Algebra Systems and their Applications (CASA) 2004 : ICCS 2004, LNCS 3039, pp.286–294. Springer, 2004. [20] K Zhou, J.C Doyle, and K. Glover. Robust and Optimal Control. Prentice Hall, 1995.

28

Algebraic General Solutions of Algebraic Ordinary Differential Equations J.M. Aroca and J. Cano

R. Feng and X.S. Gao

Department of Algebra, Geometry and Topology Fac. Ciencias. Univ. de Valladolid Valladolid 47011, Spain

Key Laboratory of Mathematics Mechanization Institute of Systems Science, AMSS, Academia Sinica, Beijing 100080, China

(aroca,jcano)@agt.uva.es

[email protected] [email protected]

ABSTRACT

homogeneous ODEs. Many other interesting results on finding the Liouvillian solutions of linear ODEs were reported in [2, 6, 23, 24]. Most of these results are limited to the linear case or some special type nonlinear equations. Work on finding closed form solutions for nonlinear differential equations is not as systematic as that for linear equations. With respect to the particular ODEs of the form y 0 = R(x, y) where R(x, y) is a rational function, Darboux and Poincar´e made important contributions [16]. More recently, Cerveau, Carnicer and Corral et al also made important progresses [4, 3, 7]. In particular, Carnicer gave the degree bound of algebraic solutions in the nondicritical case. In [21], Singer studied the Liouvillian first integrals of differential equations. In [12], Hubert gave a method to compute a basis of the general solutions of first order ODEs and applied it to study the local behavior of the solutions. In [9, 10], Feng and Gao gave a necessary and sufficient condition for an algebraic ODE to have a rational type general solution and a polynomial-time algorithm to compute a rational general solution if it exists. In this paper, the idea proposed in [9] is generalized to compute algebraic function solutions. In Section 2, we give a sufficient and necessary condition for an algebraic ODE to have an algebraic general solution, by constructing a class of differential equations whose solutions are all algebraic functions. In Section 3, by treating the variable and its derivative as independent variables, a first order autonomous ODE defines a plane algebraic curve. Using the Riemann-Hurwitz formula, we give a degree bound of algebraic function solutions of the equation. This degree bound is optimal in the sense that there is a class of first order autonomous ODEs, whose algebraic function solutions reach this bound. In Section 4, based on the above results and the theory of Hermite-Pad´e approximants, we give a polynomial-time algorithm to find an algebraic general solution for a first order autonomous ODE. dy A first order autonomous ODE F (y, dx ) = 0 can be redx duced to the form G(y, dy ) = 0, where G is also a polynomial(see the section 3.2.1, (7)). Then to find the solution of F = 0, we may first find x = φ(y) as a function in y by computing the integral of an algebraic function, and then compute the inversion y = φ−1 (x). For an algebraic R function φ(x) which satisfies G(x, φ(x)) = 0, let y = φ(x)dx dy be the integral of φ(x). Then we have G(x, dx ) = 0. By the dy same way, G(x, dx ) = 0 can be converted into a first order autonomous ODE F (x, dx ) = 0. Then to find the integral y dy

In this paper, we give a necessary and sufficient condition for an algebraic ODE to have an algebraic general solution. For a first order autonomous ODE, we give an optimal bound for the degree of its algebraic general solutions and a polynomial-time algorithm to compute an algebraic general solution if it exists. Here an algebraic ODE means that an ODE given by a differential polynomial.

Categories and Subject Descriptors I.1.2 [SYMBOLIC AND ALGEBRAIC MANIPULATION]: Algorithms—Algebraic algorithms

General Terms Algorithms, Theory

Keywords Algebraic general solution, algebraic differential equation, first order autonomous ODE, algebraic curve, Hermite-Pad´e approximants

1.

INTRODUCTION

Finding the close form solution of an ODE can be traced back to the work of Liouville. For the algorithm consideration, the pioneer work is due to Risch. In [17, 18], RRisch described a method to find the elementary integral of udx where u is an elementary function. In Trager’s Ph.D thesis [22], he gave a method to compute the integral of algebraic functions based on Risch’s ideas. In [1], Bronstein generalized Trager’s results to elementary functions. For higher order linear homogeneous ODEs, Kovacic presented an effective method to find the Liouvillian solutions for second order ODEs [14]. In [20], Singer established a general framework for finding the Liouvillian solutions for general linear

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. ISSAC’05, July 24–27, 2005, Beijing, China. Copyright 2005 ACM 1-59593-095-7/05/0007 ...$5.00.

29

of φ(x), we may first find x = ϕ(y) by computing a solution of F (x, dx ) = 0 and then compute the inversion. Hence, our dy algorithm is equivalent to a polynomial-time algorithm for finding an algebraic integral for an algebraic function.

2. 2.1

P , P (η) = 0 implies that P ∈ Σ. It is well known that an ideal Σ is prime iff it has a generic zero [19]. As a consequence of Lemma 2.1, we have Lemma 2.2. Let F ∈ K{y} \ K be an irreducible differential polynomial with a generic solution η. Then for a differential polynomial P we have P (η) = 0 iff prem(P, F ) = 0.

ALGEBRAIC GENERAL SOLUTIONS OF ALGEBRAIC ODES

The following definition of the general solution is due to Ritt.

Definition of algebraic general solutions

Definition 2.3. Let F ∈ K{y} \ K be an irreducible differential polynomial. A general solution of F = 0 is defined as a generic zero of ΣF . An algebraic general solution of F = 0 is defined as a general solution yˆ which satisfies the following equation

In the following, let K = Q(x) be the differential field of d rational functions in x with differential operator dx and y an ¯ be the algebraic closure of the indeterminate over K. Let Q rational number field Q. We denote by yi the i-th derivative of y. We use K{y} to denote the ring of differential polynomials over the differential field K, which consists of the polynomials in the yi with coefficients in K. All differential polynomials in this paper are in K{y}. Let Σ be a system of differential polynomials in K{y}. A zero of Σ is an element in a universal extension field of K, which vanishes every differential polynomial in Σ [19]. In this paper, we also assume that the universal extension field of K contains an infinite number of arbitrary constants. We will use C to denote the constant field of the universal extension field of K. Let P ∈ K{y} \ K. We denote by ord(P ) the highest derivative of y in P , called the order of P . Let o = ord(P ) > 0. We may write P as follows

G(x, y) =

(3)

P Pmj i j where ai,j are in C and n i=0 ai,j x y is irreducible in j=0 C[x, y]. When n = 1, yˆ is called a rational general solution of F = 0. For algebraic solutions of a differential equation F = 0, we have the following lemma. ¯ Lemma 2.4. Let G(y) ∈ C(x)[y] and irreducible in C(x)[y] where C¯ is the algebraic closure of C. If one solution of G(y) = 0 is a solution of F = 0, then every solution of G(y) = 0 is the solution of F = 0. ¯ Proof. Since G(y) is irreducible in C(x)[y], every solution of G(y) = 0 is a generic zero of G(y) = 0. By Lemma 2.2, prem(F, G) = 0. That is,

where ai are polynomials in y, y1 , . . . , yo−1 and ad 6= 0. ad ∂P is called the separant is called the initial of P and S = ∂y o

of P . The k-th derivative of P is denoted by P (k) . Let S be the separant of P , o = ord(P ) and an integer k > 0. Then we have

S k I l F = P G0 + QG

(4)

∂G , ∂y

I is the initial of G and k, l ∈ Z. Since where S = every solution of G(y) = 0 is a generic zero, S or I do not vanish at it. Hence every solution of G(y) = 0 is a solution of F = 0.

(1)

where Rk is of lower order than o + k. Let P be a differential polynomial of order o. A differential polynomial Q is said to be reduced with respect to P if ord(Q) < o or ord(Q) = o and deg(Q, yo ) < deg(P, yo ). For two differential polynomials P and Q, let R = prem(P, Q) be the differential pseudo-remainder of P with respect to Q. We have the following differential remainder formula for R [13, 19] X JP = Bi Q(i) + R

A general solution of F = 0 is usually defined as a family of solutions with o independent parameters in a loose sense where o = ord(F ). The definition given by Ritt is more precise. Theorem 6 in Section 12, Chapter 2 in [13] tells us that Ritt’s definition of general solutions is equivalent to the definition in the classical literature.

2.2 A criterion for existence of algebraic general solutions

i

where J is a product of certain powers of the initial and separant of Q and Bi , R are differential polynomials. Moreover, R is reduced with respect to Q. For a differential polynomial P of order o, we say that P is irreducible if P is irreducible when P is treated as a polynomial in K[y, y1 , . . . , yo ]. Let P ∈ K{y}\K be an irreducible differential polynomial and ΣP = {A ∈ K{y}|SA ≡ 0 mod {P }},

ai,j xi y j = 0

j=0 i=0

P = ad yod + ad−1 yod−1 + . . . + a0

P (k) = Syo+k + Rk

mj n X X

For non-negative integers h, α, k, let A(h,α;k) (y) be the following (h + 1) × (α + 1) matrix:      

k+1 0 k+2 0

k+h+1 0

(2)

yk+1 yk+2 . . .

k+1 yk 1 k+2 yk+1 1

. . .

yk+h+1

k+h+1 1

yk+h

··· ···

k+1 α k+2 α

··· ···

k+h+1 α

yk+1−α yk+2−α . . . yk+h+1−α

   .  

Zn ≥0 ,

Let α = (α1 , · · · , αn ) ∈ α0 ∈ Z≥0 where Z≥0 means the set of non-negative integers. Let A(α0 ;α) (y) be the (h + 1) × (h + 1) matrix

where{P } is the perfect differential ideal generated by P [13, 19]. Ritt proved that [19]

(A(h,α1 ;α0 ) (y)|A(h,α2 ;α0 ) (y 2 )| · · · |A(h,αn ;α0 ) (y n ))

Lemma 2.1. ΣP is a prime differential ideal and a differential polynomial Q belongs to ΣP iff prem(Q, P ) = 0.

where n + α1 + · · · + αn = h + 1. Let D(α0 ;α) be the determinant of A(α0 ;α) (y). Note that if n = 1, D(α0 ,α) is just equal to Dn,m in [9].

Let Σ be a non-trivial prime ideal in K{y}. A zero η of Σ is called a generic zero of Σ if for any differential polynomial

30

¯ ¯ Q{y} and G(x, y) ∈ Q[x, y] which is irreducible. We say G(x, y) is nontrivial if deg(G, x) > 0 and deg(G, y) > 0. From now on, we always assume that G(x, y) is nontrivial. When we say that G(x, y) = 0 is an algebraic solution of F = 0, we mean that one of the algebraic functions yˆ(x) defined by G(x, yˆ(x)) = 0 is a solution of F = 0.

Lemma 2.5. An element y¯ in the universal extension of K is a solution of D(α0 ;α) = 0 iff it satisfies the equation (3) with mj ≤ αj for j = 0, · · · , n. Proof. Assume that y¯ satisfies the equation (3) with mj ≤ αj where j = 0, · · · , n. Then we have αj n X X

3.1 Structure for algebraic general solutions

ai,j (xi y¯j )(α0 +1) = 0

It is a trivial fact that for an autonomous ODE, the solution set is invariant by a translation of the independent variable x. Moreover, we have the following fact.

j=1 i=0 i j (α0 +1)

i j

where (x y¯ ) means the (α0 + 1)-th derivative of x y¯ with respect to x and if i > mj then ai,j = 0. Since ai,j are constants, (xi y¯j )(α0 +1) (i = 0, · · · , αj , j = 1, · · · , n) are linearly dependent over C. That is, the Wronskian determinant W ((xi y¯j )(α0 +1) ) for (xi y¯j )(α0 +1) vanishes where j = 0, · · · , n, i = 0, · · · , αj [19]. Then y¯ satisfies the equation (3) with mj ≤ αj iff W ((xi y¯j )(α0 +1) ) = 0. By the computation process,

Lemma 3.1. Let G(x, y) = 0 be an algebraic solution of F = 0. Then G(x + c, y) = 0 is an algebraic general solution of F = 0, where c is an arbitrary constant. Proof. Assume that y¯(x) is a formal power series solution of G(x, y) = 0. Then y¯(x + c) will be a solution of G(x + c, y) = 0. Because y¯(x) is a solution of F = 0, y¯(x + c) is still a solution of F = 0. Hence G(x + c, y) = 0 is an algebraic solution of F = 0. For any T ∈ K{y} satisfying T (¯ y (x + c)) = 0, let R = prem(T, F ). Then R(¯ y (x + c)) = 0. Suppose that R 6= 0. Since F is irreducible and deg(R, y1 ) < deg(F, y1 ), there are two differential polynomials P, Q ∈ K{y} such that P F + QR ∈ K[y] and P F + QR 6= 0. Thus (P F + QR)(¯ y (x + c)) = 0. Because ¯ and c is an arbitrary constant which is trany¯(x + c) ∈ / Q scendental over K, we have P F + QR = 0, a contradiction. Hence R = 0 which means that T ∈ ΣF . So y¯(x + c) is a generic zero of ΣF . Hence G(x + c, y) = 0 is an algebraic general solution.

W ((xi y¯j )(α0 +1) ) = D(α0 ;α) (¯ y ) ∗ |diag(B0 , · · · , Bn )| where diag(B0 , · · · , Bn ) is the diagonal matrix of Bj and   αj 1

Bj =

 0  .  . . 0

x 1 .. . 0

··· ··· ··· ···

x αj xαj −1   ..  . αj !

for j = 0, · · · , n. Hence W ((xi y¯j )(α0 +1) ) = 0 if and only if D(α0 ;α) (¯ y ) = 0. By the above Lemma, we can prove the following criteria theorem easily.

Lemma 3.1 reduces the problem of finding an algebraic general solution to the problem of finding a nontrivial algebraic solution. In what follows, we will show how to find ¯ y]. First of all, we a nontrivial algebraic solution in Q[x, decide the degree of an algebraic solution.

Theorem 2.6. Let F be an irreducible differential polynomial. Then F = 0 has an algebraic general solution yˆ iff there exist α = (α1 , · · · , αn ) ∈ Zn ≥0 , α0 ∈ Z≥0 such that prem(D(α0 ;α) , F ) = 0.

3.2 Degree bound of an algebraic solution

Proof. (⇒) Let yˆ be an algebraic general solution of F = 0 which satisfies the equation (3). Let α = (m1 , m2 , · · · , mn ) and α0 = m0 . Then from Lemmas 2.1, 2.2 and 2.5

Assume that G(x, y) = 0 is an algebraic solution of the differential equation F = 0. In this subsection, we will give a bound for deg(G, x) and deg(G, y). First, we introduce some concepts concerning the algebraic function fields in one variable. ¯ Definition 3.2. Q(x, α) is called an algebraic function ¯ and α is field in one variable, if x is transcendental over Q ¯ algebraic over Q(x) [11].

y ) = 0 ⇒ D(α0 ,α) ∈ ΣF ⇒ prem(D(α0 ,α) , F ) = 0. D(α0 ,α) (ˆ (⇐) prem(D(α0 ,α) , F ) = 0 implies that D(α0 ,α) ∈ ΣF by Lemma 2.1. Then all the zeros of ΣF must satisfy the equation (3). In particular, the generic zero of ΣF satisfies the equation (3).

An irreducible algebraic curve G(x, y) = 0 where G(x, y) ∈ ¯ ¯ Q[x, y] corresponds to an algebraic function field Q(α, β) which is unique under an isomorphism where α, β satisfies ¯ It is well G(α, β) = 0 and α or β is transcendental over Q. known that two algebraic curves with isomorphic function fields have the same genus.

Given an algebraic differential equation F = 0, if we know the degree bound of the equation (3) with respect to x and y which perhaps defines an algebraic general solution of F = 0, then we can decide whether it has an algebraic general solution by computing prem(D(α0 ,α) , F ) step by step. However for ODEs of order greater than one or with variate coefficients, we do not know this bound. Even for the case P (x,y) where P (x, y), Q(x, y) ∈ Q[x, y], we have no efy 0 = Q(x,y) fective method to get the bound [3, 16]. In the following, for the first order autonomous ODEs, we give a degree bound for algebraic function solutions.

3.

3.2.1 Parametrization of a curve ¯ Let Q((t)) be the quotient field of the ring of formal power ¯ series Q[[t]]. Let G(x, y) be a nontrivial irreducible polyno¯ ¯ mial in Q[x, y]. If x(t), y(t) ∈ Q((t)) satisfy G(x(t), y(t)) = 0, we say that they are the coordinates of a parametrization ¯ There exist provided x(t) or y(t) does not belong to Q. ¯ nonzero integers q and p, and units u(t), v(t) in x0 , y0 ∈ Q, ¯ Q[[t]], such that  x(t) − x0 = tq u(t), (5) y(t) − y0 = tp v(t).

DEGREE BOUND FOR FIRST ORDER AUTONOMOUS ODES

In the following, we will always assume that F = 0 is a first order autonomous ODE in Q{y} and irreducible in

31

¯ where Ai (y), Fj (y) ∈ Q[y]. We use Res(A, B, z) to denote the Sylvester-resultant of A and B with respect to z and Z stands for “the zero set of”. Let S = Z(As (y)) ∪ Z(Fd (y)) ∪ ∂F Z(Res(G, ∂G , x)) ∪ Z(Res(G, ∂G , x)) ∪ Z(Res(F, ∂y , y1 )). ∂x ∂y 1 ¯ such Then S is a finite set. Hence we can choose a c ∈ Q that c ∈ / S. Then we have the following results:

The center of the parametrization is the point P ∈ P1 × P1 defined accordingly the following cases: (a) If q > 0 and p > 0, then P = (x0 , y0 ); (b) If q > 0 and p < 0, then P = (x0 , ∞); (c) If q < 0 and p > 0, then P = (∞, y0 ); (d) If q < 0 and p < 0, then P = (∞, ∞). If p < 0 (resp. q < 0) we agree to take y0 = 0 (resp. x0 = 0). If there exists an integer k ≥ 2 such that x(t), y(t) ∈ ¯ k )), the parametrization will be called reducible, otherQ((t ¯ wise irreducible. If t¯ ∈ Q[[t]] with order with respect to t greater than zero, then x(t¯), y(t¯) is another parametrization with the same center. If the order of t¯ is equal to one, the two parametrizations will be said to be equivalent. An equivalence class of irreducible parametrizations will be called a place B of the curve G = 0 with center the center of one of its parametrizations. Two equivalent parametrizations have the same integers q and p as defined above. Then given a place B, we define nonzero integers νx (B) and νy (B) as the integers q and p of any of its irreducible parametrizations. Let g be the genus of G(x, y) = 0 and n = deg(G, y). By the Riemann-Hurwitz formula [15] we have that 1X g =1−n+ (|νx (B)| − 1) 2 B

¯ (c, z) = 0} = {z1 , z2 , · · · , zd } has ex(a) the set {z ∈ Q|F actly d elements; ¯ (b) the set {x ∈ Q|G(x, c) = 0} = {x1 , x2 , · · · , xs } has exactly s elements; (c) since series

such that G(x, ϕi (x)) = 0 for each i = 1, · · · , s. From Lemma 2.4, ϕi (x) is a solution of F = 0. Then we have F (ϕi (x), ϕ0i (x)) = 0 which implies that F (c, gi,1 ) = 0. Suppose that s > d. Then there exist at least two of gi,1 which are equal to each other. Without lost of generalization, as∂F (c, c1 ) 6= 0, there exists sume that g1,1 = g2,1 = c1 . Since ∂y 1 only one solution ϕ(x) of F (y, y1 ) = 0 such that ϕ(0) = c and ϕ0 (0) = c1 . Hence ϕ1 (x) = ϕ2 (x + x2 − x1 ) = ϕ(x − x1 ). So (x, ϕ1 (x)) and (x + x2 − x1 , ϕ1 (x)) are two coordinates of a parametrizations of G = 0. This is a contradiction by the above lemma. Hence s ≤ d. Let G0 = y1 ∂G + ∂G and ∂y ∂x 0 H(y, y1 ) = Res(G, G , x). Then H(y, y1 ) = y1s Res(G,

B

where the sum runs over all places B of the curve G = 0 with center (α, β).

Since F is first order and autonomous, we can regard F = 0 as an algebraic curve and we will use F (y, y1 ) to denote F.

Proof. By Gauss’s lemma, we know G(x, y) is irreducible ¯ ¯ Q(y(t)) ¯ ¯ in Q(y)[x]. Since y(t) ∈ / Q, is isomorphic to Q(y) ¯ which implies that G(x, y(t)) ∈ Q(y(t))[x] is irreducible too. Now assume that x(t) is a root of G(x + c, y(t)) = 0. Then we have G(x, y(t)) divides G(x + c, y(t)). It is clear that deg(G(x+c, y(t)), x) = deg(G(x, y(t)), x) and G(x, y(t)) and G(x + c, y(t)) have the same leader coefficients. Hence, G(x, y(t) = G(x + c, y(t)). Since c 6= 0, we have that deg(G(x, y), x) = deg(G(x, y(t)), x) = 0, in contradiction with the nontriviality of G(x, y).

Lemma 3.5. Assume that G(x, y) = 0 is an algebraic solution of F = 0. Then the genus of G(x, y) = 0 equals to that of F (y, y1 ) = 0. Proof. Let α satisfy G(x, α) = 0. It is clear that α is ¯ Then Q(x, ¯ ¯ transcendental over Q. α) and Q(α, α0 ) are the algebraic function fields of G(x, y) = 0 and F (y, y1 ) = 0 ¯ ¯ respectively. We only need to prove Q(x, α) = Q(α, α0 ). ¯ ¯ ¯ From Theorem 3.4, we have [Q(x, α) : Q(α)] = [Q(α, α0 ) : ¯ Q(α)]. Since G(x, α) = 0, α0 = − ∂G (x, α)/ ∂G (x, α). which ∂x ∂y 0 ¯ ¯ ¯ implies that α ∈ Q(x, α). Hence Q(x, α) = Q(α, α0 ).

Now we are ready to give the degree bound of the algebraic solution of F = 0. First, we could determine the degree deg(G, x) exactly from the degree of F .

For convenience, we consider a new differential equation

¯ Theorem 3.4. Let G(x, y) ∈ Q[x, y] be irreducible and let G(x, y) = 0 be an algebraic solution of F = 0. Then we have

1 deg(F,y1 ) F¯ (x1 , y) = x1 F (y, ) = 0 (7) x1 ¯ 1 , y] and where x1 = dx = y11 . F¯ is irreducible in Q[x dy deg(F¯ , y) = deg(F, y), deg(F¯ , x1 ) = deg(F, y1 ). Then we have the following lemma.

deg(G, x) = deg(F, y1 ). Proof. Assume that deg(G, x) = s and deg(F, y1 ) = d. Let us write F

F0 (y) + F1 (y)y1 + · · · + Fd (y)y1d

=

∂G , x) + terms of lower order in y1 . ∂y

Since Res(G, ∂G , x) 6= 0, we have deg(H, y1 ) = s. As∂y sume that y¯(x) is a solution of G(x, y) = 0. Then we have H(¯ y (x), y¯0 (x)) = F (¯ y (x), y¯0 (x)) = 0. Because F is irreducible, we have that deg(H, y1 ) ≥ deg(F, y1 ). In the other words, s ≥ d.

Lemma 3.3. Let G(x, y) be a nontrivial irreducible poly¯ y]. Let (x(t), y(t)) be the coordinates of a nomial of Q[x, parametrization G = 0. Then, for any nonzero constant c ∈ ¯ (x(t) + c, y(t)) are not the coordinates of a parametrizaQ, tion of G = 0.

A0 (y) + A1 (y)x + · · · + As (y)xs ,

6= 0, there exists a unique formal power

ϕi (x) = c + gi,1 (x − xi ) + gi,2 (x − xi )2 + · · ·

where B runs over all places of the curve G = 0. Each place B with center (α, β) corresponds to exactly qB fractional power series y(x1/qB ) which are the solutions of ¯ ∪ {∞}. Hence, by the Puiseux G(x, y(x)) = 0. Let α ∈ Q theorem we have that X |νx (B)| = deg(G, y), (6)

G(x, y) =

∂G (xi , c) ∂y

Lemma 3.6. Let F¯ be defined as in (7) and G(x, y) = 0 an algebraic solution of F = 0. Then G(x, y) = 0 also defines an algebraic function (in y) solution of F¯ (x1 , y) = 0.

32

and νx (B) > νy (B); and we say that B ∈ (4)0 if B ∈ (4) and νx (B) < νy (B). In the following inequalities Bx , By ˜x1 and B ˜y will stand for νx (B), νy (B), νx1 (B) ˜ and νy (B) ˜ B respectively. For k = 1 and k = 4, we have that X X X ˜x1 | + (10) (|Bx | − 1) ≤ |B (|By | − 1).

Proof. From the proof of Theorem 3.4, we know that Res(G, G0 , x) = A(y)F (y, y1 ) where G0 = y1 ∂G + ∂G . In the other words, there exist ∂y ∂x ¯ two polynomials P, Q ∈ Q[x, y, y1 ] such that P G + QG0 = A(y)F (y, y1 ). Replacing y1 by x11 and multiplying some power of x1 , we have ¯ ∂G + x1 ∂G ) = xk1 A(y)F¯ (x1 , y) P¯ G + Q( ∂y ∂x

B∈(k)0

B∈(k)

(8)

B∈(k)

For k = 2 and k = 3, we have that X X ˜x1 |. (|Bx | − 1) ≤ |B

¯ ¯ ∈ Q[x, where P¯ , Q y, x1 ] and k ∈ Z≥0 . Suppose that β satisfies G(β, y) = 0. Replacing x by β and x1 by β 0 in (8) where β 0 = dβ , we have that F¯ (β 0 , y) = 0. Hence G(x, y) = dy 0 is an algebraic solution of F¯ = 0.

B∈(k)

(11)

B∈(k)

˜ is over x1 = 0. If If B ∈ (1)0 ∪ (2), then the center of B ˜ is over x1 = ∞. Hence, B ∈ (3) ∪ (4)0 , then the center of B using the formula (6), we have that X ˜x1 | ≤ 2 deg(F¯ , y). |B (12)

Lemma 3.7. Let (x(t), y(t)) be an irreducible parametriza0 (t) , y(t)) is an irreducible parametrization of G = 0. Then ( xy0 (t) tion of F¯ (x1 , y) = 0.

B∈(1)0 ,(2),(3),(4)0

0

(t) where 0 means the Proof. Let us denote x1 (t) = xy0 (t) derivative with respect to t. Since x1 (t) = dx (t), we have dy F¯ (x1 (t), y(t)) = 0. Assume that (x1 (t), y(t)) is a reducible ¯ k )). that x1 (t), y(t) ∈ Q((t parametrization. Let P k ≥ 2, such 0 0 kj−1 Then x1 (t)y (t) = j≥j0 cj t . Since x (t) = x1 (t)y 0 (t), P c tkj then we have that c0 = 0 and x(t) = c + j≥j0 jkj , for some constant c. Hence we get a contradiction because ¯ k )). x(t), y(t) ∈ Q((t

By the Riemann-Hurwitz formula, we have that X ˜y | − 1) ≤ 2(gF¯ + deg(F¯ , x1 ) − 1). (|B

(13)

B∈(1),(4)

We remark that in the inequalities (12,13) we have used the ˜ between the places of G = 0 and fact that the map B 7→ B places of F¯ = 0 is injective. By the inequalities ((9)-(13)), we have that 2(gG + deg(G, y) − 1) ≤ 2(gF¯ + deg(F¯ , x1 ) + deg(F¯ , y) − 1).

Theorem 3.8. Assume that G(x, y) = 0 is a nontrivial algebraic solution of F = 0. Then we have that

Using the above inequalities, and the facts that deg(F¯ , x1 ) = deg(F, y1 ), deg(F¯ , y) = deg(F, y) and that gG = gF¯ , gives the required inequality.

deg(G, y) ≤ deg(F, y) + deg(F, y1 ). Proof. Let F¯ be as in (7). Let gG and gF¯ be the genus of G(x, y) = 0 and F¯ (x1 , y) = 0 respectively. Let B be a place of G = 0 with center P = (α, β). Let (x(t), y(t)) be an irreducible parametrization of B. Let us denote by ˜ the place of the algebraic curve F¯ (x1 , y) = 0 given by B the irreducible parametrization (x1 (t), y(t)), where x1 (t) = ˜ be the center of B. ˜ It is obvious x0 (t)/y 0 (t). Let P˜ = (α, ˜ β) ˜ If νx (B) 6= νy (B) then ˜ and β = β. that νy (B) = νy (B) ˜ = νx (B) − νy (B). Hence, if νx (B) > we have that νx1 (B) νy (B), then α ˜ = 0; if νx (B) < νy (B), then α ˜ = ∞; if ¯ νx (B) = νy (B), then α ˜ ∈ Q. The map that sends each place B of G = 0 to the place ˜ of F¯ = 0 is injective. Let B and B 0 be two places of B ˜ = B f0 . Let (x(t), y(t)) and (z(t), v(t)) G = 0 such that B be the parametrizations of B and B 0 respectively. We may 0 assume that y(t) = y0 + tp and v(t) = v0 + tp (see [26], ˜=B f0 we have that p = p0 , Chap. 4, Theorem 2.2). Since B 0 0 y(t) = v(t) and x (t) = z (t). Hence z(t) = x(t)+c, for some constant c. By lemma 3.3 we have that c = 0, so B = B 0 . By the Riemann-Hurwitz formula we have that X 2(gG + deg(G, y) − 1) = (|νx (B)| − 1), (9)

The following example shows that the degree bound given in Theorem 3.8 is optimal. Example 3.9. Assume that n > m > 0 and (n, m) = 1. Let G(x, y) = y n − xm which is irreducible. We have that G(x, y) = 0 is an algebraic solution of F = y n−m y1m − (m )m = 0. In this case, we have that deg(G, y) = deg(F, y)+ n deg(F, y1 ).

4. A POLYNOMIAL-TIME ALGORITHM The simple degree bounds given in the preceding section allow us to give a polynomial-time algorithm to compute algebraic function solutions of a first order autonomous ODE.

4.1 Algebraic approximant Algebraic approximant is a special type of Hermite-Pad´e approximant. It uses an algebraic function to approximate a given function. Definition 4.1. Let G(x, y) be an irreducible polynomial ¯ in Q[x, y]. An algebraic function y¯(x) satisfying G(x, y¯(x)) = 0 is called an algebraic approximant to a function f (x) if

B

G(x, f (x)) = O(x(m+1)(n+1)−1 )

where B runs over all places of G = 0. We will split the right hand side of the above equation in four cases: We say that B ∈ (1) if νx (B) > 0 and νy (B) > 0; that B ∈ (2) if νx (B) > 0 and νy (B) < 0; B ∈ (3) if νx (B) < 0 and νy (B) > 0; and that B ∈ (4) if νx (B) < 0 and νy (B) < 0. Moreover, we say that B ∈ (1)0 if B ∈ (1)

where m = deg(G, x) and n = deg(G, y). More generally, we will find G(x, y) such that G(x, f (x)) = O(xN +1 )

33

(14)

where N is a positive integer. We can get the coefficients of G(x, y) with respect x and y by solving linear equations. Pn to Pm i j Let G(x, y) = j=0 i=0 bi,j x y and f (x) = a0 + a1 x + N N +1 · · · + aN x + O(x ). Let   I(m+1)×(m+1)  M0 =  (15) 0(N −m)×(m+1)

Algorithm 4.3. Input: F Output: the first N + 1 terms tion of F = 0 which is not in

¯ 2 on F (y, y1 ) = 0 such that 1. Find a point (z0 , z1 ) ∈ Q S(z0 , z1 ) 6= 0 and z1 6= 0. 2. i := 2 and ϕ(x) := z0 + z1 x. 3. while i ≤ N do

where I(m+1)×(m+1) is an m + 1 unit square matrix and 0(N −m)×(m+1) is an (N − m) × (m + 1) zero matrix. Let Mi = T M i ∗ M0 for i = 1, · · · , n where   a0 0 0 ··· 0  a1 a0 0 ··· 0     a2 a a ··· 0  1 0 TM =  (16)   .. .. .. .. ..   . . . . .  aN aN −1 aN −2 · · · a0

(a) Replace y by ϕ(x) and y1 by ϕ0 (x) in F (y, y1 ). (b) c := the coefficient of xi−1 in F (ϕ(x), ϕ0 (x)). (i−1)!c (c) zi := − S(z and ϕ(x) := ϕ(x) + 0 ,z1 )

(M0 |M2 | · · · |Mn )

B0  B1   .  .. Bn

    = 0 , Bi = 



b0,i  b1,i   .  .. bm,i

4. Return(ϕ(x)). The correctness of the algorithm comes from the following facts. Let y¯(x) be a formal power series solution of F = 0. Then by (1),

    

(F (¯ y (x), y¯1 (x)))(i−1) = S y¯i (x) + R(¯ y (x), · · · , y¯i−1 (x)) = 0. Since y¯k (x)|x=0 = zk for k = 1, 2, · · · , we have that

(17)

S(z0 , z1 )zi + R(z0 , · · · , zi−1 ) = 0. Now assume that ϕ(x) = z0 + z1 x + · · · +

for i = 1, · · · , n. Let y¯(x) = a0 + a1 x + · · · be a formal power series. When we say ϕ(x) is the first N + 1 terms of y¯(x), we mean that ϕ(x) = a0 + a1 x + · · · + aN xN . The following lemma will be used in our algorithm.

R(z0 , · · · , zi−1 ) = (F (ϕ(x), ϕ0 (x)))(i−1) |x=0 which equals to (i − 1)! times the coefficient of xi−1 in F (ϕ(x), ϕ0 (x)). Let T = tdeg(F ), the total degree of F . Theorem 9 given in [9] shows that the number of the points on F (y, y1 ) = 0 which make S(y, y1 ) or y1 vanish is at most T 2. The complexity of Algorithm 4.3 is polynomial in terms of the number of multiplications in Q needed in the algorithm. In Step 1, we can find a point (z0 , z1 ) as follows. 2 We may replace y by the integers z0 = 0, ±1, · · · ± d T2 e where T = tdeg(F ) and let L(y1 ) be a monic irreducible factor of F (z0 , y1 ) ∈ Q[y1 ]. We may take z1 to be a root of L(y1 ) = 0. Since the number of the points which make S(y, y1 ) or y1 vanish is at most T 2 , there always exists an 2 integer z0 ∈ {0, ±1, · · · ± d T2 e} such that the point (z0 , z1 ) satisfies the assumption in Step 1. Hence the complexity of Step 1 is polynomial. Then all the procedures will be executed over the number field Q(z1 ). Let D = deg(L(y1 )) ≤ T = tdeg(F ). Then any element of Q(z1 ) can be represented as a polynomial in z1 with degree ≤ T −1. Let β, γ ∈ Q(z1 ). Then there exist P (z), Q(z) ∈ Q[z] such that β = P (z1 ), γ = Q(z1 ) where deg(P ) ≤ T − 1, deg(Q) ≤ T − 1. To compute φ = β ∗ γ, we need to compute R = prem(P Q, L). Therefore, a multiplication of two elements in Q(z1 ) needs O(T 2 ) multiplications of rational numbers. Since computing the inversion of β can also be done in O(T 2 ), the division of two elements in Q(z1 ) needs O(T 2 ) multiplications of rational numbers too. In Step 3, the computation of (a0 +a1 x+· · ·+aN xN )T needs at most O(N 2 T 4 ) multiplications in Q(z1 ), and hence at most O(T 2 · N 2 T 4 ) = O(N 2 T 6 ) multiplications in Q.

where deg(Qi (x), x) ≤ m and not all of them are zero. Then (18)

¯ does not equal to zero. where λ ∈ Q Proof. Let Q(x, y) = Q0 (x) + Q1 (x)y + · · · + Qn (x)y n . ¯ There exist S, T ∈ Q[x, y] such that (19)

where deg(S, y) < n and deg(T, y) < n. If Q(x, y¯(x)) = 0, then (18) is true. Assume that Q(x, y¯(x)) 6= 0 and Res(G, Q, y) 6= 0. Then it is not difficult to know that deg(Res(G, Q, y), x) ≤ 2mn. However, substituting y¯(x) to the left side of (19), the left side will become a series with order greater than 2mn, a contradiction. Hence Res(G, Q, y) = 0 which implies (18) is true, because G(x, y) is irreducible.

4.2

Then

Since ϕ(k) (x)|x=0 = zk for k = 1, · · · , i − 1 and ϕi (x) = 0, we have that

Q0 (x) + Q1 (x)ϕ(x) + · · · + Qn (x)ϕ(x)n = O(x2mn+1 )

SG(x, y) + T Q(x, y) = Res(G, Q, y)

zi−1 xi−1 . (i−1)!

(F (ϕ(x), ϕ0 (x)))(i−1) = R(ϕ(x), · · · , ϕ(i−1) (x)).

Lemma 4.2. Let y¯(x) be a formal power series such that G(x, y¯(x)) = 0. Assume that m = deg(G, x) and n = deg(G, y). Let ϕ(x) be the first 2mn + 1 terms of y¯(x). If ¯ Q0 (x), Q1 (x), · · · , Qn (x) ∈ Q[x] such that

G(x, y) = λ(Q0 (x) + Q1 (x)y + · · · + Qn (x)y n )

zi xi . i!

(d) i := i + 1.

and ai are the coefficients of f (x). Then by the computation process, we can write (14) as the matrix form 

= 0 and a positive integer N . of a formal power series solu¯ Q.

An algorithm to compute algebraic solutions

First, we give an algorithm to compute the first N + 1 terms of a formal power series solution of F = 0 for a given positive integer N . Regarding F = 0 as an algebraic curve, find a point (z0 , z1 ) on it such that the separant S(y, y1 ) of F (y, y1 ) does not vanish at (z0 , z1 ). Then we can compute yi = zi step by step from (1). Then y¯(x) = z0 +z1 x+ z2!2 x2 + · · · is a formal power series solution of F = 0. Moreover, if ¯ z1 6= 0, then y¯(x) ∈ / Q.

34

needs O(T 8 ), because T M is an l × l matrix with l ≤ 2T 2 + 1 and M0 is a p × q matrix with p ≤ 2T 2 + 1, q ≤ T + 1. (Note that in the worst case, we have to do the operations over Q(z1 ). Hence the complexity has to increase by O(T 2 ).) In Step 2(d), we need only to solve at most 4T 2 + 1 linear equations with at most 2T 2 + 3T + 1 variables. Hence its complexity is polynomial. In Step 2(g), from ([25],p152), GCD(G, S) and GCD(G, I) can be computed in O(T 6 ). In Step 2(h), for deciding whether prem(F, G) = 0, we compute R1 = prem(F, G0 ) first. Since R1 = ( ∂G )k F (y, − ∂G / ∂G ) where k ≤ T , we can com∂y ∂x ∂y 12 pute it in O(T ) and have that deg(R1 , x) ≤ 2T 2 and deg(R1 , y) ≤ 4T 2 + T . Then we compute the GCD(R1 , G) which can be computed in O(T 10 ). If GCD(R1 , G) = G, then prem(F, G) = 0; otherwise prem(F, G) 6= 0. The number of the circulation in Step 2 is at most 2T . Hence the complexity of Step 2 is also polynomial.

Now we can give the algorithm to compute an algebraic solution of F = 0. Algorithm 4.4. Input: F = 0. Output: an algebraic solution of F = 0 if it exists. 1. d := deg(F, y1 ) and e := deg(F, y). 2. k := 1. while k ≤ d + e do (a) Compute the first 2dk + 1 terms ϕ(x) of a formal power series solution of F = 0 by Algoritm 4.3. (b) ai := the coefficient of xi in ϕ(x) for i = 0, · · · , 2dk. (c) In (15) and (16), let m = d, n = k and N = 2dk. We construct the linear equations (17). (d) If (17) has no nonzero solution or the dimension of the solution space of (17) is great than one, then go to Step (i). (e) Otherwise, choose one of nonzero solutions ¯bi,j where i = 0, · · · , d and j = 0, · · · , k. Pk Pd ¯ (f) G(x, y) := bi,j xi y j , S := ∂G and j=0

i=0

Example 4.5. Consider F = (y 6 +2y+1)y13 −(12y 5 +9y 4 −1)y12 +27y 8 +54y 7 +27y 6 +4y 3 . 1. Let d = 3 and e = 8.

∂y

I := the initial of G(x, y). (g) If GCD(G, S) 6= 1 or GCD(G, I) 6= 1, then go to Step (i). Otherwise, go to next step.

2. For the case k = 1, we get a G(x, y) = 0 which is not the solution of F = 0. Here we only give the process in the case k = 2.

(h) Let R = prem(F, G). If R = 0, then return(G(x, y) = 0).

3. The first 13 terms of the formal power series solution of F = 0 is

(i) k := k + 1.

5 2 9 1 5 41 6 65 7 x − x3 + x4 + x5 − x − x 2 4 2 4 32 64 363 8 111 9 2545 10 5141 11 5891 12 + x − x − x + x + x . 128 256 512 1024 1024

ϕ(x) = 1 − 2x +

3. If the algorithm does not return G(x, y) = 0 in Step 2, F = 0 has no algebraic solution and the algorithm terminates.

4. Let m = 3, n = 2 and N = 12. We construct the linear equations (17). Solving it, we get a nonzero solution

From Theorem 2.6 and Lemma 2.5, we know that if F = 0 has a nontrivial algebraic solution, then every formal power series solution is algebraic. From Lemma 4.2, we only need to compute the first 2dk + 1 terms of a nontrivial formal power series solution to construct the algebraic approximant. From Theorems 3.4, 3.8, if F = 0 has an algebraic solution G(x, y) = 0, then there is a k which satisfies that k ≥ 1 and k ≤ d + e such that deg(G, x) = d and deg(G, y) = k. From Lemma 4.2 again, the dimension of the solution space of (17) equals to one. If G(x, y) = 0 is an algebraic solution, then G(x, y) is irreducible. Then it is obvious that GCD(G, S) = 1 and GCD(G, I) = 1. Now assume that GCD(G, S) = 1, GCD(G, I) = 1 and prem(F, G) = 0. We will prove that G(x, y) is irreducible. Suppose that k = h. ¯ Then G(x, y) can not have a factor u(x) ∈ Q[x], because ¯ GCD(G, I) = 1. If G(x, y) = g(y) ∈ Q[y], then by (14), g(ϕ(0)) = 0 and g 0 (ϕ(0))ϕ0 (0) = 0. Since ϕ0 (0) = z1 6= 0 and g 0 (y) = S, we have GCD(G, S) 6= 1. Hence G(x, y) ∈ / ¯ Q[y]. If G(x, y) is reducible, then G(x, y) has an irreducible ˜ y) which is nontrivial and deg(G, ˜ y) < h. Since factor G(x, GCD(G, S) = 1, GCD(G, I) = 1 and prem(F, G) = 0, by ˜ y) = 0 is an algebraic solution of F = 0. Hence, we (4), G(x, ˜ y) when k was less than h and the algowould have got G(x, rithm must terminate before k = h, a contradiction with the assumption k = h. So G(x, y) is irreducible and G(x, y) = 0 is an algebraic solution. The complexity of Algorithm 4.4 is polynomial in T where T = tdeg(F ). In Step 2(a), the complexity is polynomial. In Step 2(c), we need only to compute T M 2T ∗ M0 which

(−1, 1, 0, 0, 0, 3, −3, 1, 1, 0, 0, 0). 5. Let G(x, y) = −1 + x + 3xy − 3x2 y + x3 y + y 2 and S = 2y + 3x − 3x2 + x3 , I = 1. 6. We have GCD(G, S) = 1 and GCD(G, I) = 1. 7. prem(F, G) = 0. Hence G(x, y) = −1 + x + 3xy − 3x2 y + x3 y + y 2 = 0 is an algebraic solution of F = 0.

5. REFERENCES [1] Bronstein, M., Integration of elementary functions, J. Symb. Comput., 9, 117-173, 1990. [2] Bronstein, M. and Lafaille, S., Solutions of linear ordinary differential equations in terms of special functions, Proc. ISSAC2002, ACM Press, 2002. [3] Carnicer, M.M., The Poincar´e problem in the nondicritical case, Ann. of Math., 140, 289-294, 1994. [4] Cerveau, D. and Lins Neto, A., Holomorphic foliations in CP (2) having an invariant algebraic curve, Ann. Inst. Fourier, 41(4), 883-903, 1991. [5] Cormier, O., Singer, M.F., Trager, B.M. and Ulmer, F., Linear differential operators for polynomial equations, J. Symb. Comput., 34, 355-398, 2002. [6] Cormier, O., On Liouvillian solutions of linear differential equations of order 4 and 5, Proc. ISSAC2001, 93-100, ACM Press, 2001.

35

[7] Corral, N. and Fern´ andez-S´ anchez, P., Isolated invariant curves of a foliation, to apper in Proc. Amer. Math. Soc.. [8] Davenport, J.H., On the integration of algebraic functions, Lecture Notes in Computer Science, 102, Springer-Verge, New York, 1981. [9] Feng, R. and Gao, X.S., Rational general solutions of algebraic ordinary differential equations, Proc. ISSAC2004, 155-162, ACM Press, 2004. [10] Feng, R. and Gao, X.S., A polynomial-time algorithm to compute rational solutions of first order autonomous ODEs, MM-Preprints, No.23, 54-65, December, 2004. [11] Fulton, W., Algebraic Curves, Benjamin/Cummings Publishing Company, Inc, 1969. [12] Hubert, E., The general solution of an ordinary differential equation, Proc. ISSAC1996, 189-195, ACM Press, 1996. [13] Kolchin, E.R., Differential Algebra and Algebraic Groups, ACM Press, New York, 1973. [14] Kovacic, J.J., An algorithm for solving second order linear homogeneous differential equations, J. Symb. Comput., 2(1), 3-43, 1986. [15] Lang, S., Introduction to Algebraic and Abelian Functions, second edition, Springer-Verlag, New York, 1972. [16] Poincar´e, H., Sur l’int´egration alg´ebrique des ´equations diff´erentielles du premier ordre et du premier degr´e, Rend. Circ. Mat. Palermo, 11, 193-239, 1897.

[17] Risch, R.H., The problem of integration in finite terms, Trans. Amer. Math. Soc., 139, 167-189, 1969. [18] Risch, R.H., The Solution of the problem of integration in finite terms, Bull. Amer. Math. Soc., 76, 605–608, 1970. [19] Ritt, J.F., Differential Algebra, Amer. Math. Sco. Colloquium, New York, 1950. [20] Singer, M.F., Liouillian solutions of nth order homogeneous linear differential equations, Amer. J. Math., 103(4), 661-682, 1981. [21] Singer, M.F., Liouillian first integrals of differential equations, Trans. Amer. Math. Sco., 333(2), 673-688, 1992. [22] Trager, B., Integration of Algebraic Functions, Ph.D thesis, Dpt. of EECS, Massachusetts Institute of Technology, 1984. [23] Ulmer, F. and Calmet, J., On liouvillian solutions of homogeneous linear differential equations, Proc. ISSAC1990, 236-243, ACM Press, 1990. [24] Van der Put, M. and Singer, M. Galois Theory of Linear Differential Equations, Springer, Berlin, 2003. [25] von Zur Gathen, J. and Gerhard, J. (1999). Modern Computer Algebra, Cambridge University Press, Cambridge. [26] Walker, R. J., Algebraic Curves, Princeton Unv. Press, 1950.

36

Adherence is Better than Adjacency: Computing the Riemann Index Using CAD James C. Beaumont, Russell J. Bradford, James H. Davenport & Nalina Phisanbut

∗

Dept. of Computer Science, University of Bath Bath BA2 7AY England

{J.Beaumont, R.J.Bradford, J.H.Davenport, cspnp}@bath.ac.uk ABSTRACT

Algorithms, Theory

as set equality. As pointed out in [15] not all such formulae are identities and instead require set inclusions: we have that Log(z 2 ) ⊃ 2 Log(z) for example. Many others may be found in [1], which we will use to provide a set of realistic test formulae— exercising occasional care, however, in their interpretation[16]. Given H ⊆ 0, the problem is then to decide on what regions of n does h = 0 hold? The paper [15] provides a graphic illustration that one is indeed forced to consider the geometry of n with respect to the branch cuts of h in order to answer this question. One important application of having a method to decide such questions is to the area of simplification. To obtain and define precisely what one means by a ‘simplification’, is an old and difficult problem[23]. We shall not grapple with this issue here but refer instead to recent progress in [9] for a potential approach, and to [8] for some of the problems involved with working with multi-valued transformations.

Keywords

1.1 Previous Work

Riemann Surfaces, Elementary Functions, Branch Cuts

Progress towards constructing a verification system for multi-valued formulae as described above has been reported in [4] and its precursors. This is based on the Decomposition Method first suggested in [19], which requires one to:

Given an elementary function with algebraic branch cuts, we show how to decide which sheet of the associated Riemann surface we are on at any given point. We do this by establishing a correspondence between the Cylindrical Algebraic Decomposition (CAD) of the complex plane defined by the branch cuts and a finite subset of sheets of the Riemann surface. The key advantage is that we no longer have to deal with the difficult ‘constant problem’.

Categories and Subject Descriptors I.1.2 [Symbolic and Algebraic Manipulation]: Algorithms

General Terms

1.

INTRODUCTION

The elementary functions are the field of functions obtained by applications of exp, log and the arithmetic operations to a set of variables x1 , . . . , xn and constants, . As in previous papers by the authors[4, 8], we shall focus particularly on those elementary functions which are in fact multivalued. We shall use the notation that terms such as log and

1. calculate the set of branch cuts of the proposed identity F = 0; 2. find a sample point in each of the regions in defined by the cuts; 3. evaluate the identity numerically using that point, thereby concluding whether the formula is true or not on that entire region by the Monodromy theorem.

, and more generally, f, h, denote single-valued functions from to , whilst Log, nSqrt and F, H denote multivalued functions, regarded as mapping into √ sets of values, so that Sqrt(z) = {w : w 2 = z} = {± z} for example. Numerous well-known identities for multi-valued functions exist; examples include Log(z 2 ) − Log(z) − Log(z) = {0}, and Sqrt(z 2 ) = {±z}. The = is of course to be interpreted n


Further details of how each of the steps above should be performed can be found in [4]. A key point to remember is that, as first suggested in [8], we use Cylindrical Algebraic Decomposition (CAD; see [10]) for step two; we shall assume that the reader is familiar with the basic notions involved in this algorithm. To do this, we restrict the class of formulae under consideration to those where the branch cuts are algebraic: it is sufficient to prohibit anything other than nth roots from being nested inside other elementary functions.1 It is however worth pointing out that step one has now been made computationally efficient by the recent proposal to use resultants to eliminate nth roots in the input expression; see [5] for details. Also in that paper, the use of an efficient method[18] to decide which is the best projection

∗The authors gratefully acknowledge the support of EPSRC, under grant number GR/R84139/01.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. ISSAC’05, July 24–27, 2005, Beijing, China. Copyright 2005 ACM 1-59593-095-7/05/0007 ...$5.00.

1 Note that this applies to constant functions; witness log(x − exp(2)), which has a non-algebraic branch cut.

37

order is demonstrated to be of great importance regarding the efficiency of step two. Thus in this paper, we are still strongly advocating a CAD based Decomposition Method; what we are proposing here is a new and more efficient approach to performing the final step. This step is surprisingly non-trivial as was seen in [3, 4] and so an improvement is highly desirable. We first provide a convenient summary of the most serious problems involved at this stage. Suppose for simplicity of exposition that our formula H is of one variable. Then we can compute a description of the branch cuts of h, as a semi-algebraic set in 2 real variables. Firstly suppose that the region we are testing has co-dimension > 0. In the terminology of CAD, this means that we are investigating a region that comprises a particular section, s say, of a stack. Let the sample point of s be p = (x, y), which must be interpreted as the complex number x + iy. As argued in [8], numerical evaluation of H(p) may give completely incorrect results except in the rare cases where x, y have finite floating point representations. A symbolic approach would therefore seem necessary, but in the worst case scenario, the point p will have some coordinates that cannot be expressed in terms of radicals. Any attempt to search for alternative sample points in sections where this is possible— those constructed over level 1 regions of full dimension— would result in an undesirable coupling between steps 2 and 3 of the algorithm. Even in the cases where p is expressible using radicals one runs into the ‘constant problem’ and has to generally resort to algorithms that are potentially costly and rely on the truth of number-theoretic conjectures[26]. The constant problem is in fact undecidable for sufficiently large function fields[25]. This is inhibiting as ultimately, one would like to handle non-elementary multi-valued functions as well, such as the W -Lambert function[14] for example. Secondly, in all regions, it may happen that p is an ‘unlucky sample point’. If H is a rational function of elementary functions Hi each of which has the set of branches {hi }i then p is an unlucky point if for one or more of the hi we have that hi (p) are equal for several different branches. If we use such a p, then we cannot guarantee that we can draw correct conclusions about the truth of the identity H on the region it represents. An example is afforded by p(z) Log(q(z)2 ) − 2p(z) Log(q(z)) where p, q ∈ [z] and the unlucky points are then the roots of p, q. The method of [27], as was demonstrated in [3], is an effective, although costly, solution to this problem. Finally the work reported in [5] demonstrated that the number of cells that are produced by problems of seemingly simple appearance can be extremely large. This of course exacerbates the problems mentioned above. In such cases, numerical evaluation can also be very time-consuming— assuming we are on regions where it can be applied safely. The rest of the paper is as follows. In sections 2 and 3 we describe and exemplify a method for testing the proposed formula on the branch cuts. In section 4 we propose a new method based on Riemann surfaces to make testing the identity on all cells still more efficient. Finally we summarize our contribution and consider future directions.

2.

inverse elementary functions, such as F (z) = arcsin(z), log(z). These are the only elementary functions with branch cuts, so they will be the main focus here. It will be convenient to work with functions of the form Hi = Fi (Gi (z)) where Gi ∈ rt (z) and Fi is an nth root or logarithm as our ‘building blocks’; since all of the base functions can be defined in terms of these, it follows that all our admissible formulae (as in 1.1) can be expressed recursively in terms of the Hi . The definitions we use for the base functions are those from [12], although everything which follows applies regardless of the initial choices made, as long as we are consistent. With F as above, the Riemann surface associated with F shall be denoted by RS (F ), and we recall that this is a path-connected domain for F having either n or an infinite number of sheets respectively, each of which is a domain for a particular branch of the function in question. The branches ) for F (z) in this case are of course either n g(z) exp( 2kπi n for k = 0, . . . , n − 1 or log(z) + 2kπi for k ∈ . For a fixed k, we shall refer to a particular k-th branch of F , denoted by f k which has domain which will refer to as comprising the k-th sheet of RS (F ); we denote the principal branch of F by f 0 , or simply f . Further, RI (F )|c shall be the Riemann index of F on a particular cell c of the CAD induced by the branch cuts: that is, either exp( 2kπi ) or 2kπi for some fixed n k. The branch cuts serve of course, to act as boundaries where distinct sheets are joined; we denote the set of branch cuts of f by B(f ). The important question we shall address is, which sheet does a given branch cut belong to? We now make a key definition, for the case of CADs in the plane. 

Definition 1. (Adherence) Suppose that we have a CAD for B(h), and c is a section of a stack representing part of, or all of, a particular branch cut. Let s be an adjacent sector cell to c. Then we say that the branch cut c adheres to s if c belongs to the same sheet of RS (H) as does s. Recall that, two cells of a CAD are said to be adjacent if their union is path-connected[11]. When H = F (G), c can not adhere to both adjacent cells by monodromy. A brief overview of the algorithm, for the single variable case, is presented below. Its purpose is to solve the problem of testing an identity on a branch cut by using an adjacent cell of full-dimension instead. In what follows, for any x ∈ we define sign(x) = 1 if x > 0 and sign(x) = −1 if x < 0. Given an input formula φ, we recall that the TruthValue of a cell c in the CAD with respect to φ is a boolean value, depending on whether or not φ(p) is satisfied at any p ∈ c. 

Algorithm 1. Input: H = F (g) where F is Log or nSqrt ., g ∈ (z) Output: A CAD of B(h); adherent cells determined.

2

1. Compute S = B(h) and D = CAD(S) 2. Compute adj(D) if need be 3. For each c ∈ D where T ruthV alue(c) = T rue do sign := sign(=(g(p))), p ∈ c1 , with s1 , c adjacent if sign = 1 then RI (H)|c = RI (H)|s1 else RI (H)|c = RI (H)|s2 where s2 is adjacent to c and not to s1 .

COMPUTING THE ADHERENCE

We now comment further on this algorithm. The role of steps 1 and 2 is obvious. The adherence of any branch cut

If we define 0 p(z) = p(z), then we may define rt (z) recursively to be the set of functions of the form φ( n pi (z) ) where φ is a rational function with pi ∈ rt (z) and n ∈ . By a base (inverse) function we shall mean any of the 14

2 The case of G ∈ rt (z) has extra difficulties which we return to later in sections 2.1, 3.




38

of f (g) will depend only on what we what choose to be the closure of the base function f (z). Examination of the definitions of the complex logarithm and nth root functions shows that this in turn is determined only by what definition we take for the principal branch of arg: here we choose the modern convention which requires that arg(z) ∈ (−π, π]. Thus for the branch cut of f (z), we have Counter Clockwise (CC) Closure, or, in the terminology here, the branch cut c of f (z) adheres to the cell T having cell index (1,3), which lies in the same stack in which c does in the CAD constructed with respect to this single branch cut. We now make step 3 more precise. We shall require that s1 is of full-dimension. Thus in the x—y plane, there are two possibilities for c: either (a) c is a vertical line x = a for a ∈ or (b) it is not. In the former case we choose a sector cell s1 which is adjacent to c and lies in either of the two adjacent stacks to the one in which c resides, (another reason for treating these cuts specially is given in 2.1) and in the latter case, we choose one of the two sector cells that are adjacent to c in the same stack. Notice that for two-dimensional CADs such as here, we can decide a priori whether intra-stack adjacency alone, which does not require an algorithm, will be sufficient; thus allowing us to bypass step two above. To do this we examine the CAD data structure to see if there are any cells which have truth value T rue, are of dimension one, and are constructed over level one cells of zero dimension. The idea behind step 3 is the following. Let γ : [0, 1] → be a path with γ(0) = p and γ(1) = q, with q ∈ c, and let g(q) = s where s ∈ − . Then if g(γ(t)) → s + 0+ i as t → 1 then c adheres to s1 ; otherwise, it must adhere to another cell, s2 say. In case (a) this s2 will be an adjacent sector cell to c, but in the different stack to which s1 resides; in case (b), it will be the only other adjacent sector to c in that stack. Applying an adjacency algorithm[11] to D produces output which can be thought of as comprising lists of cell indices where all indices in a list represent cells which form a single connected component. Computing the adjacencies thus gives us the cells s1 whose sample points are required for the last step. Notice that there will be any finite number of adjacent cells s1 to c in general. We only need to choose one cell of course for each branch cut that comprises a single connected component, and this can be achieved by using the adjacency information, thus avoiding the potential redundancy. In the case where g ∈ (z), the particular choice of adjacent cell does not matter. We point out that the representation used in step 1 will produce a CAD containing other sections which are not branch cuts, and so we do not wish to apply the algorithm to them. For example, if we wish to represent a semi-circular branch cut we would use {(x2 + y 2 − r2 = 0) ∧ (x > 0)}, but this will make for a CAD containing both roots of the bivariate polynomial. However these unwanted cells will always have TruthValue F alse, so can be ignored.

the CAD solution formula construction step, they comprise cells which are assigned the truth value T rue, since they satisfy the formula S. In some cases one can evaluate s = g(p) for p on the sample point of each cell in S and check whether or not s ∈ B(f ). In general, we run into the same problems as described above in (1.1). One might think that we could detect the spurious cuts by examining the signs of the imaginary parts of g(pi ) for pi ∈ si with i = 1, 2, but the fact that we cannot guarantee that s ∈ B(f ) means that the reasoning based on the continuity of g is not sufficient. For the important case of square roots, we showed in [4] how to remove the spurious branch cuts by adding polynomial constraint equalities to the system. It is a minor extension to handle nth roots which we defer to [6]. However this contributes exponentially to the number of polynomials that are used in the CAD construction. A further difficulty is that the faster methods for eliminating roots such as Gr¨ obner bases, or better, the method of [5] work as black-boxes and do not automatically generate the appropriate sets of constraints at each stage. Except in simpler cases, where it may prove to be efficient to remove the spurious cuts, we propose to apply algorithm (1) exactly as we did before; this can be done since for any p ∈ s1 , =(g(p)) < 0 or =(g(p)) > 0. (In the case where =(g(p)) = 0, we know the cut is spurious) Suppose that, as in step 3 that c is in fact spurious, although we cannot yet decide if this is so. It is easy to see that s1 , s2 will then belong to the same sheet of RS (H) as does c, and so correctness at the sample point testing stage is guaranteed. The ‘adherence information’ we obtain in this case is of course spurious but since c adheres to both the si , we cannot obtain incorrect results. Whilst this approach provides us with a generic algorithm, it is wasteful in that we must compute g(p) for each spurious c nevertheless. One further issue to be exemplified later in section 3, is that given f (g) it may happen that a (non-spurious) cut c derives from both f and g. In that case, we must first calculate the adherence of c with respect to g, before we can compute the adherence with respect to f . This is because, unlike for the case where g ∈ (z), the choice of s1 in step 3 does matter: we need to choose it so that c adheres to s1 with respect to g, and then g will be continuous onto the c, as required.





2.2 Justification In the interest of brevity a detailed proof of correctness shall not given in this paper. However the essential ideas can be sketched as follows. First, one must remember when computing with g in the manner above that g is a complex valued function, g : → , only the branch cuts are real algebraic. However working with both real and complex geometry in this way does not cause any problems provided that one stays away from regions of non-analyticity. (This is why we do not allow complex conjugation: if g(z) = z for example, then one can easily check that this does not satisfy the Cauchy-Riemann equations, and is therefore not analytic.) We have that g(p) ∈ B(f ) by necessity. Now g is continuous on a path from at least one of the pi in the adjacent sectors onto p since in following [4], at step 1, we always include the branch cuts at infinity: that is where the denominator of g vanishes; the only other potential problem would be discontinuities arising in g due to the presence of nth roots, (as in example 3, next section) but as with the singularities they will be by virtue of the CAD, not inside

2.1 Nested roots Suppose that H = F (G) where G ∈ rt (z). In [4] we pointed out (for the case of square roots) that the method to calculate B(h) will produce a semi-algebraic set S say that contains spurious branch cuts. Recall that in the simplest case, when g = n p(z), that these are sets {z | (g k (z) ∈ B(f ))∧(k 6= 0)}— although in general g may contain several nth roots. They arise as an artefact of the method to remove nth roots from the input formula. The problem is that in

39

arctan(x)

the adjacent sectors we are looking at. The only case where this does not occur is when, given f (g), the cuts for f and g coincide, but this situation is not a problem if we deal with it as described at the end of (2.1). In the case of vertical line cuts, the end points may be singularities. However, since we use sectors in adjacent stacks, as opposed to using the sector above and below such cuts, we avoid losing the notion of a continuous path onto the cut. We remark that [11] is the most recent adjacency algorithm at the time of writing; we should mention that this may fail in higher than four dimensions, albeit with a very low probability. In the case of failure, we cannot guarantee to be able to represent the section in question correctly and we may under-represent the branch cuts. Fortunately for us, the CADs derived from formulae we are most interested in have at most 4 dimensions.

π/2

+0 x

x<0

x>0

−π/2

−π

0

Figure 1:

∗

& corresponding branches for Arctan(x).

have branch cuts at infinity (example 1) and singularities (example 2) which are not just zero dimensional regions.


1. H = Arctan

x+y 1−xy 

− Arctan(x) − Arctan(y) = 0; ?

we now investigate the corresponding identity h=0, where h = h1 − h2 − h3 . For real inputs, arctan(x) only has branch cuts at ±∞. Our representation is only for finite branch cuts, so we see that the branch cuts for H derive from h1 only: that is, the set B(h) = {xy − 1 = 0}. This comprises the two branch cuts shown in figure (2) which now require investigation. By numeric or symbolic evaluation, the identity is readily verified to be true on region 2. On region 1 we have that H = −π and that H = π on region 3. We now consider the formula on the cuts themselves. Notice that the CAD data structure will tell us that we do not need to compute the adjacency of the CAD with respect to B(h) here. Clearly, a purely numeric approach here would fail, at any point on the cuts, due to the blow-up. We shall show that arctan(x) must be viewed as a function of the form, (−∞, ∞] → (− π2 , π2 ]: that is the point (∞, π2 ) belongs to the principal branch of arctan whilst the point (−∞, − π2 ) belongs to the branch, arctan(x) − π. This requires us to work on the extended real line ∗ = ∪{∞} which we recall is constructed by identifying the end points ±∞ as in figure (1).3 Thus on ∗ one works with positive infinity only,and if x → ∞+ then we see that one passes onto the point ∞ continuously (see the left-hand diagram of figure 1) whilst if we let x → ∞− then we do not. In order to preserve the continuous 1-1 correspondence between the finite domain of arctan(x), that is , and the extended domain that is ∗ , we see that −∞ does not belong to the principal branch domain of arctan(x). As always, when passing through a branch cut one passes onto the adjoining branch domain; we can see that in this case that this will be the branch arctan(x) − π, as in the right-hand diagram of figure (1). Let c1 = {(xy − 1 = 0) ∧ (x > 0)} and c2 = {(xy − 1 = x+y 0) ∧ (x < 0)}, and put g = 1−xy . Now we can use adherence to determine which branch each ci is on by determining whether g tends to +∞ or −∞: the ci will adhere to the cell where g is positive. This requires that our adjacent cells are sign-invariant for g so we must first add the line y + x = 0 to the CAD. Consider c1 , where our adjacent regions are 1 and the part of 2 above the line, and then suppose that our sample points for these are given by p1 = (1, 2) and p2 = (1, 12 ) respectively. We obtain g(p1 ) = −3 and g(p2 ) = 3, which shows that c1 adheres to region 2, and we conclude the identity is true on this branch cut. Similarly we treat c2 = {(xy −1 = 0)∧(x < 0)} using regions 2 (below the line) and 3 and p3 = (−1, −2) and p2 = (−1, − 12 ). This

2.3 Application In general the input formula H = 0 contains several of our Hi = Fi (Gi ) building blocks. To deal with this, all that one needs to add in an implementation is a piece of additional information, namely, which Hi each branch cut cell derives from, to the usual CAD cell data structure. (Such as that found in the package we have used here[24].) that is, which Hi each branch cut cell derives from. Then we should simply apply the algorithm as shown, only using this data to ensure that we apply the appropriate gi at stage 3. Of course, some branch cut sectors may derive from several Hi . Now the adherence information alone is insufficient to determine the actual RI (Hi ) on a cell. However the point is that we now have a much simpler method (modulo the remark at the end of this subsection) to determine the truth of the formula ? h=0 on the branch cuts by testing the appropriate adjacent full dimensional cells instead. Apart from the possibility of choosing unlucky sample points (this will be avoided using the method of section 4), it is now an easy task to calculate the correction factor for h = 0 on these cells by using the method of [8] which uses floating point evaluation of H at the sample point together with bounds on the accuracy derived in the manner shown there. Of course this does not tell us what RI (Hi ) is for individual Hi . A problem occurs however, on branch cut cells which derive from several Hi , for some of these functions may adhere to different cells. An example of this is discussed in section 3. It is often convenient for a user to specify a function by arccosh say, as opposed to giving its definition in terms of logs and square roots. We have therefore determined the adherence for each of the base functions using the method above so that one can now allow our input function F to be any of these 14 functions. For future reference, this information is presented in the appendix. If a particular cut of F under consideration lies on the imaginary axes of the plane, then one must make the necessary minor changes in step 3: we calculate sign := sign(<(g(p))) instead for cuts that lie on the imaginary axes of the plane, and then if the cut is from the side <(z) < 0, for example, (as given in the table) the condition should be taken to be sign = −1.

3.

+π

EXAMPLES

The three simple examples which follow are chosen so that they may be readily verified by the reader. In the first two examples, x and y shall be real. The case of 2 will be seen to exhibit different behaviour from that of , for we now 

3

40

Corresponding to a circle on the Riemann Sphere.

2

1

QE) g1 never maps any points onto the branch cuts of f1 (z). We now calculate the adherence of h2 for the branch cut ci 0 . Following (2.1), we must first calculate the adherence of the ci (with respect to g2 ). This is easily verified by algorithm 1 to yield the adherence of c1 to the region s1 containing the point (-2,1) for example, whilst the cut c2 adheres to the region s2 containing the point (2,-1). We must therefore choose these points with which to compute the adherence of the ci 0 . Comparing the results to those of h1 (from the table) we see that the adherence of the cuts for h1 is different to the adherence of the cuts for h2 . The problem arises if we try to apply this adherence knowledge directly: one clearly cannot test the identity on either of the cuts by using any single sector point! When are test? ing h1 =h2 and the adherence differs in this way on a cut, we should test the identity on the adjacent regions first. Then if we find a adjacent sector where the identity holds, then this is sufficient to conclude that the identity cannot hold on the cut itself. In this example, the identity easily proven to be true everywhere off the branch cuts, and so we conclude that it is false on the ci . To see this, we argue as follows. Suppose that we take a continuous path onto a point p ∈ c1 from s1 . Now h = h1 − h2 is analytic and h = 0, until we reach the end-point of our path, whereupon h2 only becomes discontinuous, and so h(p) 6= 0. Hence by the Monodromy theorem, h 6= 0 on c1 . The argument for c2 runs entirely analogous.

h=−π

h=0

c1

c2 h= π

3

Figure 2: decomposition & adherence for Ex.2.

time g(p3 ) is positive and g(p2 ) is negative, so c2 adheres to region 3. Thus we conclude that the identity is false on this branch cut, and requires a correction factor of −π. Observe that the example could have been written in terms of complex logarithms. Then we could proceed by using the fact that log(g) = log |g| + i arg(g) where arg(g) = =(g) arctan( <(g) ). Since the argument component determines which branch we are on, and the input to the arctan is wholly real, the problem then reduces to the previous case. We omit the details for brevity. This approach should allow us to consider more general examples, although this is work in progress. 2. H = Log(xy) − Log(x) − Log(y) = 0. The branch cuts for the first logarithm are the top-left and bottom right-hand quadrants of 2 , together with the x and y axes. The latter set comprises the singularities; notice that they are one dimensional, and that they disconnect the plane— this has implications for the algorithm in section 4. As is well known, at the point of ramification that is z = 0 ∈ , all the sheets of R = RS (Log(z)) (and also for the nth root) are joined, but this point does not actually belong to R, corresponding to the fact that exp(z) 6= 0 for all z. So unlike the previous example, it does not make sense to ask which one of adjacent sheets of Log(z) does this point adhere to. The branch cuts for the other two logarithms are immediately clear, and it therefore turns out that we only can test the identity on the branch cuts. Since these are of full dimension, there is no problem: evaluating log(xy) − log(x) − log(y) at (-2,3),(2,3) and (2,-3), for example, shows that the H is always true on the branch cuts and undefined everywhere else. 

4. WINDING SEQUENCES In [22] it is reiterated that consideration of Riemann surfaces might yield an effective algorithm and a request is made to the community for further investigation into this area. Some of the difficulties with making these objects computational was pointed out in [13]. We do not propose a new representation for computing with Riemann surfaces; rather, we propose to compute with the well-known, theoretical ‘cut-plane’ representation of Riemann surfaces. To make this computational, one needs to bear the structure of the original surface in mind as it is projected down onto . The prevailing ethos behind this approach is to capture the idea of a continuous choice of argument whilst retaining the geometry and sense of where one is on the Riemann surface; as such this method can be viewed as an effective synthesis of the work reported in [13] and the Decomposition Method. A major problem with the cut-plane approach is the multiplicity of representation: informally, each region represents more than one (and in the case of logarithm, infinitely many) sheets of the Riemann surface. A step towards fixing this problem lies in the following definition. Let H = F (G(z)) be as in section 2, and Φ a formula, although we shall not allow nested roots for the moment.

z 3. Arcsin(z) ⊂ Arctan Sqrt(1−z . 2) Notice we only have set inclusion, not equality. We now con


?

sider the often quoted ‘identity’: arcsin(z)= arctan 

√

z

1−z 2

. 

Let z = x + iy. Writing arcsin(z) in terms of its def√ ? inition, −i log( 1 − z 2 + iz), let us label this as h1 =h2 with hi = fi (gi ) for i = 1, 2. Following the branch cut finding algorithm we see that each gi (z) has branch cuts given by c1 ∨ c2 where c1 = {(x < −1) ∧ (y = 0)} and c2 = {(x > 1) ∧ (y = 0)}. For the branch cuts of h2 we obtain the three-dimensional semi-algebraic set, φ = [(y 2 − x2 + 1 ≤ 0) ∧ (y = 0) ∧ (y 2 t2 − x2 t2 + t2 − y 2 + x2 = 0)] ∧ [(t + 1 ≥ 0) ∨ (t + 1 ≤ 0)]. As described in [5] we must now apply Quantifier Elimination (QE) to the formula (∃t)φ in order to eliminate t and obtain a formula without this extraneous variable. This yields the set comprising c1 0 ∨ c2 0 , which is in fact identical to c1 ∨ c2 . Now h1 does not have any branch cuts since (by

Definition 2. (Reachable Sheet) The k-th reachable sheet of RS (H(z)) is the sheet with index k such that there exists z ∈ such that Arg(g(z)) ∈ Mg , where Mg = ((2k − 1)π, (2k + 1)π]; we refer to the set of all ˆ S (H). such sheets as reachable sheets, denoted by R ˆ S (H) analogously. We define the reachable cells of a Ri ∈ R We may compute Mg by factorising numerator and denomαj (z − αi ) for αi,j ∈ inator into the form and us



j

i

ing the multivalued rules for Arg. For example, given h = log(z − z 2 ) we obtain Arg(z − z 2 ) = Arg(z) + Arg(−1) +

41

the problem raised in [13] of what to do when one has a sum of Hi such as H1 + H2 where the Hi are two separate Riemann surfaces. First we must locate which sections derive from the L0 (Hi ) and choose one to obtain the initial cell of our path, c0 . Let us compute M = Mg (gi ). There may be several argument branches for any number of the reachable sheets, but the L0 branch(es) may be successfully singled out by computing Arg(g(pi )) where pi for i = 1, 2 are the sample points of two adjacent sectors to an Li . We do this by factorising g as we did to find Mg above and the argument of each factor is computed by floating point evaluation. We now try to determine R = RI (c0 ). As we see in (4.1.1), we may have that R 6= 0, since 0 6∈ Mg . In this case, c0 (nor any other cell in L0 ) cannot be our first cell in the WS and we must leave it until it is reached later in the path, or in another path. In this case, only one of the adjacent sectors cl and ck say, to c0 can be chosen as the next cell in the path. Again we must examine Mg to decide which. Suppose that we have now reached a sector cell cj−1 in our current path, and determined RI (Hi )|cj−1 . It remains to show how to choose a section cj so that we still have a WS. Let us call the sections that derive from the set S(hi ) under consideration critical sections; the current RI (Hi ) can change only if cj is one of these. There are 2 cases; first suppose that s = cj derives from B(hi ). Then the adjacency of the CAD of S, computed at step three, allows us to apply algorithm (1): we outline the few possibilities. If Hi = Log(g(z)) then if we approach s from the nonadherent side then we subtract −2πi from RI (Hi )|cj−1 to obtain RI (Hi )|s , and then RI (Hi )|cj+1 has the latter Riemann index for our next valid cell in the path. If we approach s from the adherent side then we add 2πi to RI (Hi )|cj+1 only. If H = n g(z), we either reduce or increase k ∈ k by 1 in the Riemann index exp( 2kπi ) each time, noting that n if RI (Hi )|cj−1 = 1 and if cj−1 does not adhere to s then ). This is efficient as no evaluation RI (Hi )|s = exp( 2(n−1)iπ n of Φ need be done on these sectors. However, if RI (Hi )|cj corresponds to a non-reachable sheet (which can be checked ˆ then we must in fact terminate since we already know Z) the path after or before including cj in the path, depending on the adherence of s. The second case occurs if cj derives from any of the other functions in S(hi ). Then, we may also have the same problem as just described, or more difficult to detect, it may happen that RI (Hi ) changes on cj even though it is a branch cut or argument line of one (or more) of the f (pi ). Thus we may obtain the wrong index if we simply include cj as our next cell in the path. In this case, there are two approaches. The first is to try to find new paths avoiding cj starting from the last valid cell in the former path. The second requires us to compute Arg(p) on the sector cj+1 by the Mg method to give us the correct RI (Hi )|cj+1 and then use adherence to determine the index for the section cj . The last step, where we simplify IA (Φ)|c , is required in order to decide whether the correction factor for the singlevalued formula φ(hi ) = 0 is zero or not on each cell. For non-nested gi it is just a case of collecting the Riemann indices from the logs together, treating π as a symbolic indeterminate, and similarly for the roots. The nested case is a little more involved and will not be described here but in both cases the general constant problem does not arise.

Arg(z−1) and then (−π, π]+π+(−π, π] = (−π, 3π]. Whence there are only 2 reachable sheets, having indices 0 and 1, although RS (H) comprises infinitely many sheets. The number of reachable sheets of RS (H) is always finite, and Zˆ ⊂ shall be the set of reachable sheet indices. With H = F (g) being one of the building blocks in a formula, we shall demonstrate that, if we construct a CAD with respect to a set S to be described below, then we can obtain a surjective correspondence ψ from the set of regions ˆ S (H). That is, each cell in the CAD correof D onto R ˆ S (H). ψ sponds to part of, or all of a unique element of R is not a bijection because one of the draw-backs of CAD is that of over-representation; it generally produces far more cells than the actual number of connected components of the decomposition induced by the algebraic curves. With g factored into linear factors pi as above we compute S(h) = {B(f (pi )), L(f (pi )), B(h), L(h)} for each i where we define ˜ the set of argument branches of h, ˜ to be the set of L(h), points {z | g˜(z) ∈ [0, ∞)}. We shall later require the prin˜ this is the set cipal argument branch, denoted by L0 (h): ˜ where |k| is the such that Arg(p) = 2kπ for all p ∈ L0 (h), ˆ We compute the semi-algebraic set repleast element of Z. resenting the argument branches by using exactly the same algorithm as we did for the branch cuts. We now make the key definition of this section. Given H as above suppose that we have computed S(h) and that we have constructed a CAD D with respect to this set. 

Definition 3. (Winding Sequence) A Winding Sequence (WS) for D shall be a finite set of ordered lists called paths li , each of them finite, and where each element of the list is a cell index ci for i ∈ 0, . . . , Nli and the following conditions are satisfied: ˆ S (H); 1. ci , ci+1 represent adjacent regions on R 2. each cell in D has an index belonging to at least one of the li ; 3. c0 represents L0 , or part of L0 in at least one of the li .



Before we give an overview of the proposed algorithm, we need to make one more definition. The index aggregate of Φ = φ(Hi ) on a cell c, denoted by IA (Φ)|c shall be the result of making a set of substitutions 4 into Φ of the form, Fi → fi k1 , and for each subexpression Gj of Gi where ˆ Gj = n Sqrt(.) , Gj → gj k2 for some fixed k1 , k2 ∈ Z.

Algorithm 2. Input: Φ = φ(Hi ) Output: A CAD D, IA (Φ)|ck for all ck ∈ D For each Hi do steps 1-3 1. compute Zˆ

2. Compute S(h), D = CAD(S) and adj(D). 3. choose li until every cell of D where Hi is defined and has been included in at least one of the li 4. For each cell c compute and simplify IA (Φ)|c . The above computes paths through the cells of the CAD which correspond to paths on the Riemann surface of each Hi in turn. It is important however to notice that ensuring the adjacency of cells in a path is a necessary, but not sufficient condition for this to be the case. We point out that computing a separate path for each Hi effectively side-steps 4

A similar notion used in a different context occurred in [19].

42

2π i

2π i

2π i

2π i i

0

2π i 0

z=1 0

z=1

z=1

2π i

0 −i

0

0

0

Figure 4: Riemann indices for Log(−1 − z 2 ).

Figure 3: The reachable sheets of RS (Log(1 − z)).

paths which complete most of the WS without performing these computations. We point out that from Mg we may determine RI (L1 ) on (1,1) and (3,1) for free as well as (3,2), even though the latter cell is a critical section deriving from F (pi ) = Log(i + z), and would generally require an argument calculation. Note that after finding (1,3), (1,2) only requires the adherence of the bi which is this cell. Finally, the points (0, ±i) but not (0,0), can be placed into the above WS in the obvious way.

4.1 Examples 1. H = Log(1 − z) − Log(−1) − Log(z − 1). The sets of indices for the reachable sheets corresponding to the three logarithms above are {0, 1}, {0} and {0} respectively. Thus we only need to compute a WS for the Log(1 − z), as this is the only one which will affect IA (H). Clearly Mg = (0, 2π]. The branch cut and argument branch for this first logarithm are given by [1, ∞) and (−∞, 1) respectively, and the adherence of these can be easily calculated to obtain the set-up shown in the left-most diagram of figure (3). The other two diagrams show R = ˆ S (Log(1 − z)), namely domains of the principal and 2πi R branches, and the shaded regions represent the reachable cells of R together with their Riemann indices. (The simple CAD comprises the 9 obvious cells but is not shown here for clarity) We now show to derive those values so that they may be used to anotate the cells in a way that matches the left-hand figure as shown. There is only one argument branch here so this must be L0 , and so we let this cell be c0 . However, the Riemann index of c0 cannot yet be determined, as 0 6∈ Mg . Its cell index is (1,2) from which adjacency gives us a choice of (1,3) or (1,1) or (2,2) for c2 . Mg tells us that we must identify the cell which has points such that Arg(g) is positive, and so using p = −1 + I for example, we see that sign(=(g(p))) < 0 showing that (1,1) is a reachable cell on the principal sheet, with Riemann index 0, whilst (1,3) is not; (2,2) is a singularity, and cannot belong to the WS either. Now we take c3 = (2, 1) and c4 = (3, 1), for which the index remains 0. Next, c5 = (3, 2) is identified by the data-structure as a branch cut, and algorithm (1) shows that the index should be 0 on it and 2πi on the next cell c6 = (3, 3). Finally the remaining steps of the WS are (2,3),(1,3), and (1,2) where the index remains at 2πi, and we discover that c0 in fact belongs to the sheet with index 1.

4.2 Remarks A major motivation for presenting algorithm (1) as a special case of algorithm (2) is due to the extra computational cost involved in steps 1 and 2; particularly the latter as we effectively double the number of polynomials in our semialgebraic set. Fortunately the cost of CAD is doubly exponential only in the number of variables[17], and is polynomial time in the number of polynomials in the input formula. If we are to apply (4.1) to examples where the Gi (z) are to admit nth roots, then we work recursively, and we use the fact that Mg ( n p(z)) = n1 Mg (p(z)). However we require that the inputs to the roots (recursively) are single ‘root monomials’, as defined in [4], only. In this case, paths must terminate on or before encountering branch cuts deriving from an nth root, depending on their adherence. The general case remains to be done. An another advantage of WS is that they solve yet another problem, not mentioned in (1.2) that occurs in the sample point testing phase. Numerical evaluation will produce in most cases an arbitrary looking floating point number. What would be more useful would be to determine a symbolic correction factor for the proposed formula H = 0 where required. As in [8] one can try to determine this by calculating error bounds and choosing the branch with the closest value at p although this is not always possible. It also requires that we prohibit inverse elementary functions appearing in denominators5 for we need to ensure separation of the set of values obtained by evaluating each branch at the point p. Working symbolically will be better suited to achieving this goal, although this will be more costly in general, and it will not reveal what branch we are on for each of the individual Hi in Φ. WS do provide this information, and it may be of great interest to the user. The cost of accessing the data-structure is the only price we pay when determing the Riemann index on many of the cells. Thus, the loss of efficiency caused by the overrepresentation of D mentioned earlier should be somewhat offset by the gains made here, without the need for clustering. This should be useful for cases where our CAD comprises several thousand cells6 .

2. H = Log(−1 − z 2 ) − Log(i − z) − Log(i + z). The sets of indices for the reachable sheets of each logarithm are {0, 1}, {0, 1} and {0} respectively, so we need to compute a WS for the first two logarithms shown. For brevity, we just focus on the first logarithm with the others being similar to (4.1.1). We compute the Mgi , argument branches ai and branch cuts bi for each logarithm. For the first one, L1 , we obtain (−2π, 3π] and the line segments a1 = (i, i∞) and a2 = (−i, −i∞), (an easy computation shows that this is L0 ) as well as b1 = (−i, i), and b2 being the x − axis. Figure (4) shows what RI (L1 ) is alone on each of the cells. Now a possible set of paths are as follows. l1 = {(2, 1), (1, 1)}, l2 = {(2, 1), (3, i), (2, 7), (1, 7), (1, 6)} where i = 1, . . . 7 and l3 = {(3, 3), (2, 3), (1, 3), (1, j)} where j = 2, . . . 5. Now l1 , l2 were terminated on finding a cell where the RI (L1 ) cannot be found without deducing, via adjacent cells, the value of Arg(p) with p ∈ L1 at its sample point. The li are efficient

5 6

43

See [8] for examples. This is not unusual, see [5] for examples.

5.

CONCLUSION & FUTURE WORK

[12] Corless, R., Davenport, J., Jeffrey, D. & Watt, S. According to Abramowitz and Stegun. SIGSAM Bulletin 34, 2 (2000), 58–65. [13] Corless, R. M. & Jeffrey, D. J. The Unwinding Number. SIGSAM Bull. 30 (1996) 2, issue 116, 28–35. [14] Corless, R., Gonnet, G., Jeffrey, D., Hare, D. & Knuth, D. On the Lambert W Function. In Adv. in Computational Mathematics 5, (1996), 329–359. [15] Davenport, J. H., The Geometry of Cn is Important for the Algebra of Elementary Functions., in Algebra, Geometry, and Software Systems 207-224,(2003). [16] Davenport, J. H., MKM from Book to Computer: A Case Study In MKM 2003 17–29, A. Asperti, B. Buchberger, J. H. Davenport, Ed., Springer. [17] Davenport, J. and Heintz, J. Real Quantifier Elimination is Doubly Exponential. J. Symbolic Computation 5 , (1988), 29–35. [18] Dolzmann. A, Seidl. A & Sturm. T. Efficient Projection Orders for CAD. In Proc. ISSAC (2004), Jaime Gutierrez, Ed., ACM, New York, 111–118. [19] Fateman, R. J. & Dingle, A. Branch Cuts in Computer Algebra. In ISSAC 1994, ACM Press, New York , 1994, 250–257. [20] Grigoriev, D. Yu. & Vorobjov, N. N.Jr. Counting connected components of a semi-algebraic set in subexponential time. Computational Complexity 2 (1992) 133–184. [21] Gabrielov, A. & Vorobjov, N. Complexity of cylindrical decompositions of sub-Pfaffian sets. J. Pure Appl. Algebra 164 (2001), 179–197. [22] Jeffrey, D. J. & Norman, A. C. Not seeing the roots for the branches: multivalued functions in computer algebra. SIGSAM Bulletin vol 38 (2004) no. 3, issue 149, 57–66. [23] Moses, J. Algebraic Simplification, a Guide for the Perplexed. In Advances in Computational Mathematics 14 , no.8 (1971). [24] H. Hong et al. Quantifier Elimination by Partial Cylindrical Algebraic Decomposition. http://www.cs.usna.edu/~qepcad/B/QEPCAD.html [25] Richardson, D. Some Unsolvable Problems Involving Elementary Functions of a Real Variable. Journal of Symbolic Logic 33(1968), 514–520. [26] Richardson, D. How to Recognize Zero. In Journal of Symbolic Computation (1994), 24(6), 627–646. [27] van der Hoeven, J. A new Zero-test for Formal Power Series. In ISSAC 2002 (2002), T. Mora, Ed., ACM, New York, 117–122.

We have seen that the adherence method in section 2 deals with some of the more difficult issues associated with the Decomposition Method in an efficient way. The only drawback in terms of complexity is that we may have to use an adjacency algorithm. It would still be nice to implement the method so that we may attempt larger examples and see what sort of run times we obtain. The Winding Sequence approach avoids the problems of unlucky sample points, the constant problem and models well the underlying cause of the problem of why multivalued formula can fail to hold; it also has good long term potential for larger function fields, provided we can still generate a decomposition with respect to the cuts. Indeed, methods do exist for the larger class of Pfaffian functions[21]. Formal justification for several variables may be harder to achieve however, for both methods. Until such a time it would be interesting to choose examples which can be verified by other means and see if any counter-examples can be found. Although the complexity (at the last step of the Decomposition Method) of producing CADs with large numbers of cells has been reduced by our method, it would clearly advantageous to produce smaller CADs if possible. One approach to this is to obtain a more accurate description of the branch cuts in the first place; we shall describe this later in [6].

6.

REFERENCES

[1] Abramowitz, M. & Stegun, I., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. J. Wiley & sons inc., May 1984. [2] Arnon, D. S., A cluster-based CAD Algorithm. In J. Symbolic Computation, 5(1/2) 189–212, 1988. [3] Beaumont, J., Bradford, R. & Davenport, J. Better Simplification of Elementary Functions Through Power Series. In ISSAC 2003 (2003), J.R. Sendra, Ed., ACM, New York, 30–36. [4] Beaumont, J., Bradford, R., Davenport, J. & Phisanbut, N. A Poly-Algorithmic Approach to Simplifying Elementary Functions. In ISSAC 2004 (2004). J. Gutierrez, Ed., ACM, New York, 27–34. [5] Beaumont, J. C., Bradford, R. & Phisanbut, N. Practical Simplification of Elementary Functions using CAD. In Algorithmic Algebra and Logic 2005. [6] —. Effective Simplification of Elementary Functions in preparation. [7] Bradford, R., Corless, R., Davenport, J., Jeffrey, D. & Watt, S. Reasoning about the Elementary Functions of Complex Analysis. Ann. Math. & Artificial Intelligence 36, (2002), 303–318. [8] Bradford, R. & Davenport, J. Towards Better Simplification of Elementary Functions. In Proc. ISSAC (2002), T. Mora, Ed., ACM, New York, 15–22. [9] Carette, J. Understanding Expression Simplification. In ISSAC 2004 (2004), Jaime Gutierrez, Ed., ACM, New York, 72–79. [10] Collins, G. Quantifier Elimination for Real Closed Fields by Cylindrical Algebraic Decomposition. Proc. 2nd. GI Conf. Automata Theory & Formal Languages (1975), vol. 33 of SLNCS. [11] Collins, G.& McCallum, S. Local box adjacency algorithms for Cylindrical Algebraic Decomposition. In J. Symbolic Computation, 33(3):321–342, 2002.

APPENDIX The adherence of the base inverse elementary functions, based on the definitions in [12], is given below. Function arcsin(z) arccsc(z) arcsinh(z) arccsch(z) arccos(z) arccosh(z) arcsec(z) arcsech(z) arctan(z) arccot(z) arctanh(z) arccoth(z)

44

Branch cut (−∞, −1) (−1, 1) (−i∞, −i) (−i, 0) (−∞, −1) (−∞, 1) (−1, 1) (−∞, 0] (−i∞, −i] [−i, i], (−∞, −1] [−1, 1]

Closure =>0 =>0 <<0 <>0 =>0 =>0 =>0 =>0 <<0 =>0 =>0 =<0

Branch cut (1, ∞) — (i, i∞) [0, i) (1, ∞) — — [1, ∞) [i, i∞) — [1, ∞) —

Closure =<0 — <>0 <<0 =<0 — — =>0 <>0 — =<0 —

Fast Algorithms for Polynomial Solutions of Linear Differential Equations Alin Bostan

Thomas Cluzeau

Bruno Salvy

Algorithms Project Inria Rocquencourt 78153 Le Chesnay (France)

LACO, Universite´ de Limoges 123, avenue Albert Thomas 87060 Limoges (France)

Algorithms Project Inria Rocquencourt 78153 Le Chesnay (France)

[email protected]

[email protected]

[email protected]

ABSTRACT

of nonzero polynomial solutions. Algorithms for computing Liouvillian solutions [23, 25, 26, 20] or definite integrals [3, 13] construct intermediate LDEs and need to check whether they possess nonzero rational solutions. In most cases no rational solution exists and it is therefore important to be able to detect this property efficiently. Another application is the desingularization of linear differential or difference equations that has been shown [14] to boil down to polynomial solutions of an adjoint linear differential equation. In order to compute polynomial solutions, the algorithms [2, 1, 4] basically compute a basis of power series solutions at a fixed point (0 or infinity) and search for a linear combination of those which is a polynomial. The complexity is related to the largest integer valuation N of Laurent series solutions at infinity, which can be computed from an indicial polynomial (this is developed in §2.2). This integer N can be exponentially large compared to the bit size of the input. For instance, the polynomial (1 + x + x2 )N is a solution of the first order differential equation

We investigate polynomial solutions of homogeneous linear differential equations with coefficients that are polynomials with integer coefficients. The problems we consider are the existence of nonzero polynomial solutions, the determination of the dimension of the vector space of polynomial solutions, the computation of a basis of this space. Previous algorithms have a bit complexity that is at least quadratic in the largest integer valuation N of formal Laurent series solutions at infinity, even for merely detecting the existence of nonzero polynomial solutions. We give a deterministic algorithm that computes a compact representation of a basis of polynomial solutions in O(N log3 N ) bit operations. We also give a probabilistic algorithm that computes√the dimension of the space of polynomial solutions in O( N log2 N ) bit operations. In general, the integer N is not polynomially bounded in the bit size of the input differential equation. We isolate a class of equations for which detecting nonzero polynomial solutions can be performed in polynomial complexity. We discuss implementation issues and possible extensions.

(1 + x + x2 )y 0 (x) − (2xN + N )y(x) = 0. The bit size of the input is of order log N , while the bit size of the output (in its monomial representation) is of order N . This shows that computing all coefficients of the polynomial solution has at least exponential bit complexity with respect to the size of the input. Another example is

Categories and Subject Descriptors: I.1.2 [Symbolic and Algebraic Manipulation]: Algorithms General Terms: Algorithms, Experimentation, Theory Keywords: Computer algebra, Linear differential equations, Polynomial solutions, Linear recurrences, Complexity

1.

(x − 1)2 (x + 1)y 0 (x) − ((x − 1)2 N + x + 1)y(x) = 0 with solution (1 + x)N exp(x/(1 − x)). The valuation at infinity is bounded by N , but there are no nonzero polynomial solutions. For this particular example, it is possible to detect the absence of nonzero polynomial solutions by an algorithm whose complexity is only polynomial in the bit size of the equation [18, Cor. 8.43]. This criterion applies to equations of order 1. To our knowledge, an extension to higher orders is known only for generalized hypergeometric equations [10, Th. 2]. We describe in §2.5 a new criterion for a restricted class of equations of arbitrary order, yielding a test that works in polynomial time in the equation bit size. In general, however, we do not know of similar criteria. Our aim is therefore to decrease the dependency on N of the complexity. By computing all the coefficients of power series solutions with index between 0 and N , previous algorithms have an arithmetic complexity (number of arithmetic operations) which is at least linear in N and a bit complexity which is at least quadratic in N because of the size of these coefficients. Our contribution is to use classical techniques

INTRODUCTION

The computation of polynomial solutions of linear differential equations (LDEs) lies at the heart of several important algorithms. Indeed, known algorithms for finding rational solutions of LDEs [22, 25, 2, 1, 4] proceed by computing a common multiple of the polar part of meromorphic solutions and then compute the numerator as a polynomial solution of a derived LDE. Detecting the existence of nonzero rational solutions is thus reduced to detecting the existence

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. ISSAC’05, July 24–27, 2005, Beijing, China. Copyright 2005 ACM 1-59593-095-7/05/0007 ...$5.00.

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for manipulating recurrences with polynomial coefficients that allow for the computation of the N th element or a slice of elements starting at index N in bit complexity roughly proportional √ to N or arithmetic complexity roughly proportional to N . From there, we derive a probabilistic algorithm for the computation of the dimension of polynomial solutions (and in particular the existence of nonzero polyno√ mial solutions) in complexity O( N log2 N ) by performing the computation modulo a suitable prime. We also derive a deterministic algorithm of bit complexity O(N log3 N ) to compute this dimension and a representation of polynomial solutions by a recurrence on the coefficients and initial conditions. Since these initial conditions have bit size O(N log N ), this is close to optimal. This compact representation of polynomials is well suited for applications; for instance, we show that it allows for fast evaluation. If all coefficients of the polynomials are required, then we also obtain a quadratic complexity, because of the bit size of the output. However, compared to previous algorithms, our method is faster by a constant factor, since only the coefficients of the polynomials are computed, instead of all those of power series solutions. This article is structured as follows. In Section 2, we recall the basic method for finding polynomial solutions. We also give a polynomial time detection criterion, for a restricted class of equations. In Section 3, we recall the binary splitting method for linear recurrences and we apply it to computing the compact representation of polynomial solutions. In Section 4, we use the baby-step/giant-step method for the case of modular computations. Extensive experiments together with implementation issues are described in Section 5. Note that, due to space limitations, some proofs will be omitted. They are given in detail in [8], together with various extensions of the present work to polynomial and rational solutions of homogeneous, inhomogeneous or parameterized systems of linear differential or recurrence equations.

with polynomial coefficients is represented by its coefficients yi in the monomial basis (xi ). An alternative data structure for y(x) is a recurrence of order o, together with a set of o initial conditions. Polynomial solutions of LDEs inherit this compact encoding. Let us explain the terminology compact on an example. Consider the LDE mxy 00 (x) − xy 0 (x) + m2 y(x) = 0. The recurrence satisfied by the coefficients yi of a polynomial solution in Z[x] is (m2 − i)yi + m(i2 + i)yi+1 = 0, with the 2 initial conditions y0 = 0 and y1 = (−1)m−1 (m2 )! mm −1 . From that, it follows easily that the equation admits a onedimensional Q-vector space of polynomial solutions of degree N = m2 ; this space is generated by the polynomial ! N X N i+m ym (x) = (−1) (N − 1)! mN −i xi . i i=1 Because of the factorials, to write down all the coefficients of ym (x), we need a number of bits linear in N 2 log(N ). In contrast, the total bit-size of the compact encoding by recurrence and initial conditions is linear in N log(N ). Complexity measures and notations. For our complexity analyzes, the measure we use is the bit (or boolean) complexity. For this purpose, our complexity model is the multi-tape Turing machine, see for instance [24]. We speak of bit operations to estimate time complexities in this model. We use a function I : N → N to denote the bit complexity of integer multiplication, i.e. such that the product of two integers of bit-size d can be computed within I(d) bitoperations. For any prime number p, the bit complexities of the operations (+, −, ×, ÷) in the finite field Fp := Z/pZ are in O(I(log p)). We denote by M : N → N a function that represents the arithmetic complexity of polynomial multiplication, i.e. such that over any ring R, the product of two polynomials of degree at most d can be computed within M(d) base ring operations (each ring operation is counted at unit cost). For computations in Fp [x], the bit complexity is bounded by multiplying the arithmetic cost estimates by the bit complexity of the basic operations in Fp . We assume that the multiplication time functions M and I are super-additive, i.e., M(d1 ) + M(d2 ) ≤ M(d1 + d2 ) and I(d1 ) + I(d2 ) ≤ I(d1 + d2 ) for all positive integers d1 and d2 ; in particular, the inequalities 2 M(d) ≤ M(2d) and 2 I(d) ≤ I(2d) hold for all d ≥ 1. We also assume that M(cd) is in O(M(d)) and that I(cd) is in O(I(d)), for all c > 0. The basic examples we have in mind are classical multiplication, for which M(n), I(n) ∈ O(n2 ), Karatsuba’s multiplication with M(n), I(n) ∈ O(n1.59 ) and the FFT-based multiplication which has M(n), I(n) ∈ O(n log(n) log(log(n))). Our references for these matters are the textbooks [17, 24]. The constant ω denotes the matrix multiplication exponent as in [17, Ch. 12], so that two n × n matrices can be multiplied within O(nω ) arithmetic operations (2 ≤ ω ≤ 3). In the sequel, log x denotes the logarithm of x in base 2; bxc and dxe respectively denote the largest integer less than or equal to x, and the smallest integer larger than or equal to x. We use the term bit-size (or simply size) of an integer a 6= 0 for λ(a) := blog(|a|)c + 1. By convention, we assume that λ(0) = 1. For p(x) ∈ Z[x], we let λ(p) denote the maximum bit-size of its coefficients. Similarly, if M is an integer matrix, we write λ(M ) for the maximum bit-size of its entries. We denote by xi the falling factorial polynomial xi = x(x−1)(x−2) · · · (x−i+1). For a polynomial f (x), the

Previous work. The naive algorithm for polynomial solutions [22, 25] is the indeterminate coefficients method: knowing a P bound N on the degree of solutions, take an i ansatz y = N i=0 yi x where the yi are unknowns, replace y in the equation and solve the homogeneous linear system for the yi . Since the coefficients yi satisfy a linear recurrence with fixed order o, the matrix of this linear system has a particular banded form of bandwidth o + 1. If o  N , this system can be solved by Gauss’s method in O(N ) arithmetic operations [19, §4.3], or O(N 2 log2 N ) bit operations. A variant of this algorithm, called hereafter the basic algorithm, was suggested in [2, 1, 4]. Roughly speaking, it computes a basis of Laurent series solutions at a fixed point (finite or infinite), then searches for a linear combination P of those series, in which the coefficients of xN +1 , . . . , xN +o are all zero (this suffices to ensure that P is a polynomial solution, since its coefficients satisfy a recurrence of order o). Regarding only the dependence in N , the complexity of the algorithms in [2, 1, 4] is asymptotically the same as that of the naive algorithm. If a solution of degree N exists, both algorithms are nearly optimal (up to logarithmic factors) for computing the monomial representation of this solution, since the bit size of the latter is of order O(N 2 log(N )). In the opposite case, none of them provide a faster way of testing the nonexistence of nonzero polynomial solutions. Compact representation of polynomial solutions. Classically, a truncated power-series solution y(x) of a LDE

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basis xi have bit size at most `, then λ(p) ≤ ` + 2n log(n). (iii) If p(x) ∈ Z[x] has degree n and if a ∈ Z, the coefficients of the polynomial p(x + a), in particular p(a), have bit-size bounded by λ(p) + n + nλ(a). (iv) Let M be a m × m matrix with entries in Z[x] of degree at most n and with coefficients of size at most `. Then λ(M (N ) · · · M (1)) ≤ N (` + n + log(m) + n log(N )).

integers val(f ) and deg(f ) stand respectively for the usual x-adic valuation and for the P degree in x of f . n i All along this text, L := i=0 ai (x) ∂ denotes a linear d differential operator of order n in ∂ := dx , where ai (x) are polynomials in Z[x], whose gcd is 1 and an 6= 0 (these hypotheses are made only to simplify complexity estimates). The integers d and ` denote a bound on the degrees of the ai (x), respectively on the bit-size of their coefficients.

2.

Using Lemma 1 and [17, Th. 15.21], we get the following bounds on the bit-size of the possible degrees of polynomial solutions and on the complexity to compute them.

DIFFERENTIAL EQUATIONS AND RECURRENCE RELATIONS

In §2.1 we recall well-known facts concerning recurrences and indicial polynomials, in §2.2 we show how to bound the degree of polynomial solutions in terms of the order of L and of the degree and bit-size of its coefficients. We add a (possibly new) complexity analysis. In §2.3, we describe the basic algorithm for polynomial solutions and estimate its bit-complexity. In §2.4, we reformulate the basic algorithm, so as to pave the way for the algorithmic improvements given in §3 and §4. Finally, in §2.5 we give a new (polynomialtime) detection criterion for nonzero polynomial solutions.

Lemma 2. We can compute a bound N on the degree of polynomial solutions of L in Olog (n2 (` + n)) bit operations. Moreover, λ(N ) ≤ ` + 2 n log(n).

2.1

In conclusion, the recurrence (2) and the bound N can be computed within Olog (n(n + o)(n + `)) bit operations, provided FFT is used. When n+o ∈ O(o) (e.g., if x = 0 is an ordinary point, so that o ≥ n), this is optimal with respect to the bit-size of the recurrence, which is in Olog (on(n + `)).

Applying again Lemma 1, we estimate the cost of computing the recurrence associated to a LDE. Lemma 3. Let L be a LDE of order n. The polynomials ui (x) defining recurrence (2) can all be computed using O(o M(n) log(n) I(` + n log(n))) bit operations.

Recurrences and indicial polynomials

To L =

i i ai ∂ we attach the nonnegative integers:

P

α := − min(val(ai ) − i) i

and

β := max(deg(ai ) − i). (1) i

P The ai (x) can thus be written ai (x) = βj=−α ai,j xi+j for some ai,j ∈ Z. For Pj ∈ {−α, i. . . , β}, we introduce the polynomials uj (x) := n i=0 ai,j x . By definition, α ≤ n, with equality if and only if x = 0 is an ordinary point. Moreover, if x = 0 is ordinary, then u−α (x) = an,−α xn has roots 0, 1, . . . , n − 1. P i If y = N i=v yi x is a polynomial solution of L satisfying yv 6= 0 and yN 6= 0, then a direct calculation shows that for all i ∈ Z, the coefficients yi satisfy the linear recurrence u−α (i + α) yi+α + · · · + uβ (i − β) yi−β = 0.

2.3

(2)

The polynomials uβ (resp. u−α ) are called the indicial polynomial of L at infinity (resp. at the origin). In what follows, we use the notation o = α + β for the order of the recurrence (2). Plugging i = N + β (resp. i = v − α) in recurrence (2), we see that the possible degrees (resp. valuations) of the polynomial solutions of L lie among the positive integer roots of the indicial polynomial of L at infinity (resp. at the origin). More generally, these roots give all the possible valuations at infinity (resp. 0) of power series solutions. This is why a root of the indicial equation at infinity may not correspond to a polynomial solution. From now on, the integer N = NL will denote the largest positive integer root of the indicial polynomial uβ ; it is a bound on the possible degrees of polynomial solutions of L. The notation Olog () indicates the presence of terms depending polynomially in log(`), log(n), log(d) and log(log(N )).

2.2

Basic Algorithm

The basic algorithm described in [2, 1, 4] uses the linear recurrence (2) to compute a basis of power series solutions of L at a point (say at x = 0), then search for linear combinations of these power series which yield polynomial solutions. For simplicity, in the rest of the paper (with the exception of §2.5) we suppose that x = 0 is an ordinary point for L. Since the number of singularities of L is at most d, this hypothesis is not restrictive: it boils down to finding an integer z that lies outside the roots of an (x), then to performing a shift by z of all the coefficients ai (x). Such a point z can be computed, together with the required shifts, using Olog (nM(d2 + d`)) bit operations [16, Th. 2.4]. Since x = 0 is ordinary, Cauchy’s theorem guarantees the existence of a basis of power series solutions yi (x) of L, of valuations 0, 1, . . . , n − 1. Taking as input the coefficients of L and an integer N , the basic algorithm outputs a basis of the polynomial solutions of L of degree at most N . The version of this algorithm which is presented in [1] proceeds as follows: (1) For 0 ≤ i ≤ n − 1, use the recurrence (2) to compute one by one coefficients of a power series solution Pthe N +o yi = xi + j=n yi,j xj + O(xN +o+1 ); ˆ ˜N +1≤j≤N +o (2) Form the n×o matrix M = yij 0≤i≤n−1 and compute a basis B ⊂ Qn of solutions of M t · c = 0; Pn−1 (3) For every (c0 , . . . , cn−1 ) ∈ B, output i=0 ci y i .

Bound on degree of polynomial solutions

The following result gathers useful bounds. We give here precise bounds rather than O() estimates. They will be necessary when we give precise bounds on probabilities in §4.2.

A direct analysis shows that, supposing that FFT is used and taking into account only the dependence in N , the bit complexities of steps (1–3) are O(N 2 log(N )), O(N log2 (N )) 2 and O(N 2 log respectively, so that the total complex` 2(N )), ´ ity reads O N log2 (N ) . Note that, if a nonzero polynomial

Lemma 1. Let n, m, ` and N be positive integers. Then: (i) λ(xn ) ≤ n log(n). (ii) If p(x) ∈ Z[x] has degree n and if its coefficients in the

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solution of degree O(N ) exists, then this complexity is almost optimal with respect to the size of the output, in its monomial representation. However, if there are polynomial solutions with degrees bounded by a certain d  N , then the previous algorithm loses its optimal character; in other words, it is not output-sensitive. Moreover, if only a partial information (e.g., the existence of nonzero polynomial solutions or their dimension) is required, the basic ` ´ algorithm cannot provide it using less than O N 2 log(N ) operations. In the variant [2, 4], the recurrence (2) is used backwards starting from indices N + 1, . . . , N + o and leads to a linear system on the yij for small j. We only analyzed precisely the algorithm presented above, but its main features are shared by this variant. Our improvements described in §3.2 (based on binary splitting) and in §4.2 (based on baby-steps / giant-steps ) apply equally well to this variant.

2.4

In the next sections, we give bit complexity estimates for this operation in the case when R = Z and respectively R = Fp , for a prime p. We then apply these results to obtain improved versions of the basic algorithm. Note that in our applications, we consider slightly more general products C(N ) · · · C(U ) with U possibly 6= 1, but there is no loss in treating only the case U = 1.

2.5

P i Corollary 1. Let L = n i=0 ai ∂ ∈ Z[x][∂] be a linear differential operator such that deg(ai ) ≤ i, for i = 0, . . . , n. Then L has a nonzero polynomial solution if and only if its indicial polynomial at infinity has a positive integer root. This can be detected using Olog (n2 (` + n)) bit operations.

Towards Improving the Basic Algorithm

It is classical that linear recurrences of arbitrary order rewrite as first order matrix recurrences. Indeed, if we let Yi := t (yi , yi+1 , . . . , yi+o−1 ), the linear recurrence (2) reads Yi+1 = R(i) Yi ,

for all

i ≥ 0,

A new detection criterion

We end this section by a corollary of Proposition 1. It provides a criterion that allows to detect existence of polynomial solutions without performing the basic algorithm.

Proof. We can suppose that x = 0 is an ordinary point, since translations preserve the degrees of the coefficients ai (x) and do not affect the dimension of the space of polynomial solutions of L. Now, since x = 0 is ordinary, it follows that α = n. The hypothesis on the degrees ensures that β = 0. In particular, the matrix I in Proposition 1 is the identity matrix. Proposition 1 ensures that L has a nonzero polynomial solution if and only if the matrix C(N ) · · · C(−β) is singular, thus if and only if one of the matrices C(N ), . . . , C(−β) is singular. Now, since u−α (i+o) 6= 0 for i = −β, . . . , N , the companion-type matrix C(i) is singular if and only if uβ (i) = 0. This leads to the first conclusion. The complexity estimate follows from Lemma 2.

(3)

1 u−α (x+o)

× C(x) and where C(x) is the folwhere R(x) = lowing o × o companion-type matrix with entries in Z[x]: 2 3 0 u−α (x + o) 0 0 6 7 .. .. .. 6 7 . 0 . . C(x) = 6 7 4 5 0 0 ··· u−α (x + o) −uβ (x) −uβ−1 (x + 1) · · · −u−α+1 (x + o − 1) The following result is an easy consequence of Lemma 1.

3.

Lemma 4. The coefficients of the entries of C(x) have bit sizes bounded by ` + n + 3n log(o) = Olog (` + n).

3.1

THE BINARY-SPLITTING METHOD

Matrix factorials over Z Let N ≥ 0 and let C(x) be an m×m matrix with entries in Z[x] of degree at most n and whose coefficients have bit-size bounded by `. We wish to compute the matrix with integer entries A = C(N ) · · · C(1). By Lemma 1, the bit `size` λ(A) is bounded by N´´(` + n + log(m) + n log(N )) = O N ` + log(m) + n log(N ) . The simplest particular case C(x) = x corresponds to the computation of factorials. The naive method (multiply i + 1 by i! for increasing i) has complexity O(N 2 log(N )), that is, quadratic in the bit size of N !. A well-known improvement mentioned in [7, Prop. 1], but possibly known recursively the formula i(i ´ ` ´ + 1) · · · (i + j) = `before, is to use e · d i+j + 1e · · · (i + j) . i(i + 1) · · · d i+j 2 2 This technique, called binary splitting (or divide-andconquer) becomes useful in conjunction with fast integer multiplication. For N ! it yields an algorithm of complexity O(I(N log(N )) log(N )) bit operations, which is almost linear in the bit-size N log(N ) of the output, if FFT is used. These facts are classical. They were rediscovered in various particular cases: base conversion, computation of places of π and e [11], computation with linear recurrences [21], [12, §6]. The earliest historic trace we could find is [5, Item 178], which suggests (without proof) that binary splitting allows for fast evaluation of solutions of LDEs. We refer to [6, §12] for a good survey on binary splitting-based algorithms. We summarize the complexity analysis for integer matrix factorials in the result below (cf [12, Th. 6.1]).

Since x = 0 is supposed to be an ordinary point, the indicial polynomial u−α (x) at the origin has roots 0, 1, . . . , n−1, so that u−α (x + o) 6= 0 for all x ≥ −β. This ensures that recurrence (3) can be used for i ≥ −β. The key point in what follows is P that the only coefficients of a power series solution y(x) = yi xi which are needed in Step (2) of the basic algorithm can be computed by the equation YN +1 = R(N ) · · · R(k) Yk , for all k ≥ −β. This formula will be exploited so as to avoid the computation of all the intermediate coefficients yn , . . . , yN −1 . The following result is a simple consequence of these considerations; it contains the theoretical basis needed in the next sections to improve the complexity of the basic algorithm. For m, n ≥ 1, we denote by Om,n the zero matrix of sizes m × n and by Idn the n × n identity matrix. Proposition 1. ˛ ˆ ˜ Let I be the transpose of the n × o matrix On,o−n ˛ Idn . Then, the matrix M t defined in Step (2) of the basic algorithm equals R(N ) · · · R(−β)I. Moreover, a vector c ∈ Qn contains the coefficients of 1, x, . . . , xn−1 of a polynomial solution of L if and only if C(N ) · · · C(−β)Ic = 0. To put this theoretical result into practice, the basic computational task is the following: Matrix factorials: given a ring R, an integer N ≥ 0 and a square matrix C with coefficients in R[x], compute the product C(N ) · · · C(1).

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by λ = O(h + ` N ). To simplify the exposition, we assume that h, `  N and that FFT is used for integer multiplication. Horner’s algorithm allows to perform the evaluation in O(λ N ) = O(N 2 ) bit operations. A better algorithm, of divide-and-conquer type, was suggested in [15]. of this algorithm ` Using ´FFT, the ` bit complexity ´ is O λ log2 (N ) = O N log2 (N ) . Neglecting log factors, this complexity is nearly optimal with P respect to the bit size i of the output. Now suppose that P = N i=0 pi x is given by a recurrence of order o (with polynomial coefficients) satisfied by the coefficients pi , together with the first o values p0 , . . . , po−1 . Let us assume that the total bit size of the recurrence is negligible compared to N . Now, P (a) equals P the N th term of the sequence vk := ki=0 pi ai , which satisfies a recurrence of order o + 1 with polynomial coefficients. Therefore, by Lemma 5, P (a) can be computed by binary splitting in O(oω I(λ) log(N )) = O(N log2 (N )) bit operations, which is again almost linear in the bit size of the output. Similar considerations can be made for evaluation of P and its derivatives at rationals or algebraic numbers.

Lemma 5. With the notations above, we ´can compute the ` matrix factorial A in O mω I(λ(A)) log(N ) bit operations.

3.2

Application to Polynomial Solutions

We now use binary-splitting to give an improved version of the basic algorithm. It computes the polynomial solutions of L, in compact representation by a recurrence and initial conditions. Neglecting logarithms, its complexity is linear in N , compared to the quadratic cost of the basic algorithm. Theorem 1. Let N ≥ d. The number of bit operations used by the algorithm BinSplitPolySols below to compute the polynomial solutions of L of degree at most N is in Olog (oω I(o`N + on N log(N )) log(N )). BinSplitPolySols Input: L ∈ Z[x][∂] with x = 0 ordinary, N ≥ 1. Output: the compact representation of a basis of polynomial solutions of L of degree ≤ N . 1. Compute the recurrence (2) and the matrix C;

4.

2. Compute the matrix Ainit = C(−1) · · · C(−β) I;

In this section, we devise a probabilistic algorithm which outputs the dimension of the Q-vector space of polynomial solutions of a linear differential operator L in Z[x][∂]. Our contribution is to use a classical technique for manipulating linear recurrences with polynomial coefficients that allow for the computation of the N th element or a slice of elements starting at index √ N in arithmetic complexity roughly proportional to M( N ). From there, we derive a probabilistic algorithm for the computation of the dimension of polynomial solutions (and in particular the existence of nonzero √ polynomial solutions) in complexity O(M( N ) I(log N )) by performing the computation modulo a suitable prime.

3. Compute the matrix A = C(N ) · · · C(0) Ainit ; 4. Compute a basis B ⊂ Qn of solutions of Ac = 0; 5. For every c = t(c1 , . . . , cn ) ∈ B, output Ainit · c. To prove Theorem 1, we need a technical lemma, whose proof is omitted. Lemma 6. The entries of Ainit have bit sizes bounded by d(6n log(o) + n log(d) + `) = Olog (`d + nd) and the entries of A have bit sizes bounded by N (6n log(o) + n log(N ) + `) + 2d(6n log(o)+n log(d)+`) = Olog (`N +nN log(N )+dn+d`). Proof of Th. 1. By Lemma 3, Step 1 can be done using O(o M(n) log(n) I(` + n log(n))) = Olog (o M(n) I(` + n)) bit operations. The matrices Ainit and A are then computed by binary splitting. By Lemma 6 and since N ≥ d, the coefficients of their entries have bit sizes bounded by a certain Γ = Olog (`N +nN log(N`)). Using Lemma ´ 5, the complexity of Steps 2 and 3 is in O oω I(Γ) log(N ) bit operations. In step 4, one has to solve a o × n linear system with integer coefficients of size Γ. Doing linear algebra ` modulo ´ a prime of size oΓ, this can be performed in Olog oω I(oΓ) bit operations. Now, the entries of c have bit size in Olog (o Γ) and the entries of Ainit have bit size in Olog (Γ). Thus, ` in the´ final step, each product Ainit ·c can be done in Olog n I(o Γ) bit operations and there are at most n ≤ o such products. Putting these estimates together concludes the proof.

3.3

BABY STEPS / GIANT STEPS

4.1

Matrix factorials in positive characteristic

Chudnovsky and Chudnovsky proposed in [12] an algorithm that determines one term of a linear sequence with polynomial coefficients without computing all intermediate terms. Using the so called baby steps / giant steps technique, it requires a number of operations which √ is roughly linear in N to compute the N th term in the sequence. The Chudnovskys rediscovered and generalized Strassen’s algorithm [27, Abschnitt 6] (used for deterministic integer factorization) which computes N ! mod p for a prime p > N . Strassen’s idea√ is to build the polynomial M (x) = (x + 1) · · · (x + √ N ), then to evaluate √ it at the arithmetic progression 0, N , . . . , N − N using fast multipoint evaluation techniques and to output √ √ N ! = M (N − N ) · · · M ( N ) M (0) (these computations are done over the field Fp ). The Chudnovskys’ result proceeds in a similar way for any recurrence, by rewriting it as a 1st order matrix recurrence uN +1 = C(N + 1)uN , where C is a matrix with rational fraction entries. We recall the complexity result (cf. [12, Th. 6.2]).

Evaluation in Compact Representation

When nonzero polynomial solutions exist, the output of our algorithm BinSplitPolySols is distinct from that of the basic algorithm: instead of the (dense) monomial representation, it returns an encoding of the polynomial solutions by a recurrence and by initial conditions for their coefficients. We show that this data structure is well suited for fast evaluation. Consider a polynomial P of degree N with integer coefficients of bit size h. Let a ∈ Z be an integer of bit size ` and consider the problem of evaluating P at a; by Lemma 1, the bit size of the output P (a) is bounded

Lemma 7. Let F be a finite field and let C be a m × m polynomial matrix over F with entries of degree at most d. Let N > d. Then, the number of operations in F required the matrix factorial A” = C(N ) · · · C(1) “ to compute √ √ is O m2 M( dN ) log(dN ) + mω M( dN ) .

49

be done in O(m2 M(k)) operations in F . Finally, the values M(i) are recovered using M(i) = M1 (i)M0 (i), for 0 ≤ i ≤ k. The complexity of the algorithm satisfies the equation C(k) ≤ C( k2 ) + O(m2 M(k) + kmω ), whence C(k) = O(m2 M(k) + mω k), as required.

Improvement of the Chudnovskys’ algorithm. The algorithm of Lemma 7 spends an important amount of computation in evaluating a polynomial matrix at points in arithmetic progression. √ Now, this matrix has the special shape M (x) = C(x + N ) · · · C(x + 1), where C is a polynomial matrix. This allows us to improve the complexity result in Lemma 7 by a factor of log(N ). Roughly, the speed-up is due to the use of the operation of extrapolation defined below, which can be performed faster than multipoint evaluation.

4.2

Application to Polynomial Solutions

Let L ∈ Z[x][∂] be a linear differential operator and let N be a bound on the degree of polynomial solutions of L. By choosing a prime number p of order O(N log N ), the method of the previous section allows us to give a probabilistic algorithm to compute the dimension of the space of polynomial √ solutions of L. It uses O(M( √ N ))2operations in Fp , so its bit complexity is of order O( N log (N )) if FFT is used. We need the next result, which easily follows from Hadamard’s bound [17, Th. 16.6] and Zippel’s Lemma [17, Th. 18.9].

Lemma 8. [9, Th. 1] Let F be a field, let d ≥ 0 and let P be in F [x] of degree at most d, such that the sequence P (0), P (1), . . . , P (d) is known. Then: • simple extrapolation: the values P (d+1), . . . , P (2d+1) can be computed using 2 M(d) + O(d) operations in F ; • extrapolation of arbitrary difference: if r ∈ F is so that the elements r − d, . . . , r + d and 1, 2, . . . , d are nonzero in F , then the sequence P (r), P (r + 1), . . . , P (r + d) can be computed using M(d) + O(d) operations in F .

Lemma 9. Let A be an m × n matrix with integer entries of size at most Ω bits. For a prime p, denote by A[p] the matrix over Fp whose entries are the reductions modulo p of the entries of A. Let B ∈ N. Then, for a prime p chosen uniformly at random in the set of prime numbers between B and 2 B, the ranks of A and A[p] are equal with probability at least 1 − (n log(n) + 2nΩ)/B.

In [9, Th. 2], the use of Lemma 8 led to an improvement of Lemma 7 in the case when all the entries of C are linear forms. We now give a straightforward extension of [9, Th. 2] to an arbitrary matrix C; for completeness, we also give a sketch of the proof. Let us briefly explain the basic algorithmic idea on the example of the usual factorial. As in Strassen’s algorithm, we need to evaluate the polynomial M (x) = (x + 1) · · · (x + n) at √ the arithmetic progression 0, n, . . . , n2 − n, where n = N . Suppose that we have already evaluated (x + 1) · · · (x + n/2) on half of the points. It then suffices to evaluate this polynomial on the other half, and also, to evaluate (x+n/2+1) · · · (x+n) on all points. Due to the form of the polynomials, both tasks can be done using only extrapolations, and without constructing the polynomials to be evaluated. Similarly, in the matrix case, the advantage of our algorithm is that, contrary to the Chudnovskys’ algorithm, it does not require polynomial matrix multiplications, but only scalar matrix multiplications.

We now state the main result of this section. Theorem 3. Let c ≥ 0 and N ≥ n + d. The algorithm ModBsGsPolySols below computes the dimension of the Qvector solutions of L of degree ≤ N”using “`space of polynomial √ √ ´ ` ´ Olog (n + d)2 M( nN ) + (n + d)ω nN I log(N ) bit operations. It chooses uniformly at random a prime number in {B, . . . , 2B}, where B = Olog (n2 `N log1+c (N )) and outputs the correct result with probability at least 1 − 2 log1c (N ) . ModBsGsPolySols Input: L ∈ Z[x][∂] with x = 0 ordinary, N ≥ 1. Output: the dimension of the Q-vector space of the polynomial solutions of L of degree at most N .

Theorem 2. Let F be a finite field of characteristic p and let C be a m × m polynomial matrix over F with entries of √ degree at most d. Let N > d and suppose that p > dN + 1. Then, the matrix A = C(N ) ·”· · C(1) can be com“ factorial √ √ puted using O m2 M( dN ) + mω dN operations in F .

1. Choose p prime in {B, . . . , 2B} (B defd below); 2. Compute the reduction C[p] of C modulo p; 3. Compute A[p] = C[p](N ) · · · C[p](−β) I[p];

Proof. For the ease of exposition, let us suppose that √ N and p d are perfect powers of 4, so that k = dN and n = N/d are powers of 2. The general case can be treated with no extra difficulty (by writing N in base 4). Let M(x) be the polynomial matrix C(nx + n) · · · C(nx + 1). Since the required scalar matrix A equals M(k − 1) · · · M(0), it suffices to evaluate M on 0, 1, . . . , k. For this, we use a recursive algorithm. Suppose that the values of the matrix M0 (x) = C(nx + n2 ) · · · C(nx + 1) at the points 0, 1, . . . , k2 are already known. Let M1 (x) = M0 (x + 21 ), so that M(x) = M1 (x)M0 (x). Since the degree of M0 is at most k/2, the values of M0 at 0, 1, . . . , k + 1 can be deduced using m2 simple extrapolations, in complexity O(m2 M(k)). The values of M1 at 0, 1, . . . , k + 1 can be deduced by two extrapolations (of difference 1/2, in degree k/2) of the values of M0 at 0, 1, . . . , k + 1. Since p > k + 1, the elements 1, 2, . . . , k + 1 and 1/2 − k/2, . . . , 1/2 + k/2 are nonzero in F , so these final two extrapolations can also

4. Return the integer n − rank(A[p]). Proof. Let us first study the correctness of the algorithm. Let, as before, o = α + β denote the order of the recurrence (2). Since x = 0 is ordinary, we have n ≤ o ≤ n + d and the indicial polynomial u−α (x) has the form ax(x − 1) · · · (x − n + 1), where a has size bounded by `. On the other hand, since p > d + n + N , none of the elements −β +o−n+1, . . . , o+N is zero in Fp . This ensures that u−α (i + o)[p] is invertible in Fp , for all −β ≤ i ≤ N , provided that p does not divide a. Therefore, if p is not a divisor of a, the recurrence (3) can be used for −β ≤ i ≤ N and the conclusion of Prop. 1 still holds for L[p] (over Fp ). In other words, the algorithm returns the dimension of the Fp -vector space of the solutions in Fp [x] of L[p]. Let us call p a good prime if p does not divide a and if, simultaneously, the matrices A and A[p] have the same rank. In short, we have just proved that if the algorithm

50

Example 1 BinSplit LFS Compact Expand 213 2.23 0.29 0.38 214 6.47 0.62 1.48 215 25.98 1.45 7.11 216 174.17 3.74 45.52 217 1358.01 10.63 318.88 218 > 4Gb 14.31 > 4Gb 220 74.52 222 398.6 24 2 > 1h Example 2 N LFS BinSplit BsGs LFS 210 0.2 0.09 0 0.54 212 1.10 0.27 0.01 3.74 214 16.27 1.11 0.02 48.80 216 > 4Gb 5.06 0.05 > 4Gb 218 24.08 0.12 220 115.39 0.27 222 416.81 0.68 224 2199.9 1.57 226 > 1h 3.77

chooses a good prime p, then the dimension of polynomial solutions over Q and over Fp coincide, and thus, the algorithm ModBsGsPolySols returns the correct output. We now estimate the probability of choosing a good prime. Using Lemma 6, the entries of A have sizes bounded by Γ = N (6n log(o) + n log(N ) + `) + 2d(6n log(o) + n log(d) + `), which is, by the assumption N ≥ d, ˚ in Olog (`N + nN log(N )). Let B be the integer B = 2 logc (N ) (` + ˇ ` ´ n log(n) + 2nΓ) , so that B = Olog n2 `N log1+c (N ) . Let us suppose that the prime p is chosen uniformly at random in the set of prime numbers between B and 2B. Then, using Lemma 9, it is easy to infer that p is a good prime with probability at least 1 − 2 log1c (N ) . We finally prove the complexity estimate. By [17, Th. 18.8], the cost of Step 1 is in Olog (I(log N ) log2 N ). Using the algorithm from Lemma 3, Step 2 can be done using O((d + n) M(n) log(n)) operations in Fp . Step 4 can be done using O(nω ) operations in Fp . Since N ≥ d + n and p > N +1, Th. 2 can be used to perform Step 3 and this concludes the complexity analysis, since ` ´ every operation in Fp costs O(I(log(p))) = Olog I(log(N )) bit operations.

N

The algorithm ModBsGsPolySols can easily be modified, so as to return also the degrees of all the polynomial solutions, within the same bit complexity bound √ Olog (M( N )I(log(N ))) and with the same probability. Combining our two algorithms leads to an algorithm for computing polynomial solutions which is output-sensitive. Indeed, suppose that the indicial polynomial of L has positive integer roots N1 < · · · < Nk = N and that the polynomial solutions of L have degrees d1 < · · · < dr = d. Using our ModBsGsPolySols algorithm, we √ compute the degrees di in bit complexity roughly linear in N ; then, using our algorithm BinSplitPolySols, we return a compact representation of solutions in bit complexity roughly linear in d. If √ d  N, this strategy has its benefits; for instance, if d ≈ N (as in Ex. 4, §5), we√compute the solutions in bit complexity roughly linear in N instead of N 2 by the basic algorithm.

5.

Total 0.67 2.10 8.56 49.26 329.51

BsGs 0 0.02 0.03 0.05 0.07 0.11 0.25 0.59 1.35 Example 3 BinSplit BsGs 0.01 0.01 0.03 0.03 0.09 0.06 0.54 0.19 3.15 0.45 16.72 1.09 93.64 2.57 506.81 6.17 > 1h 14.84

The equations used in these tables are as follows: Ex. 1 is (1 − x2 )y 00 − 2xy 0 + N (N + 1)y = 0, where N is a power of 2. The dimension of polynomial solutions is 1 and any nonzero solution has degree N ; the recurrence has order o = 2, but only two terms of the recurrence are nonzero. Ex. 2 is (x2 + 2x + 1)y 0 − (N x + N − 1)y = 0. It has no nonzero polynomial solutions, but its indicial equation at infinity has N as root; the recurrence has order o = 2. Ex. 3 (taken from [10]) is 2x3 y 00 + ((3 − 2N )x2 + x)y 0 − (N x + 1)y = 0. It has a 1-dimensional space of polynomial solutions and the recurrence has order 1. Finally, in Ex. 4 we consider a family of LDEs indexed by N of order n = 3; the recurrence (2) has order o = 7, the indicial equation is (x − d)(x − N ) = 0, the LDE has a solution of degree d, but no solution of degree N . In Exs. 1 and 4, the column Compact displays the times used by BinSplit to compute the compact representation of solutions, while the column Expand indicates the time necessary to compute the expansion of the compact representation in the monomial basis. Their sum is collected in the column Total, whose output is the same as that of LFS. All the tests have been performed on the computers of the MEDICIS resource center www.medicis.polytechnique.fr, using a 2 Gb, 2200+ AMD 64 Athlon processor. The timings (in seconds) shown in these tables prove that the theoretical complexity estimations can be observed in practice: – The cost of LFS is multiplied by more than 16 when the degree N is multiplied by 4. This is in agreement with the fact that the basic algorithm has complexity (at least) quadratic in N . Moreover, the memory requirements are also roughly proportional to N 2 , and this naturally becomes prohibitive (the mention > 4Gb means that the execution was stopped after 4Gb of memory were exhausted.) – The cost of BinSplit is multiplied by slightly more than 5 when the degree N is multiplied by 4. This accurately reflects the behavior of the GMP’s integer multiplication. – The cost of BsGs is multiplied by slightly more than 2 when the degree N is multiplied by 4. Again, this is in line with the complexity estimates and shows that the polynomial multiplication we are using is quite good. – When the recurrence has 2 terms (Exs. 1 and 3) BinSplit

EXPERIMENTS

We have implemented our algorithms BinSplitPolySols and ModBsGsPolySols in the computer algebra systems Maple v. 9.5 and respectively Magma v. 2.11-2. Our choice is motivated by the fact that Magma and Maple provide implementations of fast integer arithmetic, based on Karatsuba and FFT multiplications. They both use the GNU Multi Precision Arithmetic Library (GMP). This is important, since in our experiments over Z, the computations require sometimes up to millions of bits. Moreover, Magma employs asymptotically fast algorithms for performing arithmetic with univariate polynomials over Fp (including Karatsuba and FFTbased methods). Again, this is crucial, since in our modular baby-step / giant-step algorithm, the theoretical gains are valid only in conjunction with fast polynomial multiplication. We have also implemented the basic algorithm in Maple. The performances of our implementation are very similar to those of Maple’s function PolynomialSolutions from the LinearFunctionalSystems (LFS) package. Maple provides another implementation of the basic algorithm, namely the function polysols from DEtools package. Since LFS outperforms DEtools on the set of examples we considered, we have chosen to display only the timings of LFS for comparisons.

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Example 4 d 9 16 25 36 49 64 81 100 121 144 169

N 81 256 625 1296 2401 4096 6561 10000 14641 20736 28561

LFS 0.14 0.63 4.90 30.14 192 746.94 4098.17 >6h? > 1 day ? > 1 week ? > 1 month ?

BsGs 0.03 0.05 0.08 0.13 0.17 0.23 0.3 0.38 0.52 0.7 0.9

´ Schost. Linear [9] A. Bostan, P. Gaudry, and E. recurrences with polynomial coefficients and application to integer factorization and Cartier-Manin operator, May 2004. 29 pages. Submitted. [10] D. Boucher. About the polynomial solutions of homogeneous linear differential equations depending on parameters. In Proc. ISSAC’99, pages 261–268. ACM Press, 1999. [11] R. P. Brent. Multiple-precision zero-finding methods and the complexity of elementary function evaluation. In Analytic computational complexity, pages 151–176. Academic Press, 1976. [12] D. V. Chudnovsky and G. V. Chudnovsky. Approximations and complex multiplication according to Ramanujan. In Ramanujan revisited, pages 375–472. Academic Press, 1988. [13] F. Chyzak. An extension of Zeilberger’s fast algorithm to general holonomic functions. Discrete Mathematics, 217(1-3):115–134, 2000. [14] F. Chyzak, P. Dumas, H. Le, J. Martins, M. Mishna, and B. Salvy. Taming apparent singularities via Ore closure. Preprint, July 2004. [15] G. Estrin. Organization of computer systems—the fixed plus variable structure computer. In AFIPS conference proceedings, volume 17, pages 33–40, 1960. [16] J. von zur Gathen and J. Gerhard. Fast algorithms for Taylor shifts and certain difference equations. In Proc. ISSAC’97, pages 40–47. ACM Press, 1997. [17] J. von zur Gathen and J. Gerhard. Modern computer algebra. Cambridge University Press, 1999. [18] J. Gerhard. Modular algorithms in symbolic summation and symbolic integration. volume 3218 of LNCS. Springer–Verlag, 2005. [19] G. H. Golub and C. F. Van Loan. Matrix computations. Johns Hopkins University Press, 1996. [20] M. van Hoeij, J.-F. Ragot, F. Ulmer, and J.-A. Weil. Liouvillian solutions of linear differential equations of order three and higher. Journal of Symbolic Computation, 28(4-5):589–609, 1999. [21] P. Kogge and H. Stone. A parallel algorithm for the efficient solution of a general class of recurrence equations. IEEE Trans. Comp., C-22:786–793, 1973. [22] J. Liouville. Second m´emoire sur la d´etermination des int´egrales dont la valeur est alg´ebrique. Journal de ´ l’Ecole polytechnique, Cahier 14:149–193, 1833. [23] F. M. Marotte. Les ´equations diff´erentielles lin´eaires et la th´eorie des groupes. PhD thesis, Facult´e des Sciences de Paris, 1898. [24] A. Sch¨ onhage, A. F. W. Grotefeld, and E. Vetter. Fast algorithms. Bibliographisches Institut, 1994. A multitape Turing machine implementation. [25] M. F. Singer. Liouvillian solutions of n-th order homogeneous linear differential equations. American Journal of Mathematics, 103(4):661–682, 1981. [26] M. F. Singer and F. Ulmer. Linear differential equations and products of linear forms. Journal of Pure and Applied Algebra, 117/118:549–563, 1997. [27] V. Strassen. Einige Resultate u ¨ber Berechnungskomplexit¨ at. Jahresbericht der Deutschen Mathematiker-Vereinigung, 78(1):1–8, 1976.

BinSplit Compact Expand 0.04 0.01 0.05 0.01 0.07 0.01 0.07 0.02 0.08 0.01 0.09 0.03 0.1 0.02 0.12 0.02 0.137 0.03 0.179 0.03 0.203 0.04

essentially computes scalar factorials and there is no linear algebra step. In the opposite case (o > 1), BinSplit multiplies matrices of small size, but containing potentially huge integer entries. A further improvement (not implemented yet) is to use Strassen’s algorithm to multiply integer matrices; the gain should be already visible on 2 × 2 matrices. – In Ex. 1, expanding the compact representation has quadratic complexity, but a constant factor (between 3 and 4) is gained over LFS. At least two issues contribute to this constant factor: LFS computes two power series expansions up to order N instead of a single one; the size of the numerators and denominators are more important in the computations done by LFS than in our algorithm. – The timings in Ex. 4 clearly show the advantage of first executing the baby step/giant step algorithm to compute the possible degrees d of solutions; these values are given as input to BinSplitPolySols in order to compute the compact representation of solutions. This way, we get an algorithm that is output-sensitive. Moreover, expanding the solutions is in this case negligible. Without the information on the degrees, even though the polynomial solutions have moderate degrees (up to d = 81), LFS spends a lot of time in (uselessly) unraveling recurrences up to order N = d2 . (The entries marked by ? are estimated timings.)

6. REFERENCES [1] S. Abramov, M. Bronstein, and M. Petkovˇsek. On polynomial solutions of linear operator equations. In Proc. ISSAC’95, pages 290–296. ACM Press, 1995. [2] S. A. Abramov and K. Y. Kvashenko. Fast algorithms to search for the rational solutions of linear differential equations with polynomial coefficients. In Proc. ISSAC’91, pages 267–270. ACM Press, 1991. [3] G. Almkvist and D. Zeilberger. The method of differentiating under the integral sign. Journal of Symbolic Computation, 10(6):571–591, 1990. [4] M. A. Barkatou. On rational solutions of systems of linear differential equations. Journal of Symbolic Computation, 28(4-5):547–567, 1999. [5] M. Beeler, R. Gosper, and R. Schroeppel. HAKMEM. Artificial Intelligence Memo No. 239. MIT, 1972. [6] D. J. Bernstein. Fast multiplication and its applications. Available at http://cr.yp.to/papers.html. [7] P. B. Borwein. On the complexity of calculating factorials. Journal of Algorithms, 6(3):376–380, 1985. [8] A. Bostan, T. Cluzeau, and B. Salvy. Fast algorithms for polynomial and rational solutions of linear operator equations. Preprint, 2005.

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Non Complete Integrability of a Magnetic Satellite in Circular Orbit Delphine Boucher IRMAR Université de Rennes 1 Campus de Beaulieu F-35042 Rennes Cedex

[email protected] ABSTRACT

of meromorphic first integrals which are functionally independent and in involution (see [1], [6] and [15] for precise definitions of these notions). Here we will use a result of J.-J. Morales and J.-P. Ramis :

We consider the motion of a rigid body (for example a satellite) on a circular orbit around a fixed gravitational and magnetic center. We study the non complete meromorphic integrability of the equations of motion which depend on parameters linked to the inertia tensor of the satellite and to the magnetic field. Using tools from computer algebra we apply a criterion deduced from J.-J. Morales and J.-P. Ramis theorem which relies on the differential Galois group of a linear differential system, called normal variational system. With this criterion, we establish non complete integrability for the magnetic satellite with axial symmetry, except for a particular family F already found in [11], and for the satellite without axial symmetry. In the case of the axial symmetry, we discuss the family F using higher order variational equations ([14]) and also prove non complete integrability.

Theorem 1 ([15]). Let (S) be a Hamiltonian system, x0 (t) be a particular solution of (S), Y
(t) = A(t) Y (t) be the variational system of (S) computed along the solution x0 (t) and G be the differential Galois group of Y
(t) = A(t) Y (t). If the system (S) is completely integrable with meromorphic first integrals, then the connected component of the identity in the group G, denoted G0 , is an abelian group. In [10], using Ziglin’s theory, A. Maciejewski and K. Gozdziewski gave a numerical proof of the non complete integrability of the problem of the satellite with axial symmetry in a non magnetic field. Furthermore, A. Maciejewski ([9, 8]) and M. Audin ([2]) also gave independent (formal) proofs of non complete integrability for the satellite with axial symmetry in a non magnetic field. They both applied Morales and Ramis theorem to an order two variational equation using two different approaches. Then, the proof was extended in many ways. First in [11], A. Maciejewski and M. Przybylska proved that the system of the magnetic satellite with axial symmetry was not completely integrable except for a family F of parameters for which the answer remained open. A major improvement in this result is that they deal with real integrability and not only with complex integrability. Lastly, in [5], the author proved that the system of the satellite without axial symmetry and without magnetic field was not completely integrable.

Categories and Subject Descriptors I.1 [Computing Methodologies]: Symbolic and algebraic manipulation; I.1.2 [Symbolic and algebraic manipulation]: Algorithms; J.2 [Physical sciences and engineering]: Astronomy

General Terms Algorithms

1.

INTRODUCTION

We consider a rigid body (the satellite) moving in a circular orbit around a fixed gravitational and magnetic center ([9, 8], [10], [11]). The equations of the motion of the satellite are given by a Hamiltonian system which depends on a set P of parameters related to the inertia tensor of the body and to the magnetic field ([11]). Our goal is to find the values of the parameters for which this Hamiltonian system may be completely integrable (with meromorphic first integrals), which means that there exists a sufficient number

In this paper we deal with the case of the satellite with and without axial symmetry in a magnetic field. We use a criterion of non complete meromorphic integrability deduced from theorem 1 and established in [4, 3] : Criterion 1 ([4, 3]). Let (S) be a Hamiltonian system and (N V S) be the normal variational system computed along a particular solution of (S). If (N V S) is irreducible and has formal solutions with logarithmic terms at a singular point, then (S) is not completely integrable (with meromorphic first integrals).

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We also use a new theorem, announced in [14], on higher order normal variational systems. First for the magnetic satellite with axial symmetry, we apply criterion 1 to an equation of order 2 (instead of applying

53

Kovacic’s algorithm like in [11]). We find the system is not completely integrable except for the family F of parameters already found in [11] and for which the question of integrability remained open. For this family we use the new theorem on higher order normal variational systems ([14]) to establish the non complete integrability.

The parameter ξ is a real parameter linked to the magnetic field; the parameters A, B and C are the components of the inertia tensor which is linked to the distribution of the mass throughout the body. Without loss of generality, one can assume that they satisfy

Then for the case of the satellite without axial symmetry, we deal with a fourth order normal variational equation (or 4 × 4 normal variational system). Since Kovacic’s algorithm is restricted to second order equations, we use criterion 1. We are still in a lucky situation since our normal variational system has no exponents depending on the parameters. This allows us to avoid arithmetic conditions on the parameters and to adapt all the algorithms needed in criterion 1 to our situation. In section 2, we give the normal variational system computed along the particular solution given in [11]. In section 3 we consider the particular case when the magnetic satellite has an axial symmetry. Like in [11], we prove that this normal variational system can be reduced to a linear differential equation of order 2. We apply criterion 1 to it and we conclude to the non complete integrability except for the same particular family F as in [11]. Then we solve the remaining case of this family using higher order normal variational systems ([14]). In section 4 we deal with the general case of the magnetic satellite, without axial symmetry. We prove that the 4 × 4 normal variational system has regular formal solutions with logarithmic terms and that it is irreducible under the conditions on the parameters. Using again criterion 1 we conclude that the equations of the satellite without axial symmetry are not meromorphically completely integrable.

A < B + C, B < C + A, C < A + B,

2.

A ≥ 0, B ≥ 0, C ≥ 0,

C < B ≤ A = 1. In [11], the authors consider a particular solution to the Hamiltonian system : π π xk =t (q1 , , , 1 + w z, 0, 0) 2 2 where z = k cn(w t, k), cos(q1 ) = −k sn(w t, k),

(3)

The constraints (2) on the parameters become: w2 , 0 < w2 < 6 B − 3 ≤ 3, 3 The variational system along the solution xk is A = 1, C = B −

(4)

Y
(z) = V S(z) Y (z) where V S(z) =

dt J6 Hessian(H, xk ) dz

is a 6 × 6 matrix. We introduce a new function f defined by

NORMAL VARIATIONAL SYSTEM

f (z) = dn(w t, k) sn(w t, k). Using the following simplifications : d (cn(w t, k)) = −w dn(w t, k) sn(w t, k), dt z2 , k2 we express the coefficients of V S(z) in Q(B, w, ξ, k)(z, f (z)). As the matrix is too big, we do not give it here. Now we construct the normal variational system associated to the particular solution xk . The vector Wk = x
k =t (w z, 0, 0, −w2 k f (z), 0, 0) is a particular solution to the variational system. Thanks to it we make a symplectic transformation on the variational system, which enables to keep the symplectic structure of the differential Galois group of the variational system. We consider the following symplectic matrix P6 (such that t P6 J6 P6 = J6 ) whose first column is made of the coordinates of the vector Wk : P6 = 1 0 0 0 0 0 0 C B C B −1 0 0 0 0 C B C B 0 −1 0 0 0 C. B Wk −Wk [5] −Wk [6] Wk [2] Wk [3] 1 C B Wk [1] Wk [1] Wk [1] Wk [1] Wk [1] C B C B 0 0 0 −1 0 A @ 0 0 0 0 −1 sn(w t, k)2 = 1 −

(1)

where x =(q1 , q2 , q3 , p1 , p2 , p3 ), = −I6 , „ «2 1 s3 p 1 c2 s3 p3 1 2 + c3 p2 − H(x) = − ξ c2 + 2 2 s2 s2 „

sin(q1 ) = dn(w t, k)

w2 = 3 (B − C).

J62

1 + 2

k ∈]0, 1[

and w is a new parameter defined by

In this section we recall the equations which define the problem of the satellite using the most recent paper on the subject ([11]) and we compute the normal variational system along the particular solution given in [11] using a symplectic transformation. After some reductions detailed in [11], the equations of the rotational motion of the body can be given by a Hamiltonian system x
(t) = J6 ∇H(x(t))

(2)

«„ « c3 p1 c2 c3 p3 c3 p1 c2 c3 p3 s3 p 2 − − s3 p 2 − − s2 s2 Bs2 B Bs2 „ « s3 p 1 1 p3 2 c2 s3 p3 − + + c3 p2 − s3 s2 2 C s2 s2 « „ c3 p1 c2 c3 p3 c3 s2 − p3 c2 − − s3 p 2 − s2 s2

3 3 3 (−s3 c2 s1 + c3 c1 )2 + B (−c3 c2 s1 − s3 c1 )2 + Cs2 2 s1 2 2 2 2 and +

∀i ∈ {1, 2, 3}, ci = cos(qi ), si = sin(qi ).

54

The system satisfied by Y˜ defined by Y = P6 Y˜ is

3.1 The normal variational equation 2

If we replace C with C = 3−w , the normal variational 3 system (5) has a particular solution : t (0, 1, 0, 0). We construct a symplectic matrix P4 (such that t P4 J4 P4 − J4 = 0 where J42 = −I4 ) with t (0, 1, 0, 0) on its first row like in previous section and we reduce the equivalent system to the following one : „ « 0 α(z) µ
(z) = µ(z). β(z) 0

Y˜
(z) = V˜S(z) Y˜ (z) where V˜S(z) = P6 (z)−1 (V S(z) P6 (z) − P6
(z)). As the matrix V S is infinitesimally symplectic (i.e. J6 V S +t V S J6 = 0) and P6 is symplectic, the matrix V˜S is also infinitesimally symplectic ([15]). Furthermore as Wk is a solution of the variational system (i.e. V S(z) Wk (z) − Wk
(z) = 0), the first row of the matrix V˜S(z) is equal to zero. So V˜S(z) can be written in the following form: 0 B B B ˜ V S(z) = B B @

×

0 0 0 0 0 0

×

A1 (z) 0 0 A3 (z)

where

1

× ×

× × × 0 × ×

A2 (z) 0 0 − A1 (z) t

α(z) =

C C C C C A

β(z) =

(5)

The matrix N V S(z) has the following infinitesimal symplectic structure : „ « A1 (z) A2 (z) N V S(z) = A3 (z) A4 (z)

y

(z) −

α
(z)
y (z) − α(z) β(z) y(z) = 0. α(z)

Using the two equalities

where A4 (z) = −t A1 (z), A2 (z) =t A2 (z), A3 (z) =t A3 (z) :

A1 (z) =

A2 (z) =

(B−1) (1+w z) k w B f (z)

0 z − k1+w w f (z) −1 k w B f (z)

A3 (z) = @

f (z)2 = −

!

!

we get the same linear differential equation as in [11] :

3 k w (w2 −3 B) f (z)

a(z)

3 (B−1) −w

−3 (B−1) w

b(z)

y

(z) +

1 −

A

(w2 + 3 − 3 B) (z 2 + 1 − k2 ) + (w z + 1)2 − ξ k w f (z)

Proposition 1. When 0 < w2 < 3, the equation (6) has a formal solution with a logarithmic term at the point infinity if and only if w2 = 2 ξ. Proof. The exponents at infinity are ρ0 = −1 and ρ1 = 2. As they differ each other from an integer one can test whether there is a formal solution with logarithms using Frobenius method (chapter 16 of [7]). We first make the change of variable z = 1/t and we study the new equation around 0. We consider a formal series solution of the type X cn tn , ρ ∈ C tρ

SATELLITE WITH AXIAL SYMMETRY

We assume that two of the components of the inertia tensor are equal, let us say A and B, so the conditions (4) become : and 0 < w2 < 3

2w2 z 2 + 2wz − w2 k2 + w2 + 1 − ξ y(z) = 0. w2 (z 2 − k2 )(z 2 + 1 − k2 )

Instead of applying Kovacic’s algorithm like in [11], we are going to apply criterion 1.

In the following section we study the special case when the satellite has an axial symmetry (see also [2, 11, 5] for results in this case).

w2 3

(6)

3.2 Non complete integrability when w2 −2 ξ = 0

(B − 1) (3 B (k2 − z 2 ) + (w z + 1)2 ) b(z) = − k w B f (z)

A = B = 1, C = 1 −

(2 z 2 + 1 − 2 k2 ) z y
(z) + 1 − k2 ) (z 2 − k2 )

(z 2

whose coefficients are now in Q(w, ξ, k)(z). In next subsection we find the already known result of [11] using criterion 1.

and a(z) =

(z 2 + 1 − k2 ) (z 2 − k2 ) k2

z (2 z 2 − 2 k2 + 1) f
(z) = 2 f (z) (z + 1 − k2 ) (z 2 − k2 )

0 0

0 0

3.

2w2 z 2 + 2wz − k2 w2 + w2 + 1 − ξ . w k f (z)

According to [15] (or proposition 4.2 p.76 of [13]), this symplectic transformation enables us to apply our criterion directly on this reduced 2 × 2 linear differential system instead of applying it on the initial 4 × 4 system. We first transform it into a linear differential equation of order 2 namely the one satisfied by the first component y of µ:

One can extract from this system the following system called normal variational system ν
(z) = N V S(z) ν(z).

−1 w k f (z)

.

n≥0

55

We plug all these expressions into the differential equation and we get a system of equations which are polynomial in w, ξ and linear in p0 , p1 . Under the constraint on w and ξ, we find the following exponential solutions : 1 − wz if and only if 2ξ − w2 = 2w2 k2 − w2 − 2 = 0; p − (4 wz − 4 + w2 ) (−4 wz + w2 + 4) if and only if 2ξ − w2 = (4kw + 4 + w2 )(4kw − 4 − w2 ) = 0. So there is an exponential solution if and only if w2 − 2ξ = 0 and (2w2 k2 − w2 − 2) (4kw + 4 + w2 ) (4kw − 4 − w2 ) = 0 which ends the proof.

and compute the linear recurrence relation satisfied by the coefficients cn . f0 (ρ + n) cn + f1 (ρ + n − 1) cn−1 + f2 (ρ + n − 2) cn−2 +f3 (ρ + n − 3) cn−3 + f4 (ρ + n − 4) cn−4 = 0 with f0 (n) = w2 (n − 2) (n + 1) , (whose roots are the exponents −1 and 2 at infinity) f1 (n) = −2 w,

We can now state : f2 (n) = −2 w2 k2 n2 + w2 k2 + w2 n2 − w2 − 1 + ξ,

Proposition 3. Under the constraints on the parameters, w, ξ ∈ IR, 0 < w2 < 3, the satellite with axial symmetry is not completely integrable along the solution xk with 0 < k < 1 when

f3 (n) = 0, f4 (n) = nw2 k2 (k − 1) (k + 1) (n + 1) .

w2 − 2ξ = 0.

According to proposition 7 of [3], there exists a formal solution at t = 0 with a logarithmic term if and only if the following determinant does not cancel when ρ = ρ0 = −1 : ˛ ˛ ˛ f1 (ρ + 2) f2 (ρ + 1) f3 (ρ) ˛ ˛ ˛ Fρ1 −ρ0 (ρ) = F3 (ρ) = ˛˛ f0 (ρ + 2) f1 (ρ + 1) f2 (ρ) ˛˛ . ˛ 0 f0 (ρ + 1) f1 (ρ) ˛

Proof. This proposition is directly deduced from the two previous ones and from criterion 1. So until now we have no answer for the family F defined by F= w2 w2 ,ξ= , 0 < w2 < 3}. 3 2 One can prove that in this case the connected component of the identity in the Galois group G of the equation (6) is abelian. Indeed, the fourth symmetric equation of (6) has a rational solution so the unique exponential solution to (6) is algebraic, G is a finite subgroup of the Borel group and G0 is the additive group so it is abelian. For the family F, we are going to use higher order variational equations ([13]). {(A, B, C, ξ), A = B = 1, C = 1−

We find F3 (−1) = 4w3 (w2 − 2ξ) so if 0 < w2 < 3, it does not cancel if and only if w2 − 2ξ = 0. Proposition 2. When 0 < w2 < 3, the equation (6) is irreducible if and only if 2ξ − w2 = 0 or (2w2 k2 − w2 − 2)(4kw + 4 + w2 )(4kw − 4 − w2 ) = 0. Proof. Since the equation is of order 2, it is irreducible if, and only if, it has no factor of order one i.e. no solution y such that y
/y is in Q(w, ξ, z). Such solutions are called exponential solutions. There are many algorithms to find these solutions (see for example chapter 4 of [16]). Here we follow the ’classical’ method. We first compute the exponents at the singularities of the equation. Then for each singularity we construct sets of exponents which are equal up to integers and we keep the minimum value of each set.√ At the points s1 = k, s2 = i 1 − k2 , s3 = −s1 and s4 = −s2 the exponents are 0 and 12 so at each finite singularity we get two distinct sets of one single exponent and we keep all the exponents. At infinity, the exponents are −1 and 2 so we get one set of two exponents and we keep only the exponent −1. An exponential solution will be of the form (z − s1 )

e1

(z + s1 )

e2

(z − s2 )

e3

(z + s2 )

e4

3.3 Non complete integrability when w2 −2ξ = 0 We first make the change of variable z = 1t in equation (6) and we write the new equivalent equation like a companion system: Y1
(t) = A Y1 (t) (E1 ) with

„ A=

and

8 ˜(t) = − < a : ˜b(t) = −

0 1 a ˜(t) ˜b(t)

«

2 t2 k2 w2 −w2 t2 −4 wt−2 t2 −4 w2 2t2 (−1−t2 +k2 t2 )(kt−1)(kt+1)w2 t(1−2 k2 −2 k2 t2 +2 k4 t2 )

(−1−t2 +k2 t2 )(kt−1)(kt+1)

Then we consider the formal solutions at the point 0 of the higher order variational equations. As it is explained in [14], we stop if we find a logarithmic term in some solution. Then according to theorem 4 and lemma 3 of [14], we conclude to the non complete integrability of the Hamiltonian system. On a practical point of view, we follow the ideas of section 6 of [12]. The second order variational equation is

p(z)

where the degree of p is equal to −(e1 + e2 + e3 + e4 + e∞ ), ei is one of the previously selected exponents at the point si and e∞ is one of the previously selected exponent at the point ∞ (here e∞ = −1). As −(e1 +e2 +e3 +e4 −1) must be a natural integer, the possible exponential solutions are p0 + p1 z where p0 , p1 are unknown coefficients or (z − s1 )e1 (z + s1 )e2 (z − s2 )e3 (z + s2 )e4 where (e1 , e2 , e3 , e4 ) belongs to {(0, 0, 12 , 12 ), (0, 12 , 0, 12 ), (0, 12 , 12 , 0), ( 12 , 12 , 0, 0), ( 12 , 0, 12 , 0), ( 21 , 0, 0, 12 )}.

Y2
(t) = A Y2 (t) + B2 (E2 ) where B2 = A
Y1 (t) and Y1 is a solution of (E1 ). We replace Y1 (t) with λ1 F1 (t) + λ2 F2 (t) where (F1 , F2 ) is a basis of formal solutions of (E1 ) at 0 and λ1 , λ2 ∈ C.

56

1 where Q(z) is a diagonal matrix whose diagonal is 1, f (z) , 1 , 1. f (z) As f (z)2 is rational (k2 f (z)2 = (z 2 + 1 − k2 ) (k2 − z 2 )), this transformation does not change the (non) virtual abelianity of the group. We get the equivalent system :

Using the method of variation of constant we look for λ2,1 (t) and λ2,2 (t) such that λ2,1 (t) F1 (t)+λ2,2 (t) F2 (t) is a solution of (E2 ). If there are values of λ1 , λ2 such that the residue in the formal series of λ
2,1 (t) or λ
2,2 (t) is not equal to zero then we stop as we found a logarithmic term in a solution of (E2 ) else we go on : for k ≥ 3, we define the kth order variational equation (Ek ) by Yk
(t)

η
(z) = M (z) η(z)

= A Yk (t) + Bk (Ek )




where


Bk−1

where Bk = A Yk−1 (t) + and Yk−1 (t) is a previously computed formal series solution of (Ek−1 ). We look for a solution Yk (t) = λk,1 (t) F1 (t) + λk,2 (t) F2 (t) of (Ek ). If for all λ1 , λ2 , the residues in the formal series of λ
k,1 (t) and λ
k,2 (t) are equal to zero then we go on else we stop. These computations lead to the following proposition.

0 B m3 (z) B M (z) = @ m6 (z) m7 (z) and

Proposition 4. Under the constraints on the parameters, w, ξ ∈ IR, 0 < w2 < 3, the satellite with axial symmetry is not completely integrable along the solution xk with 0 < k < 1 when w2 − 2ξ = 0. Proof. Following the method explained below, we find that there is a logarithm in some solution of the fourth order variational equation. More precisely, a basis of formal solutions of (E1 ) at 0 is 8 1 0 −1 1 2 2 1 −w2 −2 + 3w + 2w k12w t + h.o.t. > 2 > 3 t > > A F1 (t) = @ > > > > 1 1 2w2 k2 −w2 −2 > + + h.o.t. > 3 t2 12w2 <

Lemma 1. For all values of the parameters satisfying the condition (C), the exponents at infinity of the system (8) are −1, −2, −3 and −4. Proof. We first make the change of variable z = 1/t in the normal variational system (8) then we make a Moser transformation η = T η where 0

t1 B t3 B T =@ 0 t4

The residue in λ
4,1 (t) is always non zero so there is a logarithm in some solution of the fourth order variational equation and we conclude with theorem 4 and lemma 3 of [14].

0 t3 0 0 1 0 0 0

1 t2 t C C 0 A t5

with 8 3 (w2 −3)Bt3 > t1 = w2 −6 B+3 w3 > > ( ) > > > t3 (−1+B)k(w6 −9 w2 B+3 w4 −9 w2 B 2 +27 B 2 −27 B ) > > t = > 2 > (−3 B+2 w2 +3)B(w2 −3)(−3 B+w2 +3) > < (w2 −3)(−3 B+w2 +3)Bt t3 = k(−1+B)w3 w2 −6 B+3 ( ) > > > 3 t2 B (w2 −3) > > t4 = − w2 w2 −6 B+3 > > ( ) > > 2 2 2 > > : t5 = − t (w +3)(−1+B)k(−3 B+w )w 2 2 w −3 −3 B+w +3 B ( )( )

We now extend the proof to the case of the magnetic satellite without axial symmetry.

SATELLITE WITHOUT AXIAL SYMMETRY

Now we assume that there is no axial symmetry i.e. B = 1 then the conditions (4) become w2 , 3

k (B−1) (1+w z) w (z 2 +1−k2 ) (k2 −z 2 ) B −k w B (z 2 +1−k2 ) (k2 −z 2 ) 1+w z −k w z (2 k2 −2 z 2 −1) (z 2 +1−k2 ) (k2 −z 2 ) 3 k w (w2 −3 B) 2 (w +3−3 B) (z 2 +1−k2 )+(w z+1)2 −ξ kw −3(B−1) w (1−B) k (3 B (k2 −z 2 )+(w z+1)2 ) . w B (z 2 +1−k2 ) (k2 −z 2 )

We detect formal solutions with logarithmic terms at the point infinity.

8 8(3w − 4) = − + h.o.t. t 3w

(C) : C = B −

1 m2 (z) 0 0 m5 (z) C C m4 (z) −m3 (z) A −m1 (z) 0

4.1 Formal solutions with logarithmic terms

where h.o.t. means ’higher order terms’. We consider the particular series solution Y1 (t) = λ1 F1 (t) + λ2 F2 (t) of (E1 ) and for k greater than 1, we compute a solution Yk (t) = λk,1 (t) F1 (t) + λk,2 (t) F2 (t) of (Ek ) until we get a logarithm for some value of (λ1 , λ2 ). At the fourth stage, we get, for λ1 = 0 and λ1 = 1,

4.

8 m1 (z) = > > > > > m 2 (z) = > > > > > m3 (z) = > > > < m4 (z) = m5 (z) = > > > > > > m6 (z) = > > > > m7 (z) = > > > : m (z) = 8

m1 (z) m4 (z) m7 (z) m8 (z)

One can notice that the symplectic structure has not been preserved (Q was not a symplectic matrix) but the coefficients of M (z) depend now rationally on the parameters and on z.

> > 0 2 1 > > 1 3 > t + h.o.t. t + 2w > > > @ A > > : F2 (t) = 3 2 2 t + 2w t + h.o.t.

λ
4,1 (t)

0

(8)

w2 < 6B − 3 < 3 .

To our knowledge, this case has not been studied before. We work with the normal variational system (5). We make the following transformation

The matrix T is well defined as under the constraints on the parameters, one can check that B − 1, w2 − 6 B + 3, w2 − 3, −3 B + w2 + 3 and −3 B + 2w2 + 3 do not cancel. The normal variational system (8) is equivalent to a system of the type

ν(z) = Q(z) η(z)

t η
(t) = (A0 + A1 t + · · · ) η(t)

(7)

57

There exists a matrix P0 such that P0−1 A0 P0 is the diagonal matrix with −1, −2, −3, −4 on its diagonal. We do not give the expression of the matrix P0 . Its determinant is: ` ´3 det(P0 ) = w2 − 6 B + 3 w6 k2 (−1 + B)3 ´3 ` 4 3 w B + w4 − 9 w2 B + 6 w2 − 9 w2 B 2 + 9 + 27 B 2 − 36 B ` ´ ´ ` ´ ` ` 4 4 / 12B 6 w2 − 3 −3 B + w2 + 3 −3 B + 2 w2 + 3 ´2 ´ ` 2 w − 3B .

where B0 is an upper-triangular matrix whose eigenvalues are λ1 − d1 , λ2 − d2 , . . . , λn − dn . Proof. The system satisfied by Z is of the type t Z
(t) = [D−1

∞ X

Ak D tk − t D−1 D
] Z(t)

k=0

i.e. ∞ X D−1 Ak D tk − diag(d1 , d2 , . . . , dn )] Z(t) t Z
(t) = [

Using Mathematica, one can prove that the determinant of P0 never cancels under the constraints (C) on the parameters. We introduce the variable W = w2 and we find that 3 W 2 B + W 2 − 9 W B + 6 W − 9 W B 2 + 9 + 27 B 2 − 36 B is always positive under the constraints W > 0, 6 B − 3 − W > 0, 1 − B > 0 :

k=0

Furthermore, for all k, the coefficients of the matrix D−1 Ak D (k) (k) are the ai,j tdj −di where the ai,j are the coefficients of the matrix Ak . So the orders at 0 of the non zero coefficients of the matrix D−1 Ak D are at least −1 according to (H1 ). For k ≥ 2 the non zero coefficients of the matrix D−1 Ak D tk have orders greater than 0. If k = 1, the non zero coefficients of the matrix D−1 Ak D tk have orders at least 0 and those of orders 0 are on the upper part of the matrix according to (H1 ). Lastly, when k = 0, then D−1 Ak D tk remains a uppertriangular matrix with a main diagonal λ1 , λ2 , . . . , λn and a second upper diagonal with coefficients 4i,i+1 tdi+1 −di which are in {0, 1} according to (H2 ). So the system satisfied by Z is of the type t Z
(t) = (B0 + B1 t + · · · ) Z(t) where the matrix B0 is 1 0 λ1 − d1 ∗ · · · ∗ C B .. .. B0 = @ A . . 0 λn − dn

Experimental‘ImpliesRealQ[{W>0,6*B-3-W>0,1-B>0}, {9*W*B^2-27*B^2+9*W*B+36*B-3*W^2*B-9-6*W-W^2>0}] true

So the system (8) is rationally equivalent to the following system t η
(t) = (A0 + A1 t + · · · ) η(t) where the matrix A0 is 0

1 −1 0 0 0 B 0 −2 0 0 C C A0 = B @ 0 0 −3 0 A 0 0 0 −4

According to Theorem 5.1 of [16], as the eigenvalues of A0 differ each other from integers, we can make a transformation η = P η to get an equivalent system of the type

and where the terms ∗ come from the matrix A1 (case k = 1) and from the second diagonal of A0 (case k = 0). Thanks to lemma 2 we establish the following proposition :

t η
(t) = (B0 + B1 t + · · · ) η(t)

Proposition 5. Under the conditions (C) on the parameters, the system (8) is equivalent to a system of the type

where the eigenvalues of the matrix B0 are 0, 0, 0, 0. Furthermore, the existence of logarithmic terms will be directly given by the shape of the matrix B0 . To get this transformation matrix P , we will need this small lemma :

t η
(t) = (B0 + B1 t + · · · ) η(t) If 2 w2 (w2 − 2 ξ) (3 B − w2 − 3) − 9 ξ 2 (B − 1) = 0 then




Lemma 2. Let t Y (t) = A(t) Y (t) be a n × n linear differential system where

0

0 B 0 B B0 = @ 0 0

A(t) = A0 + A1 t + · · · is the development of A(t) at 0 and where the matrix A0 is a Jordan matrix with eigenvalues λ1 , λ2 , . . . , λn : 0 1 λ1 41,2 0 · · · 0 B 0 C 0 λ2 0 · · · B C A0 = B . C @ .. 4n−1,n A 0 0 λn

1 0 0 0

0 0 0 0

1 0 0 C C. 1 A 0

If 2 w2 (w2 − 2 ξ) (3 B − w2 − 3) − 9 ξ 2 (B − 1) = 0 then 0

0 B 0 B0 = B @ 0 0

with 4i,i+1 ∈ {0, 1}. Let D be a diagonal matrix D = diag(td1 , . . . , tdn ) with

1 0 0 0

0 0 0 0

1 0 0 C C. 0 A 0

In particular, there are formal solutions with a logarithmic term at infinity for all values of the parameters satisfying (C). Proof. Step 1. We increase the three first eigenvalues of A0 of 3 and the fourth of 4 : 0 −3 1 t 0 0 0 B 0 t−3 0 0 C C D1 = B @ 0 0 A 0 t−3 0 0 0 t−4

(H1 ) 0 ≤ d1 − d2 ≤ d1 − d3 ≤ . . . ≤ d1 − dn ≤ 1 and (H2 ) 4i,i+1 = 1 ⇒ di = di+1 Let Z defined by Y = D Z. It satisfies the system t Z
(t) = (B0 + B1 t + · · · ) Z(t)

58

and we conclude like previously. Lastly, we assume

We get a new matrix A0 which we diagonalize with a transformation matrix P1 whose determinant is : det(P ` 1) = ´ − 3 w4 B + w4 − 9 w2 B + 6 w2 − 9 w2 B 2 + 9 + 27 B 2 − 36 B ´ ` ´2 ´ ` ` 2 w − 6 B + 3 (−1 + B) / 12B 2 w6 w2 − 3 B The new matrix A0 becomes diagonal with 2, 1, 0, 0 on its diagonal. Step 2. We decrease the two first eigenvalues of A0 of 1 and keep the two other ones : 1 0 t 0 0 0 B 0 t 0 0 C C D2 = B @ 0 0 1 0 A 0 0 0 1

2 w4 − 4 w2 ξ − 3 ξ 2 = 2 w6 + 6 w4 − 4 w4 ξ − 12 w2 ξ − 9 ξ 2 = 0. This is impossible as it leads to w = ξ = 0.

4.2 Irreducibility In this part, we fix k in order to simplify the computations. The parameter k is just used to parameterized the particular solution of the normal variational system, so we can fix it in ]0, 1[. We choose k = 35 . As the normal variational system does not have any symplectic structure, we choose to use a cyclic vector transformation (vector [1, 0, 0, 0]) and to work with an equivalent scalar linear differential equation. We do not detail all the computations.

The determinant of the matrix P2 is : det(P2 ) =

(−3 B +

18w2 + 3) (−1 + B)

w2

• Factors of degree 1. We look for exponential solutions. We need the minimum of the exponents up to an integer at the singularities. We find - at the four finite singularities: 0 and 12 ; -at the point infinity : −1 So if there is an exponential solution, then it is one of these expressions : q q

and the matrix A0 becomes an upper-triangular matrix : 1 0 1 0 0 0 B 0 0 0 0 C C A0 = B @ 0 0 0 1 A 0 0 0 0 Step 3. We decrease the first three other ones. 0 t B 0 B D3 = @ 0 0

eigenvalue of 1 and keep the 0 1 0 0

0 0 1 0

(z − 35 ) (z + i 45 ) , q (z − 35 ) (z − i 45 ) , (z + 35 ) (z − i 45 ) , q q (z + 35 ) (z + i 45 ) , (z + i 45 ) (z − i 45 ) or a0 + a1 z where a0 and a1 are unknown coefficients.

1

q

0 0 C C 0 A 1

The determinant of the matrix P3 is :

Plugging the six first expressions in the differential equation we get polynomial systems and prove they have no solution under the constraints C on the parameters. Plugging a0 + a1 z we get a linear system of size 8 × 2 with coefficients depending on w, B, ξ, we prove that its rank is 2 under the constraints on the parameters, so a0 = a1 = 0. To conclude, there is no non zero exponential solution and no factor of degree one.

2 w2 (w2 − 2 ξ) (3 B − w2 − 3) − 9 ξ 2 (B − 1) . 18w4 The determinant of P3 may cancel. We first assume det(P3 ) = −

2 w2 (w2 − 2 ξ) (3 B − w2 − 3) − 9 ξ 2 (B − 1) = 0 then the matrix P3 is invertible 0 0 B 0 B A0 = B0 = @ 0 0

and the matrix A0 becomes 1 1 0 0 0 0 0 C C 0 0 1 A 0 0 0

• Factors of degree 2. To compute the factors of degree two, one needs to look for rational solutions of the second exterior system, Y
(z) = Λ2 (M )(z) Y (z). We compute an equivalent linear differential equation of order 6 and find that the only possible rational solution is (z − 35 ) (z + 35 ) (z + i 45 ) (z − i 45 ). Plugging this expression in the second exterior scalar equation we get a polynomial system in the variables B, w, ξ. We pick up only four equations among the 33 equations and we choose to use the resultant to prove this polynomial system has no solution under the constraints on the parameters.

According to Theorem 5.1 of [16], as B0 is a matrix whose eigenvalues λ satisfy 0 ≤ Re(λ) < 1, the local monodromy around 0 for the system (∗) is e2πiB0 . So in both cases, as the matrix B0 is a Jordan matrix which is not diagonal, there are formal solutions at infinity with logarithmic terms for the normal variational system . Now we assume that 2 w4 − 4 w2 ξ − 3 ξ 2 = 0 and that B=

(z − 35 ) (z + 35 ) ,

• Factors of degree 3. The normal variational system has a factor of degree three if and only if its adjoint system has a factor of degree one. Let us notice here that we have lost the symplectic structure due to the transformation (7) so we cannot link factors of degree one and factors of degree three. We construct a linear differential equation associated to this adjoint system using the cyclic vector approach. If there is an exponential solution, then it is one of these : q

1 2 w6 + 6 w4 − 4 w4 ξ − 12 w2 ξ − 9 ξ 2 . 3 2 w4 − 4 w2 ξ − 3 ξ 2

In this case steps 1 and 2 remain unchanged and step 3 changes. We keep the matrix D3 and compute a new matrix P3 . Its determinant is −1 and the new matrix A0 is 0 1 0 1 0 0 B 0 0 0 0 C C A0 = B0 = B @ 0 0 0 0 A 0 0 0 0

(z − 35 ) (z + 35 ) (z + i 45 ) (z − i 45 ),

59

q

q

(z − 35 ) (z + i 45 ) (a0 + a1 z), q (z − 35 ) (z − i 45 ) (a0 +a1 z), (z + 35 ) (z − i 45 ) (a0 +a1 z), q q (z + 35 ) (z + i 45 ) (a0 +a1 z), (z + i 45 ) (z − i 45 ) (a0 +a1 z)

q

(z − 35 ) (z + 35 ) (a0 + a1 z),

[4] D. Boucher and J.-A. Weil. Application of J.-J. Morales and J.-P. Ramis’ theorem to test the non complete integrability of the planar three-body problem. In IRMA Lectures in Mathematics and Theoretical Physics, From Combinatorics to Dynamical Systems, volume 3, 2003. [5] D. Boucher. Non complete integrability of a satellite in circular orbit. submitted to Portugaliae Mathematica, to appear. [6] R.-C. Churchill. Galoisian obstructions to the integrability of hamiltonian systems. In The Kolchin Seminar in Differential Algebra. Department of Mathematics, City College of New York, 1998. [7] E.-L. Ince. Ordinary Differential Equations. Dover Publications, INC New York, 1956. [8] A. J. Maciejewski. Non-integrability in gravitational and cosmological models. introduction to ziglin theory and its differential Galois extension. The Restless Univers. Applications of Gravitational N-Body Dynamics to Planetary, Stellar and Galatic Systems, pages 361–385, 2001. [9] A.-J. Maciejewski. Non-integrability of a certain problem of rotational motion of a rigid satellite. Dynamics of Natural and Artificial Celestial Bodies, pages 187–192, 2001. [10] A.-J. Maciejewski and K. Gozdziewski. Numerical evidence of nonintegrability of certain lie-poisson system. Reports on Mathematical Physics, 44, 1999. [11] A.-J. Maciejewski and M. Przybylska. Non-integrability of the problem of a rigid satellite in gravitational and magnetic fields. Celestial Mech., pages 317–351, 2003. [12] A.-J. Maciejewski, M. Przybylska, and J.-A. Weil. Non-integrability of the generalized spring-pendulum problem. Journal of Physics A: Mathematical and general, 37:2579–2597, 2004. [13] J.-J. Morales-Ruiz. Differential galois theory and non-integrability of hamiltonian systems. Ferran Sunyer i Balaguer Award Winning Monograph, 1999. [14] J.-J. Morales-Ruiz. Kovalevskaya, liapunov, painlev, ziglin and the differential galois theory. Regular Chaotic Dynamic, 5:251–72, 2000. [15] J.-J. Morales-Ruiz and J.-P. Ramis. Galoisian obstructions to integrability of hamiltonian systems i, ii. Methods Appl. Anal., 8(1):33–95, 97–111, 2001. [16] Put . M. van der and M.-F. Singer. Galois theory of linear differential equations. Grundlehren der Mathematischen Wissenschaften. 328. Berlin: Springer, 2003.

or a0 + a1 z + a2 z 2 where a0 , a1 and a2 are unknown coefficients.

Plugging all these expressions into the adjoint equation, we find polynomial systems with no solution under the constraints (C). We can now state : Proposition 6. Under the constraints (C) on the parameters and with k = 35 , the system (8) is irreducible.

4.3 Conclusion Proposition 7. Under the constraints (C), the magnetic satellite without axial symmetry is not completely integrable along the solution xk with k = 35 . Proof. We use sections 4.1, 4.2 and criterion 1. Let us notice that the restriction k = 35 has been made only in order to simplify the big computations of section 4.2. The aim was to find one solution along which we can compute a normal variational system whose Galois group is not virtually abelian this is why we could make this restriction.

5.

CONCLUSION

We gave a proof of the non complete meromorphic integrability of the satellite (a rigid body moving in a circular orbit around a fixed gravitational and magnetic center) with and without axial symmetry. This result (propositions 3, 4 and 7) completes the results of [9], [1] and [10] on the non complete integrability of the satellite with axial symmetry. Furthermore it was obtained thanks to criterion 1 (established and used in [4]) which we hope to be useful for other problems on complete integrability, and thanks to the new theorem of [14] on higher order variational equations.

6.

ACKNOWLEDGMENTS

I thank Mich`ele Audin for discussions about the satellite with axial symmetry (section 3), Marius Van der Put for discussions about section 4.1, Andrzej Maciejewski for corrections on a previous version; Felix Ulmer and the referees for their comments.

7.

REFERENCES

[1] M. Audin. Les syst`emes hamiltoniens et leur int´egrabilit´e. Cours Sp´ecialis´es, SMF et EDP Sciences., 2001. [2] M. Audin. La r´eduction symplectique appliqu´ee ` a la non-int´egrabilit´e du probl`eme du satellite. Annales de la Facult´e des Sciences de Toulouse, 12:25–46, 2003. [3] D. Boucher. Sur les ´equations diff´erentielles lin´eaires param´etr´ees, une application aux syst`emes hamiltoniens. Th`ese de l’Universit´e de Limoges, October 2000.

60

Symmetric and Semisymmetric Graphs Construction Using G-graphs Alain Bretto

∗

Universite´ de Caen, GREYC CNRS UMR-6072, Campus II Bd Marechal Juin BP 5186, 14032 Caen cedex Caen, France

Luc Gillibert

Bernard Laget

Universite´ de Caen, GREYC CNRS UMR-6072, Campus II Bd Marechal Juin BP 5186, 14032 Caen cedex Caen, France

Ecole Nationale d’Ingénieurs de Saint-Etienne, DIPI 58, rue Jean Parot, 42023 Saint-Etienne Cedex 02. France.

[email protected] [email protected] ABSTRACT

graphs have very nice highly-regular properties. But Cayley graphs are always vertex-transitive, and that can be a limitation. In this article we present and use G-graphs introduced in [4, 5]. G-graphs, like Cayley graphs, have highlyregular properties, consequently G-graphs are a good alternative tool for constructing some symmetric graphs. After the definition of these graphs we give a characterization of bipartite G-graphs. Then, using this characterization, we build a powerful algorithm based on G-graphs for computing symmetric graphs. The classification of symmetric graphs is a very interesting problem. Ronald M. Foster started collecting specimens of small cubic symmetric graphs prior to 1934, maintaining a census of all such graphs. In 1988 the current version of the census was published in a book containing some graphs up to the order 512 [7]. But symmetric graphs are not the only interesting graphs. There exist regular graphs which are edge-transitive but not vertex-transitive [6], they are called semisymmetric graphs, and it is quite difficult to construct them [14, 11]. Indeed, Cayley graphs are always regular and vertex-transitive, so they cannot be semisymmetric, but G-graphs can be either regular or non-regular, vertextransitive or not vertex-transitive. In this paper we exhibit an efficient algorithm, based on G-graphs, constructing cubic semisymmetric graphs. So, with G-graphs, it becomes easy not only to extend the The Foster Census up to order 800, but also to construct cubic semisymmetric graphs, quartic symmetric and semisymmetric graphs, quintic symmetric and semisymmetric graphs and so on.

Symmetric and semisymmetric graphs are used in many scientific domains, especially parallel computation and interconnection networks. The industry and the research world make a huge usage of such graphs. Constructing symmetric and semisymmetric graphs is a large and hard problem. In this paper a tool called G-graphs and based on group theory is used. We show the efficiency of this tool for constructing symmetric and semisymmetric graphs and we exhibit experimental results.

Categories and Subject Descriptors G.2.2 [Discrete Mathematics]: Graph Theory—Graph algorithms

General Terms Algorithms, Theory

Keywords Symmetric graphs, semisymmetric graph, graphs from group, G-graphs.

1.

[email protected]

INTRODUCTION

A graph that is both edge-transitive and vertex-transitive is called a symmetric graph. Such graphs are used in many domains, for example, the interconnection network of SIMD computers. But to construct them is a very hard problem. Usually, the graphical representation of groups is used for the construction of those graphs. A popular representation of a group by a graph is the Cayley representation. A lot of work has been done about these graphs [3, 10]. Cayley

2. BASIC DEFINITIONS We define a graph Γ = (V ; E; ) as follows:

∗Authors by alphabetical Order.

• V is the set of vertices and E is the set of edges. •  is a map from E to P2 (V ), where P2 (V ) is the set of subsets of V having 1 or 2 elements.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. ISSAC’05, July 24–27, 2005, Beijing, China. Copyright 2005 ACM 1-59593-095-7/05/0007 ...$5.00.

In this paper graphs are finite, i.e., sets V and E have finite cardinalities. For each edge a, we denote (a) = [x; y] if (a) = {x, y} with x = y or (a) = {x} = {y} if x = y. If x = y, a is called loop. The set {a ∈ E, (a) = [x; y]} is called multiedge or p-edge, where p is the cardinality of the set. We define the degree of x by deg(x) = card({a ∈ E, x ∈

61

(a)}). The line graph of a graph G is the graph obtained by associating a vertex with each edge of G and connecting two vertices with an edge if and only if the corresponding edges of G shares an extremity. In this paper, groups are also finite. We denote the unit element by e. Let G be a group, and let S = {s1 , s2 , . . . , sk } be a nonempty subset of G. S is a set of generators of G if any element θ ∈ G can be written as a product θ = si1 si2 si3 . . . sit with i1 , i2 , . . . it ∈ {1, 2, . . . , k}. We say that G is generated by S = {s1 , s2 , . . . , sk } and we write G = s1 , s2 , . . . , sk . Let H be a subgroup of G. We denote by Hx a right coset of H (with respect to x) in G. A subset TH of G is said to be a right transversal for H if {Hx, x ∈ TH } is precisely the set of all cosets of H in G.

3.

(e a2 a4)

(e a3)

(a a3 a5)

(a a4)

(a2 a5)

3.1 Algorithmic procedure The following algorithm constructs a G-graph from the list L of the cycles of a group: Group_to_graph_G(L) for all s in L Add s to V for all s’ in L for all x in s for all y in s’ if x=y then

GROUP TO GRAPH PROCESS

Let (G, S) be a group with a set of generators S. For any s ∈ S, we consider the left action of the F subgroup H = s on G. Thus, we have a partition G = x∈Ts sx, where Ts is a right transversal of s. The cardinality of s is o(s) where o(s) is the order of the element s. Let us consider the cycles

Add (s,s’) to E

For the construction of the cycles we use the following algorithm, written in the GAP programming language [12]: InstallGlobalFunction ( c_cycles, function(G, ga) local ls1,ls2,gs,k,x,oa,a,res,G2; res:=[]; G2:=List(G); for a in ga do gs:=[]; oa:=Order(a)-1; ls2:=Set([]); for x in G do if not(x in ls2) then ls1:=[]; for k in [0..oa] do; Add(ls1, Position(G2, (a^k)*x)); AddSet(ls2, (a^k)*x); od; Add(gs, ls1); fi; od; Add(res, gs); od; return res; end);

(s)x = (x, sx, s2 x, . . . , so(s)−1 x) of the permutation gs : x −→ sx. Notice that sx is the support of the cycle (s)x. One cycle of gs contains the unit element e, namely (s)e = (e, s, s2 , . . . , so(s)−1 ). We now define a new graph denoted by Φ(G; S) = (V ; E; ) as follows: • The vertices of Φ(G; S) are the cycles of gs , s ∈ S, i.e., V = s∈S Vs with Vs = {(s)x, x ∈ Ts }. • For all (s)x, (t)y ∈ V , {sx, ty} is a p-edge if: card(sx ∩ ty) = p, p ≥ 1 Thus, Φ(G; S) is a k-partite graph and any vertex has a o(s)˜ loop. We denote Φ(G; S) the graph Φ(G; S) without loops. By construction, one edge stands for one element of G. We can remark that one element of G labels several edges. Both ˜ graphs Φ(G; S) and Φ(G; S) are called graph from a group or G-graph and we say that the graph is generated by the group (G; S). Finally, if S = G, the G-graph is called a canonic graph. Example: Let G be the cyclic group of order 6, G = {e, a, a2 , a3 , a4 , a5 }, it is known that G can be generated by an element of order 3 and an element of order 2. Let S be {a2 , a3 }, (a2 )3 = (a3 )2 = e. The cycles of the permutation ga2 are:

4. PROPERTIES OF THE G-GRAPHS We now introduce some useful results: Proposition 1. Let Φ(G; S) = (V ; E; ) be a G-graph. This graph is connected if and only if S is a generator set of G.

(a2 )e = (e, a2 e, a4 e) = (e, a2 , a4 )

Proof. If card(S) = 1, G = s. The graph has only one vertex. So it is connected. AAssume that card(S) ≥ 2. Let (s)x ∈ Vs and (s )y ∈ Vs
, because G = S, there exists s1 s2 s3 . . . sn ∈ S such that y = s1 s2 s3 . . . sn x.

(a2 )a = (a, a2 a, a4 a) = (a, a3 , a5 ) The cycles of ga3 are: (a3 )e = (e, a3 )

x ∈ sx ∩ sn x

(a3 )a = (a, a3 a) = (a, a4 )

sn x ∈ sn x ∩ sn−1 sn x

(a3 )a2 = (a2 , a3 a2 ) = (a2 , a5 )

sn−1 sn x ∈ sn−1 sn x ∩ sn−2 sn−1 sn x

˜ The graph Φ(G; S) is the following:

...

62

that for all t ∈ {1, 2, . . . , p − 1}, for all k ∈ {1, 2, . . . , p − 1} we have:

s2 . . . sn x ∈ s2 s3 . . . sn x ∩ s1 s2 . . . sn x y ∈ s1 s2 . . . sn x ∩ s y

st1 = sk2 ,



Consequently there exists a chain from (s)x ∈ Vs to (s )y ∈ Vs
. So Φ(G; S) is a connected graph. Conversely, let x ∈ G. There exists si1 ∈ S and x1 ∈ Tsi1 such that x ∈ si1 x1 , with x = sti11 x1 . The graph is connected so there exists a chain from (si1 )x1 to (sik )e:

because if st1 = sk2 , p being a prime number, sk2 is generator of s1 , and the following equality becomes true: st1  = s1  = sk2  = s2  = s1 , s2  = G Consequently the only edge between (s1 )e and (s2 )e is the edge corresponding to e. More generally, let (s1 )x and (s2 )y be two cycles. If x = y, let us suppose that there exists t ∈ {1, 2, . . . , p − 1}, and k ∈ {1, 2, . . . , p − 1} such that st1 x = sk2 y. We have st1 = sk2 , and that led us to the first case. So there can be only one edge between (s1 )x and (s2 )y: the edge corresponding to the element x. Let us consider the case where x and y are different. If there is a multi-edge between (s1 )x and (s2 )y, then st1 x = sk2 y and sl1 x = si2 y. We can suppose that l = t + n and i = k + m. So we have the two following equalities:

t

x = sti11 x1 , x1 = sti22 x2 , . . . , xk−1 = sikk xk With xk = e, so x = sti11 sti22 . . . stikk . Proposition 2. Let h be a morphism between (G1 , S1 ) and (G2 , S2 ), then there exists a morphism, Φ(h), between Φ(G1 ; S1 ) and Φ(G2 ; S2 ). Proof. We define Φ(h) = φ = (f, f # ) in the following way: • f : s∈S1 V1,s −→ s∈S2 V2,s

t sl1 x = st+n x = sn 1 (s1 x) 1

(s)x −→ (h(s))h(x)

k y = sm sl1 x = si2 y = sk+m 2 (s2 y) 2

#

• f : E1 −→ E2

t m k t k n So sn 1 (s1 x) = s2 (s2 y), but s1 x = s2 y, consequently s1 = m s2 , and that led us to the first case.

([(s)x, (t)y]; u) −→ ([(h(s))h(x), (h(t))h(y)]; h(u))

We will use this well-known result:

It is easy to verify that φ is a morphism from Φ(G1 ; S1 ) to Φ(G2 ; S2 ). So, any group morphism gives rise to a graph morphism.

Theorem 2. [14] Let Γ be a simple graph. Then Aut1 (Γ)  Aut∗ (Γ) if and only if

For abelian groups we have the following:

(a) not both G1 and G2 , are components of Γ

Theorem 1. Let G1 and G2 be two abelian groups. These two groups are isomorphic if and only if Φ(G1 ; G1 ) and Φ(G2 ; G2 ) are isomorphic. Proof. It has been proved that group isomorphism leads to graph isomorphism. Now, suppose that Φ(G1 ; G1 ) is isomorphic to Φ(G2 ; G2 ). These two graphs have the same degree sequence. Hence the two groups have the same number of elements of the same order. It is known that two abelian groups are isomorphic if and only if they have the same number of elements of the same order. That leads to our assertion.

G1

G2

(b) and none of the graphs Gi , i ∈ {3, 4, 5}, is a component of Γ. x1

x4

x2

x2

We also have: x3

Proposition 3. Let Γ be a connected bipartite and regular G-graph of degree p, p being a prime number, then either Γ is simple or Γ is of order 2.

x4 G3

x1

x1

x3 G4

x2 x4

x3 G5

We are now in position to characterize the bipartite Ggraphs:

Proof. The graph Γ is bipartite and regular of degree p, ˜ so Γ = Φ(G, {s1 , s2 }) with s1 and s2 two different elements of order p. But Γ is a connected graph, so the family {s1 , s2 } generates the group G, in other words G = s1 , s2 . We can notice that the groups s1  and s2  are isomorphic to the cyclic group of order p called Cp . If s1  and s2  are not different we have:

Theorem 3. Let Γ = (V1 , V2 ; E) be a bipartite connected semi-regular simple graph. Let (G, {s1 , s2 }) be a group with o(s1 ) = deg(x), x ∈ V1 and o(s2 ) = deg(y), y ∈ V2 . The three following properties are equivalent: ˜ (i) The graph Γ = (V1 , V2 ; E) is a G-graph, Φ(G; {s1 , s2 }).

s1  = s2  = s1 , s2  = G

(ii) The line graph L(Γ) is a Cayley graph Cay(H; A), (where A = a1  ∪ a2  \ {e}) with (G, {s1 , s2 })  (H, {a1 , a2 })

Therefore Γ is the graph of the cyclic group Cp generated by a family S, with S containing two elements of order p, so the order of the graph Γ is 2. Now let us consider the case where s1  and s2  are different. It is equivalent to saying

(iii) The group G is a subgroup of Aut∗ (Γ) which acts regulary on the set of edges E.

63

˜ Proof. Suppose that Γ is a G-graph Φ(G; {s1 , s2 }). We show that (iii) is true. An edge stands for a unique element of G and an element of G stands for a unique edge. So it is sufficient to show that the action (left multiplication) of G ˜ on itself preserves the graph Φ(G; {s1 , s2 }). Let e1 and e2 be ˜ two adjacent edges of Φ(G; {s1 , s2 }) and let g −1 ∈ G. The images of e1 and e2 are e1 .g −1 and e2 .g −1 . Because e1 and e2 are adjacent we have e1 = sk .u and e2 = sl .u. Hence, e1 .g −1 = sk .u.g −1 = sk .v and e2 .g −1 = sl .u.g −1 = sl .v. Consequently, e1 .g −1 and e2 .g −1 are adjacent and g induced ˜ an automorphism of Aut∗ (Φ(G; {s1 , s2 })). By construction we can remark that, for all x ∈ V1 , an edge ex incident to x is adjacent to s1 .ex , and for all y ∈ V2 , an edge ey incident to y is adjacent to s2 .ey . Assume now that (iii) is true and show that the line-graph of L(Γ) is a Cayley graph verifying the properties of (ii) Let L(Γ) = (L(V ); L(E)) be the line-graph of Γ. It is easy to see that Aut1 (Γ)  Aut(L(Γ)). Moreover, from Theorem 2, we have Aut1 (Γ)  Aut∗ (Γ), that leads to Aut∗ (Γ)  Aut(L(Γ)). Consequently the action of Aut∗ (Γ) on E is equivalent to the action of Aut(L(Γ)) on E. Hence Aut(L(Γ)) contains a subgroup (H, {a1 , a2 }) isomorphic to (G, {s1 , s2 }) which acts regularly on the set of vertices of L(Γ) which characterize the fact that L(Γ) is a Cayley graph. Moreover it is easy to see that o(a1 )−1

A = {a1 , a21 , . . . , a1

o(a2 )−1

, a2 , a22 , . . . , a2

are sorted by their orders and they are listed up to isomorphism. For computing the list of the groups generated by two elements of order 4 we use the following algorithm: result=[]; for all g, group of order n order4=[]; for all x in g if order(x)=4 then add x to order4 end if end for all for all x1 in order4 for all x2 in order4, x2>x1 if <x1,x2>=g add (g,<x1,x2>) to result end for all end for all end for all return result; After the list is established, it is easy to generate all the corresponding G-graphs and to compute their automorphism group with Nauty [17]. If there is only one orbit in the vertex automorphism group, then the graph is vertex-transitive and symmetric. Otherwise, the graph is semisymmetric and there are two orbits, because there is a theorem [14] which affirms that every semisymmetric graph is bipartite. With that algorithm we establish a list of almost all quartic symmetric graphs up to the order 126. The following tables are not exhaustive. When two or more groups give isomorphic quartic graph the name of the two groups are given in the column G. For more information the full tables are on-line at: http://users.info.unicaen.fr/˜bretto (in Publications).

}

Let us suppose that (ii) is true. From L(Γ) we are going to build Γ. For all u ∈ Ta1 , {ai1 .u; aj1 .u} ∈ L(E), 0 ≤ i < j ≤ o(a1 ) − 1 and for all v ∈ Ta2 , {ak2 .v; al2 .v} ∈ L(E), 0 ≤ k < l ≤ o(a2 ) − 1. We have a bijection from H = L(V ) on E such that two elements of L(V ) are adjacent if and only if the corresponding edges in Γ are adjacent. Let us denote o(a )−1 (a1 ).u = (u, a1 .u, a21 .u, . . . , a1 1 .u), for all u ∈ Ta1 , and o(a )−1 (a2 ).v = (v, a2 .v, a22 .v, . . . , a1 2 .v), for all v ∈ Ta2 . Now, let us put an edge between (a1 ).u and (a2 ).v if and only if card(a1 .u ∩ a2 .v) = 1. By construction this graph is a graph which has L(Γ) as line-graph and it is a H-graph. Moreover it has been shown that, if (G; S)  (H; A) then ˜ (G; S)  Φ ˜ (H; A). Φ

5.

THE CONSTRUCTION OF SYMMETRIC AND SEMISYMMETRIC GRAPHS

5.1 Quartic G-graphs Let G be a group of order 4n, G = C4 , and S a family such that G =< S > and S = {a, b}, with a4 = e and ˜ S) is bipartite, edge-transitive b4 = e. Then the graph Φ(G; and quartic. So there are two possibilities: ˜ 1. Φ(G; S) is vertex-transitive, so it is a symmetric quartic graph; ˜ 2. Φ(G; S) is not vertex-transitive, so it is a semisymmetric quartic graph. Therefore G-graphs are a very interesting tool for constructing quartic edge-transitive graphs, especially semisymmetric graphs. We establish a list of all small groups G of order 4n such that G is generated by two elements of order 4. For that we use GAP and the SmallGroups library. This library gives access to all groups of certain small orders. The groups

64

O(Γ) 8 10 12 16

Quartic O(G) 16 20 24 32

16 18 20 24

32 36 40 48

26 30 32 32 32

52 60 64 64 64

34 .. . 120 120 120 120 120

68 .. . 240 240 240 240 240

122 126

244 252

symmetric simple G-graphs G O(Aut(Γ)) sg-16-2, sg-16-3, sg-16-4 1152 sg-20-3 240 sg-24-12 768 sg-32-2, sg-32-10 4096 sg-32-11, sg-32-13 sg-32-6 384 sg-36-9 144 sg-40-12 80 sg-48-11, sg-48-12 98304 sg-48-19, sg-48-48 sg-52-3 104 sg-60-7 720 sg-64-18 4096 sg-64-34 256 sg-64-8, sg-64-21 2097152 sg-64-39 sg-64-41, sg-64-48 sg-68-3 136 .. .. . . sg-240-189 960 sg-240-120 480 sg-240-121 480 sg-240-124 480 sg-240-72, sg-240-73 1.3835e20 sg-240-80, sg-240-197 sg-244-3 488 sg-252-32 1008

Quartic semisymmetric simple G-graphs O(Γ) O(G) G O(Aut(Γ)) 24 48 sg-48-30 3072 32 64 sg-64-9, sg-64-20 294912 sg-64-23, sg-64-32 sg-64-33, sg-64-35 48 96 sg-96-185, sg-96-186 3145728 sg-96-194, sg-96-195 64 128 sg-128-122, sg-128-134 25165824 sg-128-136, sg-128-139 sg-128-141 64 128 sg-128-144, sg-128-146 2048 64 128 sg-128-26, sg-128-71 268435456 sg-128-72, sg-128-75 sg-128-76, sg-128-80 sg-128-118 72 144 sg-144-115, sg-144-116 37748736 72 144 sg-144-120 144 72 144 sg-144-33 2.416e9 80 160 sg-160-83, sg-160-84 83886080 80 160 sg-160-234 335544320 96 192 sg-192-185 192 96 192 sg-192-26, sg-192-32, 1.649e12 sg-192-33, sg-192-34 sg-192-35, sg-192-86 sg-192-114, sg-192-957 sg-192-960, sg-192-971 sg-192-972 96 192 sg-192-955, sg-192-963 1.288e10 sg-192-964, sg-192-969 sg-192-970, sg-192-987 sg-192-989, sg-192-991 100 200 sg-200-42 400 120 240 sg-240-107 1.055e15 120 240 sg-240-192 1440 120 240 sg-240-91 2.577e11 120 240 sg-240-95, sg-240-97 7.731e11

3

10

5

(3 12 6)

(4 10 7)

(1 11 6)

(2 9 7)

(3 10 5)

(4 12 8)

12

6

9

2

11

7

8

5.2 Cubic and quintic symmetric or semisymmetric graphs By the same process it is easy to establish some tables of all quintic and cubic symmetric and semisymmetric Ggraphs. Such tables can be found on-line at: http://users.info.unicaen.fr/˜bretto (in Publications).

O(Γ) 54 112 120 144 216 240 294 336 336 378 384 400 432 432 448 486 546 576 672 702 702 720 784 784 784 798

Cubic O(G) 81 168 180 216 324 360 441 504 504 567 576 600 648 648 672 729 819 864 1008 1053 1053 1080 1176 1176 1176 1197

semisymmetric G-graphs G Aut(Γ) sg-81-7 2 orbits; Order=1296 sg-168-43 2 orbits; Order=168 sg-180-19 2 orbits; Order=720 sg-216-153 2 orbits; Order=432 sg-324-58 2 orbits; Order=648 sg-360-51 2 orbits; Order=1440 sg-441-9 2 orbits; Order=882 sg-504-157 2 orbits; Order=2016 sg-504-158 2 orbits; Order=504 sg-567-21 2 orbits; Order=567 sg-576-5129 2 orbits; Order=2304 sg-600-150 2 orbits; Order=600 sg-648-102 2 orbits; Order=1296 sg-648-702 2 orbits; Order=1296 sg-672-1257 2 orbits; Order=672 sg-729-100 2 orbits; Order=1458 sg-819-6 2 orbits; Order=819 sg-864-2666 2 orbits; Order=1728 sg-1008-517 2 orbits; Order=2016 sg-1053-30 2 orbits; Order=1053 sg-1053-51 2 orbits; Order=1053 sg-1080-487 2 orbits; Order=2160 sg-1176-42 2 orbits; Order=1176 sg-1176-215 2 orbits; Order=1176 sg-1176-220 2 orbits; Order=1176 sg-1197-9 2 orbits; Order=1197

Notice that the the following well-known cubic symmetric or semisymmetric graphs are G-graphs. The corresponding groups are indicated between parenthesis:

Example: Let G be the group sg-12-3 generated by a family S = {a, b} of order 2 with a3 = b3 = e. For information the group ˜ S) is named sg-12-3 is isomorphic to A4 . The graph Φ(G; the following: (2 11 8)

4

It is a symmetric quartic graph isomorphic to the cuboctahedral graph. It is also a G-graph generated by the group G = C2 × C2 × C2 .

These two table have been built in 38 minutes on a 2 gigahertz Athlon (a 32 bit processor). By the implication (i) ⇒ (ii) of Theorem 3, the linegraph of a cubic G-graph is a quartic Cayley graph. Our table of cubic symmetric and semisymmetric G-graphs gives us a table of quartic Cayley graph.

(1 9 5)

1

1. The cube (G = A4 , S = {(1, 2, 3), (1, 3, 4)}) 2. The Heawood’s graph (a, b | a7 = b3 = e, ab = baa, S = {b, ba}) 3. The Pappus’s graph (G = a, b, c | a3 = b3 = c3 = e, ab = ba, ac = ca, bc = cba, S = {b, c})

It is a cubic symmetric G-graph on 8 vertices isomorphic to the skeleton of a cube. If we compute the linegraph of ˜ the G-graph Φ(G; S) we found the following quartic Cayley graph:

4. The Mobius-Kantor’s graph (G =SmallGroup(24,3), S = {f 1, f 1 ∗ f 2})

65

502 528 542 550 562 600 610 622 640 662 682 682 682 682 704 710 722 800

5. The Gray graph (G =SmallGroup(81,7), S = {f 1, f 2}) 6. The Ljubljana graph (G =SmallGroup(168,43), S = {f 1, f 1 ∗ f 2 ∗ f 4}) A census of all symmetric and semisymmetric cubic graphs up to 768 vertices has already been established [8, 9], but our non-exhaustive algorithm is faster. For computing almost all cubic symmetric and semisymmetric G-graphs up to the order 800, except order 768, 1 hour and 56 minutes are necessary on a 2 Gigahertz Athlon. No list was established for quintic symmetric or semisymmetric graphs. The two following non-exaustive lists, built in 32 minutes on a 2 gigahertz Athlon, are the fist one: Quintic semisymmetric G-graphs O(Γ) O(G) G Aut(Γ) 120 300 sg-300-22 1200 240 600 sg-600-54 2400 250 625 sg-625-7 10000 720 1800 sg-1800-555 14400

1255 1320 1355 1375 1405 1500 1525 1555 1600 1655 1705 1705 1705 1705 1760 1775 1805 2000

sg-1255-1 sg-1320-13 sg-1355-1 sg-1375-9 sg-1405-1 sg-1500-112 sg-1525-3 sg-1555-1 sg-1600-6786 sg-1655-1 sg-1705-3 sg-1705-4 sg-1705-5 sg-1705-6 sg-1760-1139 sg-1775-3 sg-1805-2 sg-2000-488

2510 5280 2710 2750 2810 12000 3050 3110 12800 3310 3410 3410 3410 3410 3520 3550 7220 16000

6. REFERENCES [1] Fred Annexstein, Marc Baumslag, and Arnold L. Rosenberg. Group action graphs and parallel architectures, SIAM J. Comput, 19:544–569, 1990. [2] Sheldon Akers and Balakrishnan Krishnamurthy. Group graphs as interconnection networks, In Proc. 14th Int. Conf. Fault Tolerant Comput, pages 422–427, 1984. [3] L. Babai. Automorphism groups, isomorphism, reconstruction. Chapter 27 of Handbook of combinatorics, 1994. [4] A. Bretto and B. Laget. A new graphical representation of a group. Tenth International Conference on Applications of Computer Algebra.(ACA-2004), Beaumont, USA, 23-25 July 2004. National Science Foundation, (NSF), 2004 25-32, (ISBN: 0-9759946-0-3). [5] A. Bretto and L. Gillibert. Graphical and computational representation of groups, LNCS 3039, Springer-Verlag pp 343-350. Proceedings of ICCS’2004. [6] I. Z. Bouwer, An Edge but Not Vertex Transitive Cubic Graph, Bull. Canad. Math. Soc. 11, 533-535, 1968. [7] I.Z. Bouwer, W.W. Chernoff, B. Monson and Z. Star, The Foster Census, Charles Babbage Research Centre, 1988. [8] Marston Conder and Peter Dobcsanyi, Trivalent symmetric graphs on up to 768 vertices, J. Combinatorial Mathematics and Combinatorial Computing 40 (2002), 41-63. [9] Marston Conder, Aleksander Malnic, Dragan Marusic and Primoz Potocnik, A census of semisymmetric cubic graphs on up to 768 vertices, preprint, March 2004. [10] G. Cooperman and L. Finkelstein and N. Sarawagi. Applications of Cayley Graphs. Appl. Algebra and Error-Correcting Codes. Springer Verlag. Lecture Notes in Computer Sciences, Vol. 508 1991, 367–378. [11] J. Folkman, Regular line-symmetric graphs, J. Combinatory Theory, 3:215-232, 1967.

Quintic symmetric G-graphs O(Γ) O(G) G Aut(Γ) 10 25 sg-25-2 28800 22 55 sg-55-1 1320 24 60 sg-60-5 480 32 80 sg-80-49 3840 48 120 sg-120-5 960 50 125 sg-125-3 4000 62 155 sg-155-1 310 64 160 sg-160-199 640 82 205 sg-205-1 410 110 275 sg-275-3 550 122 305 sg-305-1 610 128 320 sg-320-1012 2560 142 355 sg-355-1 710 144 360 sg-360-118 5760 160 400 sg-400-213 3200 162 405 sg-405-15 19440 202 505 sg-505-1 1010 242 605 sg-605-1 1210 242 605 sg-605-5 1210 242 605 sg-605-6 2420 262 655 sg-655-1 1310 264 660 sg-660-13 5280 288 720 sg-720-409 2880 302 755 sg-755-1 1510 310 775 sg-775-3 1550 320 800 sg-800-1065 6400 352 880 sg-880-214 1760 362 905 sg-905-1 1810 382 955 sg-955-1 1910 384 960 sg-960-11357 7680 384 960 sg-960-11358 46080 410 1025 sg-1025-3 2050 422 1055 sg-1055-1 2110 432 1080 sg-1080-260 4320 482 1205 sg-1205-1 2410 486 1215 sg-1215-68 4860

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[12] The GAP Team, GAP - Reference Manual, Release 4.3, www.gap-system.org, May 2002. [13] R. Greenlaw and R. Petreschi, Cubic Graphs. ACM Computing Surveys, 27(4):471–495, 1995. [14] Joseph Lauri and Raffaele Scapellato, Topics in Graphs Automorphisms and Reconstruction, London Mathematical Society Student Texts, 2003.

[15] Joseph Lauri, Pseudosimilarity in graphs - a survey, Ars Combinatoria, 36:171-182, 1997. [16] Joseph Lauri, Constructing graphs with several pseudosimilar vertices or edges, Discrete Mathematics, Volume: 267, Issue: 1-3, p. 197-211. June 6, 2003. [17] Brendan D. McKay, Computer Science Department, Australian National University, (1981), Practical graph isomorphism, Congressus Numerantium 30, p. 45-87.

67

Picard–Vessiot Extensions for Linear Functional Systems Manuel Bronstein

∗

∗

Ziming Li

†

Min Wu

∗,†

†

INRIA – Cafe´ 2004 Route des Lucioles 06902 Sophia Antipolis, France

Key Lab of Math.-Mechan. Acad. of Math. and Syst. Sci. Zhong Guan Cun, Beijing (100080), China

[email protected]

[email protected]

ABSTRACT

[email protected] [email protected]

Categories and Subject Descriptors

the field of coefficients has characteristic 0 and has an algebraically closed constant field, then Picard-Vessiot extensions for such systems contain no new constants. In this paper, rings are not necessarily commutative and have arbitrary characteristic, unless otherwise specified. Ideals and modules are left ideals and left modules. Fields are however always commutative. The notation (·)τ denotes the transpose of vectors or matrices, while R p×q denotes the set of p × q matrices with entries in (the ring) R. The commutator of a, b ∈ R is [a, b] = ab − ba. We write 1R for the identity map on R and 0R for the zero map on R, and we omit the subscripts when the context is clear.

I.1.2 [Computing Methodologies]: Symbolic and Algebraic Manipulation—Algorithms

2.

Picard-Vessiot extensions for ordinary differential and difference equations are well known and are at the core of the associated Galois theories. In this paper, we construct fundamental matrices and Picard-Vessiot extensions for systems of linear partial functional equations having finite linear dimension. We then use those extensions to show that all the solutions of a factor of such a system can be completed to solutions of the original system.

Let R be a ring and σ be an endomorphism of R. A σderivation ([4]) is an additive map δ : R → R satisfying δ(ab) = σ(a)δ(b) + δ(a)b for all a, b ∈ R. A ∆-ring (R, Φ) is a ring R together with a set Φ = {(σ1 , δ1 ), . . . , (σm , δm )}, where each σi is an automorphism of R, each δi is a σi derivation of R, and [σi , σj ] = [δi , δj ] = [σi , δj ] = 0 for all i 6= j. If R is also a field, then (R, Φ) is called a ∆-field. An element c of R is called a constant if σi (c) = c and δi (c) = 0 for all i. The set of all the constants of R is denoted CR and is clearly a subring of R, and a subfield when R is a field. Remark that a ∆-ring is a (partial) differential ring if σi = 1 for all i, and a (partial) difference ring if δi = 0 for all i.

General Terms Algorithms

Keywords Linear functional systems; Picard-Vessiot extensions; Fundamental matrices; Modules of formal solutions.

1.

INTRODUCTION

A linear functional system is a system of form A(Z) = 0 where A is a matrix whose entries are (partial) linear operators, such as differential, shift or q-shift operators or any mixture thereof, and Z denotes a vector of unknowns. A common special case consists of integrable systems, which are of the form {∂i (Z) = Ai Z}1≤i≤m , and correspond to the matrix A given by the stacking of blocks of the form (∂i − Ai ). We show in this paper that fundamental matrices1 and Picard-Vessiot extensions1 always exist for linear functional systems having finite linear dimension1 , which include in particular all integrable systems. In addition, if 1

FULLY INTEGRABLE SYSTEMS

Definition 1. We say that the ∆-ring (R, Φ) is orthogonal if δi = 0 for each i such that σi 6= 1. By reordering the indices, we can assume that there exists an integer ` ≥ 0 such that σi = 1 for 1 ≤ i ≤ ` and δi = 0 for ` < i ≤ m. We write (R, Φ, `) for such an orthogonal ∆-ring. All the δi are usual derivations in an orthogonal ∆-ring. Mixed systems of partial linear differential, difference and qdifference equations can be represented by matrices with entries in Ore algebras ([4]) over orthogonal ∆-rings. Let (F, Φ) be a ∆-field, and suppose that for each i such that σi 6= 1, there exists ai ∈ F such that σi (ai ) 6= ai and σj (ai ) − ai = δj (ai ) = 0 for all j 6= i. Replacing the xi by the ai in the proof of Theorem 1 in [6], one sees that linear functional equations over F can be rewritten as equations over an orthogonal ∆-field. There are however orthogonal ∆rings that do not contain such ai ’s, for example F = C(x) together with Φ = {(1, d/dx), (σx , 0)} where σx is the automorphism of F over C that sends x to x − 1. This field is used in modeling differential-delay equations, and does not match the definition of orthogonality given in [6].

To be defined precisely in Sect. 3 and 5.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. ISSAC’05, July 24–27, 2005, Beijing, China. Copyright 2005 ACM 1-59593-095-7/05/0007 ...$5.00.

68

via δi (U ) = Ai U and for ` + 1 ≤ j ≤ m, the σj are extended to automorphisms of R via σj (U ) = Aj U (σj is bijective because Aj is invertible). It follows from the conditions (2) that these extended maps turn R into a welldefined orthogonal ∆-extension of F and that ∂i (U ) = Ai U for each i. Let D = det(U ) and R be the localization of R with respect to D. Extend the δi and σj via the formulas δi (1/D) = −δi (D)/D2 and σj (1/D) = 1/σj (D), respectively (note that σj (D) = det(Aj )D for j > `). Then R becomes an orthogonal ∆-extension of F , and U is a fundamental matrix of the system. 2 The following proposition reveals that any two fundamental matrices differ by a constant matrix.

Let (F, Φ, `) be an orthogonal ∆-field. We say that a commutative ring E containing F is an orthogonal ∆-extension of (F, Φ, `) if the σi and δi can be extended to automorphisms and derivations of E satisfying: (i) the commutators [σi , σj ] = [δi , δj ] = [σi , δj ] = 0 on E for 1 ≤ i 6= j ≤ m; (ii) σi = 1E for i ≤ ` and δi = 0E for j > `. ˜ be two orthogonal ∆-extensions of F . A Let E and E ˜ is called a ∆-morphism if φ is a ring homap φ from E to E momorphism leaving F fixed and commuting with all the δi and σi . Two orthogonal ∆-extensions of F are said to be isomorphic if there exists a bijective ∆-morphism between them. Definition 2. A system of form δi (Z) = Ai Z for i ≤ `,

σi (Z) = Ai Z for i > ` ,

Proposition 1. Let {∂i (Z) = Ai Z}1≤i≤m be a fully integrable system of size n over F and U ∈ E n×n be a fundamental matrix where E is an orthogonal ∆-extension of F . If V ∈ E n×d with d ≥ 1 is a matrix whose columns are solutions of the system then V = U T for some T ∈ CEn×d . In particular, any solution of {∂i (Z) = Ai Z}1≤i≤m in E n is a linear combination of the columns of U over CE .

(1)

where Ai ∈ F n×n and Z = (z1 , . . . , zn )τ is called an integrable system if the following conditions are satisfied: σi (Aj )Ai + δi (Aj ) = σj (Ai )Aj + δj (Ai )

for all i, j. (2)

The integrable system (1) is said to be fully integrable if the matrices A`+1 , . . . , Am are invertible.

Proof. Let T = U −1 V . A straightforward calculation implies that δi (T ) = 0 for i ≤ `, and σj (T ) = T for j > `. Hence all the entries of T belong to CE . 2 In [10, 11], Picard-Vessiot rings for linear ordinary differential and difference systems are defined. Picard-Vessiot fields for integrable systems of partial differential equations have been studied by Kolchin who proved their existence and developed the associated Galois theory [2, §2][5]. PicardVessiot extension fields have also been defined in [1] for fields with operators, which are more general ∆-fields where the operators do not necessarily commute. While the associated Galois theory was developed there, the existence of Picard-Vessiot extensions was not shown. Indeed, with automorphisms allowed, there are fully integrable systems for which no Picard-Vessiot field exists. Generalizing the definition of Picard-Vessiot rings used for difference equations [10, (Errata)], we obtain Picard-Vessiot rings together with a construction proving their existence. Our definition is compatible with the previous ones: for differential systems, Picard-Vessiot rings turn out to be integral domains, and the Picard-Vessiot fields of [5] are their fields of fractions; For ∆-rings, the Picard-Vessiot rings are generated by elements satisfying linear scalar operator equations, which is the defining property of the Picard-Vessiot fields of [1]. An ideal I of a commutative ∆-ring R is said to be invariant if δi (I) ⊂ I and σi (I) ⊂ I for all 1 ≤ i ≤ m. The ring R is said to be simple if its only invariant ideals are (0) and R.

Using Ore algebra notation, we write {∂i (Z) = Ai Z}1≤i≤m for the system (1) where the action of ∂i is meant to be δi for i ≤ ` and σi for i > `. Note that the conditions (2) are derived from the condition ∂i (∂j (Z)) = ∂j (∂i (Z)) and are exactly the matrix-analogues of the compatibility conditions for first order scalar equations in [6]. Example 1. Let F = C(x, k) and δx and σk denote respectively the ordinary differentiation w.r.t. x and the shift operator w.r.t. k. Then {δx (Z)=Ax Z, σk (Z)=Ak Z} where   2 2 Ax = 

 Ak = 

x −kx−k x(x−k)(x−1)

x −kx+3k−2x kx(x−k)(x−1)

k(kx+x−x2 −2k) (x−k)(x−1)

x3 +x2 −kx2 −2x+2k x(x−k)(x−1)



k+1+kx2 −xk2 −x (x−k)(x−1)

−x − k+1+kx−k k(x−k)(x−1)



x(k+1)(k+1+kx−k2 −x) (x−k)(x−1)

(k+1)(x2 −2kx−x+k2 ) k(x−k)(x−1)



2

is a fully integrable system.

3.

FUNDAMENTAL MATRICES AND PICARD-VESSIOT EXTENSIONS

A square matrix with entries in a commutative ring is said to be invertible if its determinant is a unit in that ring. Let the orthogonal ∆-ring (F, Φ, `) be as in the previous section and {∂i (Z) = Ai Z}1≤i≤m be a fully integrable system of size n over F . An n × n matrix U with entries in an orthogonal ∆-extension E of F is a fundamental matrix for {∂i (Z) = Ai Z}1≤i≤m if U is invertible and ∂i (U ) = Ai U for each i, that is, each column of U is a solution of the system.

Definition 3. Let {∂i (Z) = Ai Z}1≤i≤m be a fully integrable system over F . A Picard-Vessiot ring for this system is a (commutative) ring E such that: (i) E is a simple orthogonal ∆-extension of F . (ii) E = F [U, det(U )−1 ] for some fundamental matrix U for the system.

Theorem 1. For every fully integrable system, there exists a fundamental matrix whose entries lie in an orthogonal ∆-extension of F .

We now construct Picard-Vessiot rings by the same approach used in the ordinary differential and difference cases [10, 11].

Proof. Let {∂i (Z) = Ai Z}1≤i≤m be a fully integrable system of size n over F , U = (ust ) be a matrix of n2 distinct indeterminates and R = F [u11 , . . . , u1n , . . . , un1 , . . . , unn ]. For 1 ≤ i ≤ `, the δi are extended to derivations of R

Lemma 1. Let R be an orthogonal ∆-extension of F and I a maximal invariant ideal in R. Then, (i) E := R/I is a

69

Let E be the orthogonal ∆-extension F [T, T −1 ] such that δi (T ) = ai T for i ≤ ` and σj (T ) = aj T for j > `. Case 1. There does not exist an integer k > 0 and r ∈ F ∗ such that δi (r) = kai r for i ≤ ` and σj (r) = akj r for j>`. Then E is a Picard-Vessiot ring of (3). Case 2. Assume that the integer k > 0 is minimal so that δi (r) = kai r and σj (r) = akj r for some r ∈ F ∗ and for
all i ≤ ` and j>`. Then E/ T k − r is a Picard-Vessiot ring of (3). The verification of the above two assertions is similar to that in Example 1.19 in [11]. Unlike in the differential case, the elements of PicardVessiot rings cannot always be interpreted as complex functions: the system {dy/dx = y(x), y(x + 1) = y(x)} is in Case 1 above and has a Picard-Vessiot ring over C(x), but has no nonzero complex function solution. Next, we show that a Picard-Vessiot ring of the system in Example 1 is F [ex , e−x , Γ(k), Γ(k)−1 ] where F = C(x, k). Note that the change of variable1 Z = M Y , where   x−k x2 x , M= 2 (x − k)k x k

simple orthogonal ∆-extension of F . (ii) CE is a field. (iii) If F has characteristic 0, CF is algebraically closed and E is a finitely generated algebra over F , then CE = CF . k

`+1 km Proof. Let I = {σ`+1 · · · σm (a)| a ∈ I, k`+1 , . . . , km ∈ Z}. One can verify that I is an invariant ideal containing I but 1 ∈ / I, and hence I = I since I is maximal. The δi and σj can be viewed as derivations and surjective endomorphisms on E = R/I via the formulas δi (a + I) = δi (a) + I and σj (a + I) = σj (a) + I for all a in R, respectively. If σj (a + I) = I then σj (a) ∈ I = I and thus a ∈ I. So the σj are automorphisms of E and E is a simple orthogonal ∆-extension of F . To show the second statement, let c be a nonzero constant of E. Then the ideal (c) is invariant. Since E is simple, (c) contains 1. To show the last statement, suppose that b ∈ CE but b ∈ / CF . By the argument used in the proof of Lemma 1.8 in [10], there exists a nonzero monic polynomial g overF with minimal degree d such that  P k = 0. Apply the δi and σj to g(b), g(b) = bd + d−1 k=0 gk b  P d−1 k = 0 for i ≤ ` , and respectively, we obtain k=0 δi (gk )b  P d−1 k = 0 for j > `. The minimality of d k=0 (σj (gk ) − gk )b then implies gk ∈ CF for 0 ≤ k < d. So b ∈ CF since CF is algebraically closed, a contradiction. 2 The existence of the Picard-Vessiot extensions is stated in the next theorem.

transforms the system into B : {δx (Y )=Bx Y, σk (Y )=Bk Y },     1 0 1 0 where Bx = and Bk = . 0 0 0 k Thus we need only to find a Picard-Vessiot ring of B. First, let U be a 2 × 2 matrix with indeterminate entries u11 , u12 , u21 and u22 . Define δx (U ) = Bx U and σk (U ) = Bk U . This turns R = F [u11 , u12 , u21 , u22 , 1/ det(U )] into an orthogonal ∆-extension of F . Clearly, I = (u12 , u21 ) is an invariant ideal of R and σk−1 (I) is contained in I. Hence R/I is an −1 orthogonal ∆-ring. As the ∆-rings E = F [u11 , u22 , u−1 11 , u22 ] and R/I are isomorphic, it suffices to show that E is simple. Suppose that J is a nontrivial invariant ideal of E. Let f be a nonzero polynomial in I ∩ F [u11 , u22 ] with the smallest number of terms. It cannot be a monomial, for otherwise J −1 would be E since u−1 11 and u22 are in E. We write

Theorem 2. Every fully integrable system over F has a Picard-Vessiot ring E. If F has characteristic 0 and CF is algebraically closed, then CE = CF . Furthermore, that extension is minimal, meaning that no proper subring of E satisfies condition (ii) of Definition 3. Proof. Let {∂i (Z) = Ai Z}1≤i≤m be a fully integrable system over F . By Theorem 1, it has a fundamental matrix U = (ust ) with entries in the orthogonal ∆-extension R = F [u11 , . . . , unn , det(U )−1 ].

f = ud111 ud222 + rue111 ue222 + other terms,

Let I be a maximal invariant ideal of R and E = R/I. Then E is a simple orthogonal ∆-extension of F by Lemma 1. Clearly, E is generated over F by the entries of the matrix U := (ust + I) and by det(U )−1 . Since U is a fundamental matrix for the system, E is a Picard-Vessiot ring for the system. Assume further that F has characteristic 0 and CF is algebraically closed. Then CE = CF by the third assertion of Lemma 1. Let S = F [V, det(V )−1 ] be a subring of E where V is some fundamental matrix of the system. By n×n such that V = U T . Proposition 1, there exists T ∈ CE Since CE = CF , all the entries of U and the inverse of det(U ) are contained in S. Hence S = E. 2 Assume that the ground field F has characteristic 0 with an algebraically closed field of constants. Let E be a PicardVessiot ring for a fully integrable system of size n over F . Then Proposition 1 together with CE = CF implies that all the solutions of this system in E n form a CF -vector space of dimension n. A direct generalization of Proposition 1.20 in [11] and Proposition 1.9 in [10] reveals that any two Picard-Vessiot rings for a fully integrable system over F are isomorphic as orthogonal ∆-extensions. We present a few examples for Picard-Vessiot rings. Consider the fully integrable system of size one: ∂i (z) = ai z

where ai ∈ F and i = 1, . . . , m.

where r ∈ F with r 6= 0, and (d1 , d2 ) 6= (e1 , e2 ). It follows from δx (u11 ) = u11 and δx (u22 ) = 0 that δx (f ) = d1 ud111 ud222 + (δx (r) + e1 r)ue111 ue222 + other terms, in which each monomial has already appeared in f . Thus (δx (f ) − d1 f ) must be zero, because it is in I but has fewer terms. It follows that (δx (r) − (d1 − e1 )r) is equal to
zero. In the same way, one can show that σk (r) − kd2 −e2 r = 0, because σk (u11 ) = u11 and σk (u22 ) = ku22 . But the existence of such a rational function r would imply d1 =e1 and d2 =e2 , a contradiction. Thus E is simple, and so a Picard-Vessiot ring of B, hence also of the system in Example  1.x If we un e 0 x derstand u11 as e and u22 as Γ(k), then V = 0 Γ(k) is a fundamental matrix for B in E, and hence M V is for the system in Example 1. Last, we describe a simple orthogonal ∆-extension that contains a solution of the inhomogeneous system δi (z) = ai for i ≤ `

and

σj (z) = z + aj for j > `,

(4)

1 which can be found, for example, by computing the hyperexponential solutions of the system ([6, 12])

(3)

70

where the ai and aj are in a simple orthogonal ∆-ring E with characteristic zero. This is an extension of Example 1.18 in [11]. Note that the ai and aj have to satisfy some compatibility conditions due to the commutativity of the δi and σj . A more general form for these conditions are given in (8) in the next section. If (4) has a solution in E, then there is nothing to do. Otherwise, let R = E[T ] and extend the δi and σj on R by the formulas δi (T ) = ai and σj (T ) = T + aj . The compatibility conditions imply that R becomes a well-defined orthogonal ∆-ring. If R has a nontrivial invariant ideal I, let f = fd T d + fd−1 T d−1 + · · · + f0 be a nonzero element in I with minimal degree. Let J be the set consisting of zero and leading coefficients of elements in I with degree d. Our extensions of δi and σj imply that J is an invariant ideal of E. Hence 1 ∈ J and, therefore, we may also assume d > 0 and fd = 1. Since d is minimal, both δi (f ) −fd−1 is a solution and (σj (f ) − f ) are 0. Consequently, d of (4), a contradiction. Thus R is simple and contains a solution T of (4).

4.

which implies that the Bi and Di also satisfy the compatibility conditions (2). Therefore B and D are both fully integrable. The first statement is proved. The second is immediate from (6). From Theorem 1, there exist an orthogonal ∆-extension E of F and a fundamental matrix U with entries in E for D. Let η = (η1 , . . . , ηd )τ be a solution of B in some orthogonal ∆-extension R of F . Viewing E and R as commutative F -algebras, we can extend the δi and σj to the commutative E-algebra E ⊗F R via δi (e ⊗ r) = δi (e) ⊗ r + e ⊗ δi (r) and σj (e ⊗ r) = σj (e) ⊗ σj (r) for i ≤ ` and j > `. Then (1 ⊗ η1 , . . . , 1 ⊗ ηd )τ is also a solution of B, so, replacing R by E ⊗F R, we can assume without loss of generality that R contains E. Substitute η into (6) to get ∂i (Y ) = Di Y +Ci η for each i. Let v = (v1 , . . . , vn−d )τ , where the vk are distinct indeterminates over R, and G = R[v1 , . . . , vn−d ]. We extend the δi and σj to G via δi (v)=bi and σj (v)=v + bj where b1 , . . . , bm ∈ Rn−d are given by bi = U −1 Ci η for i ≤ ` and bj = U −1 Dj −1 Cj η for j > `. To turn G into an orthogonal ∆-extension of R, all the δi and σj on G should commute, which is equivalent to the following integrability conditions:  for 1 ≤ i, j ≤ `,  δi (bj ) = δj (bi ), δi (bj ) = σj (bi ) − bi , for i ≤ `, j > `, (8)  σi (bj ) − bj = σj (bi ) − bi , for ` + 1 ≤ i, j ≤ m.

COMPLETING PARTIAL SOLUTIONS

We now consider reducible systems, i.e. systems that can be put into simultaneous block-triangular form by a change of variable Y = M Z for some M ∈ GLn (F ). Factorization algorithms for modules over Laurent–Ore algebras [12] yield such a change of variable for reducible systems, and we motivate them by showing that the solutions of a factor can always be extended to solutions of the complete system.

Although the conditions (8) are generally not satisfied for arbitrary bi ’s, we show that they are satisfied in our case. Since the Ai satisfy the compatibility conditions (2), it follows from the bottom-left block in (7) that, for all i, j,

Theorem 3. Let A: {∂i (Z) = Ai Z}1≤i≤m be a fully integrable system of size n over F , and suppose that there exist a positive integer d < n and matrices Bi in F d×d , Ci in F (n−d)×d and Di in F (n−d)×(n−d) such that   Bi 0 Ai = for 1 ≤ i ≤ m . (5) Ci Di

σi (Cj )Bi +σi (Dj )Ci +δi (Cj )=σj (Ci )Bj +σj (Di )Cj +δj (Ci ). (9) For 1 ≤ i, j ≤ `, we have
δi (bj ) = δi U −1 Cj η

Then

= =

(i) B : {∂i (X)=Bi X}1≤i≤m and D : {∂i (X)=Di X}1≤i≤m are both fully integrable systems.

which, together with σi = σj = 1 for 1 ≤ i, j ≤ `, and (9) implies δi (bj ) = δj (bi ). The last two integrability conditions in (8) are verified with similar calculations, using the fact that the Di satisfy the compatibility conditions (2). Therefore G is an orthogonal ∆-extension of R, hence of F . Let ζ = U v ∈ Gn−d . Then, for i ≤ `,

(ii) (0, . . . , 0, ζd+1 , . . . , ζn )τ is a solution of A whenever (ζd+1 , . . . , ζn )τ is a solution of D. (iii) For any solution (η1 , . . . , ηd )τ of B in an orthogonal ∆extension of F , there exists an orthogonal ∆-extension of F containing η1 , . . . , ηd as well as ηd+1 , . . . , ηn such that (η1 , . . . , ηn )τ is a solution of A.

∂i (ζ) = δi (ζ) = δi (U )v + U δi (v) = Di U v + U bi = Di ζ + Ci η,

Proof. Let X = (z1 , . . . , zd )τ and Y = (zd+1 , . . . , zn )τ . The system A can then be rewritten into a homogeneous system and an inhomogeneous system:  ∂i (X) = Bi X, for 1 ≤ i ≤ m . (6) ∂i (Y ) = Di Y + Ci X,

and, for j > `, ∂j (ζ) = σj (ζ) = σj (U )σj (v) = Dj U (v + bj ) = Dj ζ + Cj η. So (η τ , ζ τ )τ is a solution of the initial system A. 2 We point out here (but omitting the detailed explanation) that in the differential case, the quotient systems of [7] yield an alternative approach to completing solutions of factors.

Since A is fully integrable, the matrices Ai satisfy (2) and Aj is invertible for j > `. Hence, the Bj and Dj for j > ` must also be invertible since det(Aj ) = det(Bj ) det(Dj ). In addition, a routine calculation shows that for all i, j,

Example 2. Let F , δx and σk be as in Example 1, and consider the fully integrable system       Bx 0 Bk 0 δx (Z) = Z, σk (Z) = Z Cx Dx Ck Dk (10)

σi (Aj )Ai + δi (Aj ) = 

σi (Bj )Bi + δi (Bj ) 0 σi (Cj )Bi + σi (Dj )Ci + δi (Cj ) σi (Dj )Di + δi (Dj )

−U −1 δi (U )U −1 Cj η+U −1 δi (Cj )η+U −1 Cj δi (η) −U −1 (Di Cj − δi (Cj ) − Cj Bi ) η ,

 , (7)

71

where Z = (z1 , z2 , z3 )τ , Bx =  2 Cx = 

 Ck = 

x+k , x

Bk =

(k+1)x , k

2x −k2 +2x−kx x(x−k)

x3 −x2 k+2x2 −kx+2x−k2 (x−k)x

,

(k+1)(x3 −2x2 k−3x2 +k2 x+4kx+x−k2 ) k(x−k−1)2 x2 (k+1) k

−

(k+1)(x−k)2 k(x−k−1)2

5.



− xk(x − 1)

We now generalize the previous notions and results to systems of the form A(Z) = 0 where A is a matrix of linear operators. As in previous sections, let (F, Φ, `) be an orthogonal ∆-field and S = F [∂1 ; σ1 , δ1 ] · · · [∂m ; σm , δm ] be the corresponding Ore algebra [4]. In the differential case, an S-module is classically associated to such a system [8, 11]. In the difference case, however, S-modules do not have appropriate dimensions, so modules over Laurent algebras are used instead [9, 10, 13]. It is therefore natural to introduce in our setting the following extension of S: let θ`+1 , . . . , θm be indeterminates independent of the ∂i . Since the σj−1 are also −1 −1 , 0] · · · [θm ; σm , 0] is automorphisms of F , S = S[θ`+1 ; σ`+1 −1 also an Ore algebra. Since (∂j θj )a = ∂j σj (a)θj = a∂j θj for any j > ` and any a ∈PF , ∂j θj is in the center of S. Therefore the left ideal I = m j=`+1 S(∂j θj −1) is a two-sided ideal of S, and we call the factor ring R=S/I the Laurent-Ore algebra generated by Φ over F . Writing ∂j−1 for the image of θj in R, we can also write R (by convention) as

 ,

and Dx =

−2−x+k x−k

0

−2x−x2 +k2 (x−k)x

k x

(k+1)(x−k)2 k(x−k−1)2



! ,

Dk = 

(k+1)(x−k)2 k(x−k−1)2

0



. − kx xk

We complete the solution η1 = kex xk of the system given by Bx and Bk to a solution of (10). Note that ! ke−x 0 (x−k)2 U= ke−x Γ(k)xk (x−k) 2 is a fundamental matrix for the system given by Dx and Dk . By the proof of Theorem 3, we let ! kx x e Γ(k) , b1 = (x − k)(2x2 − k2 + 2x − kx) xk−1 e2x 2

b2 =

2

R

(x3 − 2kx2 − 3x + k x + 4kx + x − k2 )xk e2x

.

We find that v=

Γ(k)−kex +xkex Γ(k) k+1 2x k 2 2x

xk+2 e2x − 2x

ke

+x k e

:= F [∂1 ; 1, δ1 ] · · · [∂` ; 1, δ` ] −1 −1 [∂`+1 , ∂`+1 ; σ`+1 , 0] · · · [∂m , ∂m ; σm , 0]

and view it as an extension of S. For linear ordinary difference equations, R = F [σ, σ −1 ], is the algebra used in [10]. For linear partial difference equations with constant coefficients, R is the Laurent polynomial ring used in [9, 13]. Laurent-Ore algebras allow us to construct fundamental matrices and Picard-Vessiot extensions for linear functional systems of finite linear dimension, a concept that we now define precisely. For our purposes, a linear functional system is a matrix A=(aij ) ∈ S p×q ⊂ R p×q . For any R-module N , we can associate to A a CF -linear map λ : N q → N p given by

!

x+kx+k −xk −k−1 x e Γ(k+1) 2 2

MODULES AND PICARD-VESSIOT RINGS FOR GENERAL LINEAR FUNCTIONAL SYSTEMS

! +1

satisfies δx (v) = b1 and σk (v) − v = b2 . Therefore,   kex xk   η1 ke−x   kex xk + (x−k) = 2  U −1 v −x k ke xk+1 kex + (x−k) 2 + Γ(k)x

  Pq  ξ1 j=1 a1j ξj     .. ξ :=  ...  → 7 Aξ =  . Pq . ξq a ξ pj j j=1 

is a solution of (10).

We therefore say that ξ ∈ N q is a solution “in N ” of the system A(Z) = 0 if A(ξ) = 0, and write solN (A(Z) = 0) for all its solutions in N . Clearly, solN (A(Z) = 0) is a vector space over CF . As in the case of D-modules [8], we can associate to A an R-module as follows: the matrix A ∈ Rp×q induces the R-linear map ρ : R1×p → R1×q given by ρ(r1 , . . . , rp ) = (r1 , . . . , rp )A. Let M = coker(ρ) = R1×q /R1×p A, which is simply the quotient of R1×q by the submodule generated by the rows of A. Then

Theorem 3 also yields fundamental matrices for reducible systems. Let {∂i (Z) = Ai Z}1≤i≤m be a fully integrable system where the Ai are as in (5). Suppose that U =(uij )∈Rd×d and V ∈ E (n−d)×(n−d) are fundamental matrices for the systems {∂i (X)=Bi X}1≤i≤m and {∂i (X)=Di X}1≤i≤m respectively, where R and E are orthogonal ∆-extensions of F . As in the procedure of completing solutions, we can assume without loss of generality that R contains E. Then a fundamental matrix for the initial system can be constructed as follows: for each 1 ≤ i ≤ d, following the procedure of completing solutions, we can find an orthogonal ∆-extension Gi of R and ξi ∈ Gn−d such that (u1i , . . . , udi , ξiτ )τ ∈ Gn i is i a solution of {∂i (Z) = Ai Z}1≤i≤m . Viewing all the entries of U ,  V and the ξi as elements of G = G1 ⊗F · · · ⊗F Gd ,  U 0 n×n W = ∈ G is easily seen to be a ξ1 . . . ξd V fundamental matrix for {∂i (Z) = Ai Z}1≤i≤m (it is invertible because det(W ) = det(U ) det(V )).

ρ

π

R 1×p −→ R 1×q −→ M −→ 0

(11)

is an exact sequence of R-modules where π : R 1×q → M is the canonical map. For every s ≥ 1 and 1 ≤ i ≤ s, let eis be the unit vector in R 1×s with 1 in the ith position and 0 elsewhere. Then e1p , . . . , epp and e1q , . . . , eqq are canonical bases of R 1×p and R 1×q , respectively. Set ej = π(ejq ) for 1 ≤ j ≤ q. Since π is surjective, e1 , . . . , eq generate M

72

as an R-module. Since ρ(eip ) is the i-th row of A, we have ! q q q X X X 0 = π(ρ(eip )) = π aij ejq = aij π(ejq ) = aij ej , j=1

j=1

every element of R is congruent to an element of S modulo I (note that every element of R can be written as an element of S multiplied by the inverse of an element of H from the right-hand side). Let φ be the map from S/J¯ to R/I that sends a + J¯ to a + I for a ∈ S. The map is well-defined, injective and linear over F because J¯ = S ∩ I. By the conclusion made in the previous paragraph, for every element (b + I) of R/I with b ∈ R, there exists b0 in S such that b ≡ b0 mod I.
0 ¯ Thus φ b + J = b + I. The map φ is surjective. 2

j=1 τ

for 1 ≤ i ≤ p, which implies that (e1 , . . . , eq ) is a solution of A(Z) = 0 in M . Given two R-modules N1 and N2 , let HomR (N1 , N2 ) denote the CF -vector space of all the R-linear maps from N1 to N2 . We next show that the proof of Proposition 1.1 of [8] remains valid when D is replaced by R.

Example 3. Consider a p×1 matrix A = (L1 , . . . , Lp )τ , where the Li are in S. The system A(z) = 0 corresponds to scalar equations L1 (z) = · · · = Lp (z) = 0, whose R-module of formal P solutions is M = R/ρ(R1×p ) = R/I, where P I is the left ideal pi=1 RLi of R. Let J be the left ideal pi=1 SLi of S. Then, by Lemma 2, dimF M is finite if dimF S/J is finite and J contains no monomial in ∂`+1 , . . . , ∂m . Consider the case ` = 0 and m = 2. If J is S(∂1 + 1), then dimF (M ) is not finite. On the other hand, if J is equal to S(∂1 ∂2 (∂1 + 1)) + S(∂1 ∂2 (∂2 + 1)), then dimF S/J is not finite, but dimF M = 1, because I = R(∂1 + 1) + R(∂2 + 1).

Theorem 4. Let M =R1×q /R1×p A. Then solN (A(Z)=0) and HomR (M, N ) are isomorphic as CF -vector spaces for any R-module N . Proof. Applying the functor HomR (·, N ) to the exact sequence (11) of CF -vector spaces and using the isomorphism HomR (R 1×s , N ) → N s given by f 7→ (f (e1s ), . . . , f (ess ))τ , we get the exact sequence: π∗

λ

0 −→ HomR (M, N ) −→ N q −→ N p , in which π ∗ (f ) = (f (e1 ), . . . , f (eq ))τ and λ ((n1 , . . . , nq )τ ) = A(n1 , . . . , nq )τ for n1 , . . . , nq in N . Since π ∗ is injective, HomR (M, N ) ' Im(π ∗ )= ker(λ)=solN (A(Z)=0). 2 Theorem 4 reveals that e := (e1 , . . . , eq )τ ∈ M q is a “generic” solution of the system A(Z) = 0 in the sense that any solution of A(Z) = 0 is the image of e under some homomorphism. This means that M describes the properties of all the solutions of A(Z) = 0 “anywhere”. So we define

Example 4 (Integrable systems). Let A1 ,. . . , Am be in F n×n , 1n be the identity matrix in F n×n and   ∂1 · 1n − A1   .. mn×n A= . ∈S . ∂m · 1n − Am The system A(Z) = 0 corresponds to {∂i (Z) = Ai Z}1≤i≤m , which is not necessarily fully integrable. Let M be its module of formal solutions and e = (e1 , . . . , en )τ ∈ M n be as above. Then A(e) = 0 implies that ∂i e P = Ai e for each i. Since the entries ofPAi are in F , ∂i ej ∈ n i, j, and s=1 F es for Pall n n thus Re ⊂ F e for all j. Hence M = j s s=1 s=1 Res = Pn s=1 F es . In particular, dimF M ≤ n.

Definition 4. Let A ∈ S p×q ⊂ Rp×q . We call the Rmodule M = R1×q /R1×p A the module of formal solutions of the system A(Z) = 0. The dimension of M as an F -vector space is called the linear dimension of the system. The system is said to be of finite linear dimension if 0 < dimF M < +∞.

To check in practice whether a system is of finite linear dimension, we need to compute dimF M . As seen in Example 4, when the system is given as an integrable system, we have a set of generators for M over F , so computing dimF M can be done by linear algebra over F as in Example 5. Note that in the purely differential case, we have dimF M = n if the matrices Ai satisfy (2), dimF M = 0 otherwise. When the system is given by an ideal in S, then Lemma 2 shows that either M = 0 (if the ideal contains a monomial in ∂`+1 , . . . , ∂m ) or an F -basis of M can be computed via Gr¨ obner bases of S-modules. There are algorithms and implementations for this task [3, 4]. For more general matrices A ∈ S p×q , computing an F -basis of M involves computing Gr¨ obner bases of R-modules. In the purely differential case, this is again Gr¨ obner bases of S-modules. When difference operators are involved, the algorithms developed in [9, 13] for pure difference equations with constant coefficients are generalized in [12] to produce Gr¨ obner bases of R-modules. Let A ∈ S p×q and M be the R-module of formal solutions for A(Z)= 0. Suppose that dimF M = n and b1 , . . . , bn form a basis of M over F . Then, for b := (b1 , . . . , bn )τ there exists Bi ∈ F n×n such that ∂i (b) = Bi b for each i. We can regard M as the module of formal solutions for the integrable system {∂i (X) = Bi X}1≤i≤m . Indeed, suppose we find, as described in Example 4, its module MB of formal

Note that we choose to exclude systems with dimF M = 0 in our definition since such systems cannot have nonzero solutions in any R-module (which includes all orthogonal ∆extensions of F ). The next lemma is used to describe modules of formal solutions for finite-rank left ideals in S ([6]). Lemma 2. Let J be a left ideal of S. Assume that J does not contain any monomial in ∂`+1 , . . . , ∂m , and that S/J is finite dimensional over F . Let I be the left ideal generated by J in R and J¯ = I ∩ S. Then S/J¯ and R/I are isomorphic as vector spaces over F . In particular, R/I is finite dimensional over F . Proof. Let H be the set of all monomials in ∂`+1 , . . . , ∂m . Since every element of H is invertible in R, J¯ = {a ∈ S | ha ∈ J for some h ∈ H}. (12)
¯ ¯ Since J ⊂ J, dimF S/J is finite. Let fj be a nonzero polynomial in F [∂j ] ∩ J¯ with minimal degree for j > `. Then each fj is of positive degree with a nonzero coefficient of ∂j0 = 1, for otherwise, J¯ would contain 1, and, hence, J would have a nonempty intersection with H by (12), a contradiction to our assumption. Since ∂j−1 fj ∈ I, ∂j−1 is congruent to an element of F [∂j ] modulo I. It follows that

73

P solutions and f := (f1 , . . . , fn )τ such that MB = n s=1 F fs n and ∂i (f ) = Bi f for each i. Since b ∈ M is a solution of {∂i (X) = Bi X}1≤i≤m , there exists ϕ ∈ HomR (MB , M ) such that b = ϕ(f ) by Theorem 4. Since the bi are linearly independent over F , so are the fi . Hence MB = ⊕n s=1 F fs and ϕ is an isomorphism of R-modules. Since ∂i and ∂j commute for any i and j, ∂i (∂j (b)) = ∂j (∂i (b)). From ∂i (b) = Bi b and the linear independence of b1 , . . . , bn over F , it follows that σi (Bj )Bi + δi (Bj ) = σj (Bi )Bj + δi (Bj ),

Although this is not stated in the definition, it follows from Proposition 2 that the columns of a fundamental matrix form a CE -basis of the CE -module solE (A(Z) = 0): denote solE (A(Z)=0) and solE ({∂i (X) = Bi X}1≤i≤m ) by WA and WB respectively. Then the columns of V = P U are n×1 in WA by Proposition 2. Let c ∈ CE be such that 0=V c = P U c. Since U c ∈ WB , we have U c = 0 by Proposition 2, hence c=0 since U is invertible. Thus the columns of V are linearly independent over CE . For any η ∈ WA there exists ξ ∈ WB such that η = P ξ. By Proposition 1 there n×1 exists c ∈ CE such that ξ = U c. Hence η = P U c = V c. Let b1 , . . . , bn and d1 , . . . , dn be two bases of M over F . Write b = (b1 , . . . , bn )τ and d = (d1 , . . . , dn )τ , and let T ∈ GLn (F ) be given by d = T b. For each i, let Bi , Di ∈ F n×n be such that ∂i (b) = Bi b and ∂i (d) = Di d. If E is a PicardVessiot ring for {∂i (X) = Bi X}1≤i≤m and U ∈ E n×n is a corresponding fundamental matrix, then T U is a fundamental matrix for {∂i (Y ) = Di Y }1≤i≤m by Theorem 4, so E is a Picard-Vessiot ring for that system too. This justifies the second part of Definition 5. As a final consequence of Theorems 1 and 2, we have

1 ≤ i, j ≤ m,

i.e. B1 , . . . , Bm satisfy the compatibility conditions (2). Suppose that Bt is singular for some t > `. Then, there exists a nonzero v ∈ F 1×n such that vBt = 0 and thus v∂t (b) = vBt b = 0. Since M is an R-module on which ∂t−1 acts, we have 0 = ∂t−1 (v∂t (b)) = σt−1 (v)∂t−1 (∂t (b)) = σt−1 (v)b, which implies that b1 , . . . , bn are linearly dependent over F , a contradiction. So the Bj are invertible for ` + 1 ≤ j ≤ m and the system {∂i (X) = Bi X}1≤i≤m is fully integrable. We call it2 the fully integrable system associated to M w.r.t. the basis b1 , . . . , bn . Since any orthogonal ∆-extension E of F is turned into an R-module via the action ∂i (e) = δi (e) for i ≤ ` and ∂i (e) = σi (e) for i > `, solE (A(Z) = 0) is well-defined. We now set up a correspondence between the solutions in E of A(Z) = 0 and those of its associated fully integrable system.

Theorem 5. Every system A(Z)=0 of finite linear dimension has a fundamental matrix and has a Picard-Vessiot ring E. If F has characteristic 0 and CF is algebraically closed, then CE = CF . Proof. Let A ∈ S p×q be such that A(Z) = 0 is of finite linear dimension n > 0, M be its module of formal solutions, e1 , . . . , eq be R-generators for M and b1 , . . . , bn be an F basis of M such that A(e1 , . . . , eq )τ =0 and ∂i (b1 , . . . , bn )τ = Bi (b1 , . . . , bn )τ for each i. Let P ∈ F q×n be given by (e1 , . . . , eq )τ = P (b1 , . . . , bn )τ . Since {∂i (X) = Bi X}1≤i≤m is a fully integrable system, there exists, by Theorem 1, a fundamental matrix U ∈ E n×n for that system where E is some orthogonal ∆-extension of F . Then V := P U ∈ E q×m is a fundamental matrix for A(Z) = 0. The existence of the Picard-Vessiot ring and the second statement follow directly from Theorem 2. 2 Assume that F has characteristic 0 with an algebraically closed field of constants. Let E be a Picard-Vessiot ring for the system A(Z) = 0. As mentioned after Theorem 2, solE ({∂i (X) = Bi X}1≤i≤m ) is of dimension n over CF . But that space is isomorphic to solE (A(Z) = 0) by Proposition 2. Therefore the dimension of solE (A(Z) = 0) as a CF -vector space equals n, the linear dimension of A(Z) = 0.

Proposition 2. Let A(Z) = 0 with A ∈ S p×q be a system of finite linear dimension, M be its module of formal solutions, e1 , . . . , eq be R-generators for M and b1 , . . . , bn be an F -basis of M such that A(e1 , . . . , eq )τ = 0 and ∂i (b1 , . . . , bn )τ = Bi (b1 , . . . , bn )τ q×n

for each i . τ

Let P ∈ F be given by (e1 , . . . , eq ) = P (b1 , . . . , bn )τ . Then, for any orthogonal ∆-extension E of F , the correspondence ξ 7→ P ξ is an isomorphism of CE -modules between solE ({∂i (X) = Bi X}1≤i≤m ) and solE (A(Z) = 0). Proof. To simplify notation, we denote solE (A(Z)=0) and solE ({∂i (X) = Bi X}1≤i≤m ) by WA and WB , respectively. Write e = (e1 , . . . , eq )τ and b = (b1 , . . . , bn )τ . According to Theorem 4, for any ξ ∈ WB , there exists ϕ ∈ HomR (M, E) such that ξ = ϕ(b). Hence A(P ξ) = A(P ϕ(b)) = ϕ(A(P b)) = ϕ(A(e)) = 0, so P ξ belongs to WA . Thus the correspondence ξ 7→ P ξ is a homomorphism of CE -modules from WB to WA . For every η ∈ WA there exists ψ ∈ HomR (M, E) such that η = ψ(e) = ψ(P b) = P ψ(b). The correspondence ξ 7→ P ξ is then surjective, because ψ(b) belongs to WB . If ξ ∈ WB and P ξ = 0, then there exists ϕ ∈ HomR (M, E) such that ξ = ϕ(b). Hence 0 = P ξ = ϕ(P b) = ϕ(e). It follows that ϕ maps everything to 0 as M is generated by e1 , . . . , eq over R. Thus ξ = 0 and the correspondence is bijective. 2

Example 5. Let F , δx and σk be as in Example 1, and the system A is given by   x+1 k(x+1−k) − k(x+1−k) x x2 (k−1) x2 (k−1) 2 2 2  +kx2 +k−1 +2x2 +kx2  − xk−kx(k−1) Ax =  x + 1 xk−k +2x , x(k−1) 2 2 2 2 xk+2x +kx −2k +k xk+2x +kx2 −2k2 +1 x+1 − x(k−1) x(k−1) 

Definition 5. Let A, M, b1 , . . . , bn and P be as in Proposition 2. A q × n matrix V with entries in an orthogonal ∆-extension E of F is called a fundamental matrix for A(Z) = 0 if V = P U where U ∈ E n×n is a fundamental matrix of the fully integrable system associated to M w.r.t. the basis b1 , . . . , bn . A Picard-Vessiot ring for any fully integrable system associated to M is called a Picard-Vessiot ring for A(Z) = 0. 2

 Ak = 

k+1 k x(k+1) k x(k+1) k

k+1−xk−x x(k−1) 1−2x+k−xk+x3 k−1 1−2xk−2x+k+x3 k−1

xk+x−k−1 x(k−1) 2x+xk−x3 −k−1 k−1 2xk+2x−k−x3 −1 k−1

  .

Note that Ax and Ak satisfy the compatibility conditions (2) but Ak is singular, so the system is not fully integrable. Let S = [∂x ; 1, δx ][∂k ; σk , 0] and R be the corresponding Laurent-Ore algebra. Let A ∈ S 6×3 be the matrix corresponding to the system given by Ax and Ak (see Example 4), M = R1×3 /R1×6 A be the module of formal solutions

It is also called an integrable connection.

74

for the system A(Z) = 0, and {e1 , e2 , e3 } be a set of Rgenerators of M such that ∂x (e1 , e2 , e3 )τ = Ax (e1 , e2 , e3 )τ and ∂k (e1 , e2 , e3 )τ = Ak (e1 , e2 , e3 )τ . Solving the linear system (v1 , v2 , v3 )Ak = 0 over F , we see that Ak has rank 2, and ∂k (e1 ), ∂k (e2 ) and ∂k (e3 ) are linearly dependent over F (so are e1 , e2 and e3 by an application of ∂k−1 ). A nontrivial solution of (v1 , v2 , v3 )Ak = 0 and an application of ∂k−1 yield       1 0 e1  e2  =  0 1  e1 , e2 x(k−1) x2 −k e3 x2 −1 x2 −1 | {z }

[2] Cassidy, P. and Singer, M. Galois Theory of Parameterized Differential Equations and Linear Differential Algebraic Groups. Preprint, 2005. [3] Chyzak, F., Quadrat, A., and Robertz, D. OreModules: A symbolic package for the study of multidimensional linear systems. In Proc. of MTNS’04, Leuven (Belgium) (2004), CDRom. [4] Chyzak, F. and Salvy, B. Non-commutative elimination in Ore algebras proves multivariate identities. J. Symbolic Comput. 26, 2 (1998), 187–227. [5] Kolchin, E. Differential algebra and algebraic groups. Academic Press, New York and London, 1973. [6] Labahn, G. and Li, Z. Hyperexponential solutions of finite-rank ideals in orthogonal Ore rings. In Proc. ISSAC’2004 (2004), J. Gutierrez, Ed., ACM, 213–220. [7] Li, Z., Schwarz, F., and Tsarev, S. P. Factoring systems of linear PDEs with finite-dimensional solution spaces. J. Symbolic Comput. 36, 3–4 (2003), 443–471. [8] Malgrange, B. Motivations and introduction to the theory of D-modules. In Computer Algebra and Differential Equations (1994), E. Tournier, Ed., vol. 193 of LMS Lecture Note Series, Cambridge University Press, pp. 3–20. [9] Pauer, F. and Unterkircher, A. Gr¨ obner bases for ideals in Laurent polynomial rings and their applications to systems of difference equations. Appl. Algebra in Eng., Comm., and Comp. 9 (1999), 271–291. [10] Singer, M. and van der Put, M. Galois Theory of Difference Equations. LNM 1666. Springer, 1997. [11] Singer, M. and van der Put, M. Galois Theory of Linear Differential Equations, vol. 328 of Grundlehren der Mathematischen Wissenschaften. Springer, Heidelberg, 2003. [12] Wu, M. Factorization and decomposition of modules over Laurent-Ore algebras. Th`ese de math´ematiques, Universit´e de Nice, expected 2005. [13] Zampieri, S. A solution of the Cauchy problem for multidimensional discrete linear shift-invariant systems. Linear algebra and appl. 202 (1994), 143–162.

P

which, together with Ax and Ak , implies that ∂x (e1 , e2 )τ = Bx (e1 , e2 )τ and ∂k (e1 , e2 )τ = Bk (e1 , e2 )τ where ! 3 2 2 k(x+1−k) Bx =

−x+x −1+x −xk−k+k x(x2 −1) −x−xk+x3 −1−x2 +k2 −kx2 x2 −1

Bk =

xk+x+k2 +2k+1 k(x+1) 2 2 −2k−1)x − (kx −x−k k(x+1)

x2 (x2 −1) −k2 +xk+kx2 +3x2 −1 x(x2 −1) k+1 − x(x+1) x2 +x−1−k x+1

,

! .

Since Bk is invertible, the system B given by Bx and Bk is fully integrable, and, hence, e1 and e2 form an F -basis of M . The same method to construct a fundamental matrix for the system in Example 1 yields a fundamental matrix for B:   xkex −kxk , U= 2 x 2 k+1 kx e (x − k − 1)x hence P U is for A. In addition, a Picard-Vessiot ring of B is a Picard-Vessiot ring of A. Acknowledgments: We would like to thank an anonymous referee for his useful and constructive remarks. The second and third authors were supported in part by a 973 key project (no. 2004CB31830), and by a Bourse du Gouvernement Fran¸cais (BGF no. 2002915), respectively.

6.

REFERENCES

[1] Bialynicki-Birula, A. On Galois theory of fields with operators. Amer. J. Math. 84 (1962), 89–109.

75

On Using Bi-equational Constraints in CAD Construction Christopher W. Brown

Scott McCallum

Department of Computer Science, Stop 9F United States Naval Academy Annapolis, MD 21402, USA

Department of Computing Macquarie University NSW 2109 Australia

[email protected]

[email protected]

ABSTRACT

1. INTRODUCTION

This paper introduces an improved method for constructing cylindrical algebraic decompositions (CADs) for formulas with two polynomial equations as implied constraints. The fundamental idea is that neither of the varieties of the two polynomials is actually represented by the CAD the method produces, only the variety defined by their common zeros is represented. This allows for a substantially smaller projection factor set, and for a CAD with many fewer cells. In the current theory of CADs, the fundamental object is to decompose n-space into regions in which a polynomial equation is either identically true or identically false. With many polynomials, one seeks a decomposition into regions in which each polynomial equation is identically true or false independently. The results presented here are intended to be the first step in establishing a theory of CADs in which systems of equations are fundamental objects, so that given a system we seek a decomposition into regions in which the system is identically true or false — which means each equation is no longer considered independently. Quantifier elimination problems of this form (systems of equations with side conditions) are quite common, and this approach has the potential to bring large problems of this type into the scope of what can be solved in practice. The special case of formulas containing two polynomial equations as constraints is an important one, but this work is also intended to be extended in the future to the more general case.

Cylindrical Algebraic Decomposition (CAD) provides a data structure for representing semi-algebraic sets. It is a data structure that is particularly useful for performing quantifier elimination in elementary real algebra, and it is in this context that Collins invented CAD in the early 1970s [9]. However, people have pointed out many other interesting uses of CAD including the simplification of elementary functions [4], non-linear optimization, topologically reliable curve plotting [2], and simplification of large quantifier-free formulas in elementary real algebra [6]. In the context of quantifier elimination, and many other applications of CAD, one frequently encounters input formulas consisting of a system of equations along with a side condition given by an arbitrary formula. In this case, one would like to take advantage of the constraints imposed by this system of equations. This paper proves some results that lead us to more efficient CAD construction when the system in such an input contains two equation, in which case we say there is a bi-equational constraint. We provide some example computations to show that when this improvement is applicable, its benefits are enormous.

1.1 Previous work An application of CAD generally starts with a formula — a boolean combination of integral polynomial equalities and inequalities — which describes a semi-algebraic set. We extract the set A ⊂ [x1, . . . , xr ] of all the polynomials occurring in the formula. The CAD algorithm then constructs a decomposition of r into cylindrically arranged cells such that the signs of the elements of A are constant inside any given cell. This cylindrical arrangement means that the projections onto j , 0 < j < r, of any two cells are either identical or disjoint. Clearly the formula is identically true or identically false in any cell of this CAD. Thus, by marking the cells of the CAD appropriately, we represent the set defined by the formula. CAD construction proceeds in two phases, projection and lifting. The projection phase computes a set of polynomials called the projection factor set. The projection factor set contains the irreducible factors of the set A, and, in general, other polynomials as well. The maximal connected regions in which the projection factors have invariant signs are the cells of the CAD that is to be constructed. Thus, the projection factor set provides an implicit representation of the CAD. The lifting phase then constructs an explicit representation of this CAD. General descriptions of CAD construction may be found in [11], [3], and [9].

Categories and Subject Descriptors G.4 [Mathematics of Computing]: Mathematical Software—Algorithm design and analysis




General Terms




Algorithms, Theory

Keywords CAD, polynomial systems

Copyright 2005 Association for Computing Machinery. ACM acknowledges that this contribution was authored or co-authored by a contractor or affiliate of the U.S. Government. As such, the Government retains a nonexclusive, royalty-free right to publish or reproduce this article, or to allow others to do so, for Government purposes only. ISSAC’05, July 24–27, 2005, Beijing, China. Copyright 2005 ACM 1-59593-095-7/05/0007 ...$5.00.

76

Many people have pointed out that one of the problems with a CAD is that it contains a lot more information about the polynomials occurring in the input formula than is typically needed for the problem at hand. In fact, a CAD built for a particular formula is capable of representing any set that can be defined by any formula in the same polynomials, and many more sets as well. Partial CAD [11] was an attempt to ameliorate this by a sort of lazy approach to lifting. The method of equational constraints [10] was an attempt to address this problem for inputs of the form we have discussed — a system of equations with an arbitrary side condition. The idea is as follows: if an input formula includes the constraint f = 0, then decompose r into regions in which f has invariant sign, and then refine the decomposition so that the other polynomials have invariant sign in those cells in which f = 0. The signs of the other polynomials in cells in which f 6= 0 are, after all, irrelevant. Additionally, the method of equational constraints seeks to deduce and use constraints that are not explicit in the input formula, but rather arise as consequences of two or more explicit constraints (e.g. if f = 0 and g = 0 are explicit constraints, then res(f, g) = 0 is also a constraint).

was originally introduced in [10] without proof. Subsequent papers [17, 18] have partially validated the method initially proposed. It is suggested that the reader first consult [10], especially Sections 4 and 8, for a readable and intuitive account of the idea. It may be helpful to consider a special case which is relevant to the present paper. Assume r ≥ 2 and let x denote (x1 , . . . , xr−1 ). Let f and g be squarefree, relatively prime elements of the polynomial ring [x,xr ] both of which are primitive and have positive degree in xr . Suppose we wish to decide whether or not the sentence (∃x)(∃xr )[f (x, xr ) = 0 ∧ g(x, xr ) = 0]




of Tarski algebra is true. (Let us call this a bi-equational existence problem.) The McCallum projection of the set A = {f, g} is the set P (A) ⊂ Z[x] consisting of all of the nonzero coefficients with respect to xr of f and g, together with discrxr (f ), discrxr (g) and resxr (f, g). Notice that the equations f (x, xr ) = 0 and g(x, xr ) = 0 are constraints of our problem. Let us nominate f (x, xr ) = 0 to be the pivot constraint, with respect to which the set P (A) will be reduced, and put E = {f }. Then the restricted projection of A relative to E, denoted by PE (A) in [17], is the smaller set consisting of all of the nonzero coefficients of f only, together with discrxr (f ) and resxr (f, g). (An astute reader will note that this definition of PE (A) differs slightly, but not significantly, from that given in [17].) The main result of [17] implies that PE (A) can be safely used in place of P (A) for the first projection step (that is, elimination of xr ). The survey paper [10] proceeds to describe how equational constraints could be propagated, and to suggest the use of such propagated constraints to simplify subsequent projection steps. For the bi-equational existence problem described in the previous paragraph, this amounts in part to observing that the (r −1)-variate equation resxr (f, g) = 0 is an implied constraint for our problem, and to suggesting that this implied constraint could be used to reduce the second application of McCallum’s projection operation (that is, elimination of xr−1 ). The issue of propagation is discussed in [18], which regrettably does not succeed in rigorously justifying the use of propagated constraints for subsequent projection steps, as suggested in [10]. (The weaker notion of a semi-restricted projection of a set relative to a constraint is introduced and applied in [18].) A second important recent development in CAD theory and practice, independent of the concept of equational constraints, was the introduction of the Brown-McCallum projection operation [5]. For our bi-equational existence problem discussed in the previous two paragraphs, the BrownMcCallum projection of A is the set

1.2 Our contribution That part of the equational constraints method that takes advantage of the constraint f = 0 in projection has been successful. However, the part that tries to take advantage of more than one explicit constraint by deducing new, implicit constraints has been problematic. The original scheme proposed by Collins has never been proved correct, and the version proposed in [18] is much weaker. Thus, we can only really say that for inputs of the form of a system of equations with an arbitrary side condition, only the case of a system of one equation has been adequately addressed. This paper addresses the case of two equations. More fundamentally, this paper proposes a break with the way CADs have always been constructed. The projection factor set in CAD construction has always contained polynomials1 . We propose a model of projection where the two constraint polynomials act as a single object during projection.

1.3 Organization of the paper In Section 2 we present some technical background information, followed by Sections 3, 4, and 5 in which we state and prove our main results. Section 6 describes two 4variable quantifier elimination problems and walks through the process of applying the theorems of the previous sections to efficiently construct CADs and find solution formulas. These experiences then motivate the algorithm proposed in Section 7.

2.

Proj(A) =

BACKGROUND MATERIAL

In this section we provide a short synopsis of some relatively recent developments in the theory and practice of CAD which are relevant to the present paper. First we discuss the application of equational constraints to simplify (that is, reduce) the McCallum projection operation [14, 15] used in the projection phase of CAD. The idea

ldcf xr (f ), ldcf xr (g), discrxr (f ), discrxr (g), resxr (f, g)

.


Note that Proj(A) includes only the leading coefficients of f and g. Certain algorithm modifications are required to use the Brown-McCallum projection for CAD construction. Subject to such modifications, this projection is currently the best (that is, smallest) general purpose projection operation which has been proved valid for CAD. Finally, combining the above two developments, one could contemplate a restricted equational version of the BrownMcCallum projection. For our bi-equational existence problem, in which f is chosen as the pivot constraint so that

1 The Brown-McCallum projection [5] could be viewed as having a projection factor set that includes polynomials and points. That is less of an issue, however, since the points play no role during the projection process.

77

Suppose we want to construct a CAD for f = g = 0 where

E = {f }, this is the set

f = (x1 + x2 )x23 − 3(2x2 + x1 − 1)x3 + x1 x2 + x2 − 3 g = (x1 − x2 )x23 + 2(x2 x1 − 1)x3 + x21 − x2 − 1.

ProjE (A) = {ldcf xr (f ), discrxr (f ), resxr (f, g)}. This is the smallest projection set which, based upon the existing literature, we could be confident to use for our example.

3.

(1) When x1 = x2 = 0, f and g have no common zeros, even though (0, 0) lies on the resultant curve and there are solutions above all points on the resultant curve near (0, 0). Thus, (0, 0) must be a separate cell in the CAD of 2 , but if the resultant alone is included in projection, it will not be. What causes the problem is that the leading coefficients both vanish at (0, 0). The simultaneous vanishing of both leading coefficients is not the only way that a projection based purely on resxr (f, g) can fail. Consider constructing a CAD for f = g = 0 where

STATEMENT OF MAIN RESULTS




This paper assumes that the reader is familiar with the McCallum projection and the notions of order-invariance and analytic delineability on which it relies. A summary of the key technical terms and results can be found in [17]. The reader is referred to [14, 15] for a complete presentation.

3.1 Delineability of V (f, g)

f = (x3 + x1 )(x3 − x1 ) + x1 − x2 g = (x3 + x2 )(x3 − x2 ) + x2 − x1

For this subsection and the next, unless stated otherwise, we shall let r ≥ 2, x denote (x1 , . . . , xr−1 ) and f and g be squarefree, relatively prime elements of the polynomial ring [x,xr ] both of which are primitive have positive degree in xr . We could construct a CAD representing f = g = 0 using the Brown-McCallum projection Proj(A) of the set A = {f, g}. (See the previous section for the definition of Proj(A).) The main result of [5] implies that over any cell in a CAD of r−1 in which discrxr (f ), discrxr (g), resxr (f, g) are order-invariant and ldcf xr (f ), ldcf xr (g) are sign-invariant, both f and g are delineable and stack construction can proceed — provided f and g are not nullified in the cell. As suggested in the previous section, we could apply equational constraints to the Brown-McCallum projection. We could choose a pivot, say f , put E = {f } and use the projection set ProjE (A) defined in the previous section. The main results of [5] and [17] imply that over any cell in a CAD of r−1 in which discrxr (f ), resxr (f, g) are order-invariant and ldcf xr (f ) is sign-invariant, f is delineable and g has constant sign in each section of f , which means stack construction can proceed — assuming that f is not nullified in the cell. Thus, the method of equational constraints is based on the observation that g doesn’t need to be delineable over a cell to construct a CAD representing f = g = 0. It suffices for f to be delineable, and for g to have constant sign in the sections of f . This allows us to reduce the size of the projection. The idea behind what we propose is that neither f nor g needs to be delineable over the (r − 1)-level2 cells to produce a CAD representing f = g = 0. It suffices for the common roots of f and g to be delineable, which we define precisely as follows. Where f and g are real polynomials in x and xr we shall denote by V (and sometimes by V (f, g)) the real variety of f and g

The leading coefficients of f and g are constant, so they are not an issue. However, once again, (0, 0) must be a separate cell in the CAD of 2 in order for V (f, g) to be delineable, because it is only over (0, 0) that f and g have exactly one common real root. The Brown-McCallum projection produces from resxr (f, g) a CAD of 2 such that resxr (f, g) is order-invariant in every cell, and yet (0, 0) is not its own cell. Thus, even when the simultaneous vanishing of leading coefficients is not a problem, the projection consisting only of resxr (f, g) may not suffice. Why does it fail in this case? The problem is that f, g, ∂f /∂xr , and ∂g/∂xr have a common zero above (0, 0). It turns out that the previous two examples of how a projection consisting solely of the resultant of f and g can fail show the only two ways it can fail. This is formalized in Theorem 3.1. What is not yet addressed is how to deal with the polynomials occurring in the side condition H. In order to represent the set f = g = 0 ∧ H it suffices that each polynomial h occurring in H is sign-invariant in each section of V (f, g). One would hope that it would be enough to include resxr ,xr−1 (f, g, h), the multipolynomial resultant of f , g and h (for each h in H) in the projection to ensure this. In fact, Theorem 3.2 shows that as long as each such multipolynomial resultant is not the zero polynomial, it does suffice.











V = V (f, g) = {(x, xr ) ∈

r


(2)

3.3 Statements of two main theorems We define the number of common roots at infinity of f and g at p ∈ r−1 to be the minimum of degxr f − deg f (p, xr ) and degxr g − deg g(p, xr ).


Theorem 3.1. Let f and g be real polynomials in the variables x1 , . . . , xr of positive degrees in xr . Let R(x) be the resultant of f and g with respect to xr , and assume that R is not the zero polynomial. Let S ⊂ r−1 be connected. Suppose that R is order-invariant in S, the number of common roots at infinity of f and g is constant and finite in S, and there is no common zero of the polynomials f, g, ∂f /∂xr , ∂g/∂xr in the cylinder over S. Then V is delineable on S.

| f (x, xr ) = g(x, xr ) = 0}

and say that V is delineable on a connected subset S of r−1 provided that the portion of V which lies in S × consists of the union of the graphs of some k ≥ 0 continuous functions θ1 < . . . < θk from S to .











3.2 Towards a projection for V (f, g) It is clear that if V (f, g) is to be delineable over the cells of the induced CAD of r−1 we will in general have to include resxr (f, g) in the projection. The question is does this suffice? The answer, unfortunately, is no.

Theorem 3.2. Let f and g be real polynomials in in the variables x1 , . . . , xr of positive degrees in xr . Let R(x) be the resultant of f and g with respect to xr , and assume that R is not the zero polynomial. Let S 0 ⊂ r−2 be connected, suppose that R is delineable on S 0 , and let S be a section of




2 The level of a cell is the dimension of the Euclidean space of which it is a subset.




78

R over S 0 . Assume the hypotheses of Theorem 3.1. Then V is delineable on S. Let h(x, xr ) be an element of [x, xr ] which has positive degree in xr . Denote (x1 , . . . , xr−2 ) by x ˜, let ρ(˜ x) = resxr−1 ,xr (f, g, h) and suppose that ρ(˜ x) 6= 0. Suppose further that ρ is order-invariant in S 0 . Then h is sign-invariant in each section of V over S.

of f (x, xr ) ( which is here considered as a polynomial in xr alone). To prove the delineability of V on S near p it suffices to show that for each i, 1 ≤ i ≤ k, for which g(p, αi ) = 0, there exists a neighbourhood Ni ⊂ N0 of p such that for every fixed x ∈ S ∩ Ni , there is exactly one common root, say θi (x), of f (x, xr ) and g(x, xr ) contained in the interior of Ci ; and for each i, 1 ≤ i ≤ k, for which g(p, αi ) 6= 0, there exists a neighbourhood Ni ⊂ N0 of p such that for every fixed x ∈ S ∩ Ni , there is no common root of f (x, xr ) and g(x, xr ) contained in the interior of Ci . (For if this has been shown, put N = ki=0 Ni and argue as in the proof of Theorem 3.2 of [14].) We now proceed to prove that for each i, 1 ≤ i ≤ k, for which g(p, αi ) = 0, there exists a neighbourhood Ni ⊂ N0 of p such that for every fixed x ∈ S ∩ Ni , there is exactly one common root of f (x, xr ) and g(x, xr ) (as polynomials in xr ) contained in the interior of Ci . By hypothesis, (p, αi ) is not a common zero of f, g, ∂f /∂xr , ∂g/∂xr . Therefore either ∂f /∂xr (p, αi ) 6= 0 or ∂g/∂xr (p, αi ) 6= 0. In the former case, we have mi = 1, so we can take Ni = N0 . Denote by σi the graph of the real root function θi defined and continuous in S ∩ N0 . Then f is order-invariant with respect to xr in σi since the order of f with respect to xr at (x, θi (x)) is 1, for every fixed x ∈ S ∩ N0 . Hence, by Theorem 4.1 (in which we take S to be S ∩ N0 ), g is sign-invariant in σi . Thus, for every fixed x ∈ S ∩ N0 , θ(x) is a common root of f (x, xr ) and g(x, xr ), indeed the unique such common root inside Ci . In the latter case, (∂g/∂xr (p, αi ) 6= 0), the implicit function theorem can be applied to g at (p, αi ) yielding a neighbourhood Ni ⊂ N0 of p and an analytic function θi : Ni → whose graph σi is identical to the real variety of g, near (p, αi ). Observing that g is order-invariant with respect to xr in σi , Theorem 4.1 can be applied with the roles of f and g reversed to deduce that f is sign-invariant in σi . Thus, for every fixed x ∈ S ∩ Ni , θi (x) is the unique common root of f (x, xr ) and g(x, xr ) inside Ci . It remains to show that for each i, 1 ≤ i ≤ k, for which g(p, αi ) 6= 0, there exists a neighbourhood Ni ⊂ N0 of p such that for every fixed x ∈ S ∩ Ni , there is no common root of f (x, xr ) and g(x, xr ) inside Ci . We simply choose Ni ⊂ N0 such that g(x, xr ) 6= 0, for all (x, xr ) with x ∈ Ni and xr inside Ci . Such Ni exists by continuity of g and compactness of the closed disk bounded by Ci , since g(p, xr ) 6= 0, for all xr in this closed disk. The proof is complete.




Subsequent sections (4 and 5) contain the proofs of the above theorems.

4.

PROOF OF THEOREM 3.1

We first present a key lemma for the proof of Theorem 3.1. The lemma’s statement makes use of the concept of the order-invariance of a polynomial with respect to a given variable in some set, which we now define. We say that f (x, xr ) is order-invariant with respect to xr in a subset σ of r if the order of f with respect to xr at (p, pr ) is constant as (p, pr ) varies in σ.







Theorem 4.1. Let R(x) be the resultant of f and g with respect to xr . Let S ⊂ r−1 be connected. Let σ be a section over S contained in the real variety of f such that f is orderinvariant with respect to xr in σ. Suppose that R is orderinvariant in S. Then g is sign-invariant in σ.


REMARK CONCERNING PROOF. This theorem is in a sense a rewording and a slight generalization of Theorem 2.2 of [17], the proof of which carries over almost word for word. We now supply the proof of Theorem 3.1. We assume that S has positive dimension. (The dimension 0 case is trivial.) By connectedness of S it suffices to show that V is delineable on S near an arbitrary point p of S. Let p be a point of S. Assume without loss of generality that the number of common roots at infinity of f and g equals degxr f −deg f (p, xr ). By hypothesis, this number is finite. Hence f (p, xr ) 6= 0. Also, as a consequence of an hypothesis, f is degree-invariant in S near p. (For deg f (q, xr ) ≥ deg f (p, xr ), for all q sufficiently near p, by continuity of the coefficients of f at p. Therefore




degxr f − deg f (q, xr ) ≤ degxr f − deg f (p, xr ), for all q sufficiently near p. But the number of common roots at infinity of f and g at q is at most the left hand side of the above inequality, by definition, and is equal to the right hand side of the inequality, by hypothesis. Therefore the inequality is in fact an equation, from which the claim follows immediately.) Denote the degree of f (p, xr ) by l. Let α1 < . . . < αk , k ≥ 0, be the real roots of f (p, xr ), let αk+1 , . . . , αt , k ≤ t, be the distinct non-real roots of f (p, xr ), and let mi be the multiplicity of αi , for 1 ≤ i ≤ t. Observe that ti=1 mi = l. Let

5. PROOF OF THEOREM 3.2 The proof of Theorem 3.2 will require the use of some results about real and complex analytic functions. One such result, a key lemma for our proof, we state at the outset. Theorem 5.1. Let ρ and ρ∗ be functions analytic in the polydisk ∆ about the origin in complex n-space n . Suppose that the zero set of ρ∗ is contained in the zero set of ρ in ∆. Then, for some polydisk ∆0 ⊂ ∆ about the origin, and some m ≥ 1, ρm is divisible by ρ∗ in ∆0 . That is, ρm = ρ∗ ρ0 , for some analytic ρ0 in ∆0 .

κ = min({|αi − αj | : 1 ≤ i < j ≤ t} ∪ {1}).



Let  = κ/2 and let Ci be the circle of radius  centred at αi , for 1 ≤ i ≤ t. Refine  > 0 as necessary to ensure that, for each i, 1 ≤ i ≤ k, for which g(p, αi ) 6= 0, Ci and its interior contain no root of g(p, xr ). By root continuity (Theorem (1,4) of [13]) and degree-invariance of f on S near p, there exists a neighbourhood N0 ⊂ r−1 of p such that for every fixed point x of S ∩ N0 , deg f (x, xr ) = l and the interior of each Ci contains exactly mi roots (multiplicities counted)

REMARK CONCERNING THE PROOF. This theorem is a relatively straightforward consequence of an important result concerning the divisibility of one analytic function by another, which is stated as Theorem 9J of Chapter 1 of [19].




79

(If ρ∗ does not vanish identically near the origin, then one puts m = ord0 ρ∗ and shows that the hypothesis of Theorem 9J is satisfied for ρm and ρ∗ in a suitable polydisk ∆0 ⊂ ∆ about the origin.) PROOF OF THEOREM 3.2. Throughout the rest of this section we shall denote xr−1 by y and xr by z. Let σ be an arbitrary section of V over S. By connectedness of S 0 (hence S, hence σ), it suffices to show that h is locally signinvariant in σ. Take a point p = (˜ p, β, γ) in σ. That h is sign-invariant in σ near p follows by continuity of h in case h(˜ p, β, γ) 6= 0. So henceforth assume that h(˜ p, β, γ) = 0. By an hypothesis, either ∂f /∂z or ∂g/∂z does not vanish at p. Without loss of generality assume the former and that p is the origin. We aim to construct a function ρ∗ , analytic near ˜ 0 in complex (r − 2)-space r−2 , whose zero set is the projection onto r−2 of the portion of the variety of f , g and h near the origin in r . By assumptions the univariate polynomial f (˜ 0, 0, z) is divisible exactly once by z. Therefore, by Hensel’s Lemma, (Theorem 3.1 of [17]) there is a polydisk ∆1 about the origin in complex (r −1)-space and polynomials in z, f1 (˜ x, y, z) = z − θ(˜ x, y) and f2 (˜ x, y, z), whose coefficients are elements of the formal power series ring [[˜ x , y]], absolutely convergent in ∆1 , such that θ(˜ 0, 0) = 0, f2 (˜ 0, 0, 0) 6= 0 and f = f1 f2 . Since a function defined as the sum of a convergent power series is analytic, θ(˜ x, y) and the coefficients of f2 are analytic in ∆1 . For any δ > 0, we denote by ∆(0; δ) the disk in about the origin of radius δ. Put P (˜ x, y) = resz (f1 , g), an element of [[˜ x , y]], absolutely convergent in ∆1 . Then P (˜ 0, y) 6= 0, since R(˜ 0, y) 6= 0. Therefore, by the (r−1)-variable analogue of the Weierstrass preparation theorem (as presented in Lec˜ 2 × ∆(0; δ) ⊂ ∆1 ture 16 of [1]), there is a polydisk ∆2 = ∆ about the origin in complex (r − 1)-space, a polynomial

f = f1 f2 yields f (α, ˜ β, γ) = 0. Therefore, by Theorem 2.4 of [16], ρ(α) ˜ = 0. The claim is proved. ˜4 ⊂ ∆ ˜3 Hence, by Theorem 5.1, there exist a polydisk ∆ 0 m ∗ 0 ˜ and an analytic function ρ in ∆4 such that ρ = ρ ρ , for some m ≥ 1. Since ρ and ρ∗ have real power series representations, so does ρ0 (because the imaginary part of the power series expansion for ρ0 about the origin must be ˜4 = ∆ ˜ 4 ∩ r−2 . 0). Therefore, ρ0 is analytic in the box B By Lemma A.3 of [14], since ρ is order-invariant in S 0 by ˜4 . But ρ∗ (˜ hypothesis, ρ∗ is order-invariant in S 0 ∩ B 0) = 0, since f1 , g and h all vanish at the origin. Hence, ρ∗ (˜ x) = 0, ˜4 . for all x ˜ ∈ S0 ∩ B We conclude our proof in the following way. Let φ denote the continuous function on S 0 whose graph is the section S of ˜4 , φ(˜ R over S 0 . We claim that for every fixed x ˜ ∈ S0 ∩ B x) is a root of P1 (˜ x, y) of multiplicity k, hence the unique root of P1 (˜ x, y). The claim is proved as follows. Now R is delineable on S 0 , by hypothesis, and the identity R = P1 (P2 P¯ ) is valid, where P¯ = resz (f2 , g). It is straightforward to show that P2 P¯ is a polynomial in y (with analytic coefficients). Hence, ˜4 . (The by Lemma A.7 of [14], P1 is delineable on S 0 ∩ B reader will note that we have used the notion of delineability from [14, 17], and have extended the notion to polynomials with analytic coefficients.) This proves our claim. We next claim that h(˜ x, φ(˜ x), θ(˜ x, φ(˜ x))) = 0, for all x ˜ ∈ S 0 ∩ B˜4 . The claim is proved as follows. Let α ˜ ∈ S 0 ∩ B˜4 . Then ρ∗ (α) ˜ = 0, by the last sentence of the previous paragraph. Therefore, there exists β ∈ ∆(0; δ 0 ) such that P1 (α, ˜ β) = T ∗ (α, ˜ β) = 0. But φ(α) ˜ is the unique root of P1 (α, ˜ y), proved above. Hence β = φ(α). ˜ As in our proof of the claim that the zero set of ρ∗ is contained in the zero set of ρ, we deduce that (α, ˜ β, γ) is a common zero of f , g and h, where γ = θ(α, ˜ β). In particular, h(α, ˜ β, γ) = 0. This proves our claim. The proof that h vanishes throughout σ, near the origin, is complete.
















P1 (˜ x, y) = y k + a1 (˜ x)y k−1 + · · · + am (˜ x)

6. EXAMPLES

˜ 2 , and an x ]], absolutely convergent in ∆ with the ai (˜ x) ∈ [[˜ element P2 (˜ x, y) of [[˜ x , y]], absolutely convergent in ∆2 , such that P1 (˜ 0, y) = y k , P2 (˜ 0, 0) 6= 0 and P = P1 P2 . Let T (˜ x, y) = resz (f1 , h). By the Weierstrass division theorem (Lecture 16 of [1]), there exists a polydisk ∆3 = ˜ 3 × ∆(0; δ 0 ) ⊂ ∆2 , an element Q(˜ x , y]], abso∆ x, y) of [[˜ lutely convergent in ∆3 , and an element T ∗ (˜ x, y) of [[˜ x ]][y], of degree in y at most k −1, whose coefficients are absolutely ˜ 3 , such that T = P1 Q + T ∗ . By root conconvergent in ∆ tinuity (Theorem (1,4) of [13]) and the analyticity (hence ˜3 continuity) of the coefficients ai (˜ x) of P1 , after refining ∆ to a smaller polydisk about the origin, if necessary, we can ˜ 3 , each root of further assume that for every fixed x ˜ ∈ ∆ P1 (˜ x, y) is in ∆(0; δ 0 ). We now complete our construction of ρ∗ : we put ρ∗ (˜ x) = resy (P1 , T ∗ ). We claim that the zero set of ρ∗ is contained ˜ 3 . The proof is as follows. Let α in the zero set of ρ in ∆ ˜ be ˜ 3 and suppose that ρ∗ (α) an element of ∆ ˜ = 0. Then there exists β ∈ such that P1 (α, ˜ β) = T ∗ (α, ˜ β) = 0. Since each root of P1 (α, ˜ y) is in ∆(0; δ 0 ), we have β ∈ ∆(0; δ 0 ). Hence we can legally substitute (α, ˜ β) for (˜ x, y) in the power series identity T = P1 Q + T ∗ , from which we deduce T (α, ˜ β) = 0. The same substitution in the power series identity P = P1 P2 yields P (α, ˜ β) = 0. Hence, with γ = θ(α, ˜ β), we have g(α, ˜ β, γ) = h(α, ˜ β, γ) = 0. Substitution of (α, ˜ β, γ) into

Here we consider two examples to demonstrate how the results proved earlier can be used to construct CADs more efficiently. They are chosen so that different hypotheses of theorems of Subsection 3.3 are not satisfied globally. Example computations are performed using Qepcad b [7], a system that performs quantifier elimination and formula simplification based on CAD, to construct CADs, and Maple’s Gr¨ obner basis facilities to compute multipolynomial resultants.













6.1 An example involving complex roots Consider the following question: when does p(z) = z 3 + az +b have a non-real root x+iy such that xy < 1. This can be expressed as (∃x)(∃y)[f = g = 0 ∧ y 6= 0 ∧ xy − 1 < 0], where f g

= =

Re(p(x + iy) Im(p(x + iy))/y

= =

x3 − 3xy 2 + ax + b 3x2 − y 2 + a.

Applying Qepcad b to this problem in this form produces the solution 27b2 + 4a3 > 0 after 342 seconds of CPU time (on a 650 MHz Sun Blade 150). Using equational constraints as Collins originally proposed, Qepcad b returns the formula 27b2 + 4a3 > 0 after less than 0.1 seconds, although we cannot be sure that the use of equational constraints is valid3 .



3

80

[18] gives criteria for determining that the full use of equa-

If Theorem 3.2 can be applied to this problem, we would be able to construct a CAD with a projection factor set consisting of J3 = {resy (f, g), resy,x (f, g, y), resy,x (f, g, xy − 1)} and J2 = Proj(J3 ). However, we need to be sure that the hypotheses of the theorems are satisfied. It would suffice to show that

1. resx,w (f, g, h) is not the zero polynomial, 2. there is no common zero of f, g, ∂f /∂x, ∂g/∂x in any stack we construct in 4-space, and 3. the leading coefficients of f and g are not simultaneously zero in any cell over which we lift.

1. neither resy,x (f, g, y) nor resy,x (f, g, xy − 1) are the zero polynomial,

The first condition is easily verified. For the second condition, we must verify that f, g, ∂f /∂x, ∂g/∂x have no common zeros. From these four polynomials, Maple computes a Gr¨ obner Basis with respect to the lexicographical order whose first element is u2 + 4. Since this is never zero over the reals, there are no real common zeros. The final condition is problematic, since both leading coefficients are zero when u = v = 0. The easiest way to deal with this is to simply assume u 6= 0 ∨ v 6= 0 (so that in constructing a partial CAD we will never lift over a cell in which u = v = 0) and treat the u = v = 0 case as a separate problem — a trivial separate problem since substituting zero for u and v yields 1 = 0 as a constraint in the formula. Having verified that, as long as we add the assumption u 6= 0 ∨ v 6= 0, the reduced projection described above suffices, we can use Qepcad b to perform CAD construction by manually removing polynomials introduced during its projection that are not part of the reduced projection described above. This takes approximately 8 seconds on a 650 MHz Sun Blade 150. The Maple computations that justified the reduced projection took less than 1 second. The resulting solution formula is quite large, presumably reflecting the fact that this is an artificial problem created, in part, to demonstrate that short, simple input formulas can swamp CAD-based quantifier elimination (see Figure 1). What this example demonstrates for us, however, is that taking advantage of bi-equational constraints in an input formula can make the difference between a CAD construction that is utterly infeasible and one that can be accomplished fairly quickly.

2. there is no common zero of f, g, ∂f /∂x, ∂g/∂x in any stack we construct in 4-space, and 3. the leading coefficients of f and g are not simultaneously zero in any cell over which we lift. Point 1 is checked with simple calculation. Point 3 is clearly satisfied, since the leading coefficient of g is constant. Computing a Gr¨ obner basis for f, g, ∂f /∂x, ∂g/∂x with an elimination order, we get 27b2 + 4a3 , so we cannot verify Point 2 globally. Therefore, we can proceed with the reduced projection as long as we assume 27b2 +4a3 6= 0. The 27b2 +4a3 = 0 case can then be treated as a separate (and hopefully simpler!) computation. We can use Qepcad b with the assumption 27b2 +4a3 6= 0 to do the quantifier elimination, and interactively remove all the projection factors it creates from the first projection step except for resy (f, g), and add as projection factors resy,x (f, g, y) and resy,x (f, g, xy − 1). The computation yields 27b2 + 4a3 > 0 in 0.11 seconds. The Maple computations required to do the verification above took less than 0.1 seconds. This leaves us with the 27b2 + 4a3 = 0 case to consider. However, this is a constraint for the last projection, so as [17] points out, we can use equational constraints for the first and last projections. This computation, in which our limited use of equational constraints is valid, proceeds in less than 0.1 seconds and tells us that the formula is never satisfied when 27b2 + 4a3 = 0.

6.2 Hong’s example In [12], Hong considered the formula (∃x)[f = 0 ∧ g = 0 ∧ h ≤ 0], where f g h

= = =

ux2 + vx + 1 vx3 + wx + u wx2 + vx + u.

What made this example interesting for demonstrating the algorithm in that paper is that CAD-based quantifier elimination performs so poorly! Even using Collins’ originally proposed equational constraints method (just hoping that it is valid for this example), Qepcad b runs for several minutes before aborting due to a system limitation on the number of primes that can be used in a modular algorithm. This resource limit is arrived at before any lifting over zero dimensional cells (which require computations over the highest degree extensions) is even attempted. If Theorem 3.2 can be applied to this problem, we would be able to construct a CAD with a projection factor set consisting of J3 = {resx (f, g), resx,w (f, g, h)} and J2 = Proj(J3 ). However, we need to be sure that the hypotheses of Theorem 3.2 are satisfied. It would suffice to show that:

Figure 1: The solution set for Hong’s example is of the form r = 0 ∧ K, where r is quadratic in w. Being without good tools for producing a 3D visualization of this set, we’ve provided plots of the regions in which there is exactly one solution for a given point (u, v) (on the left), and where there are exactly two (on the right). This provides some indication that the solution set is inherently complex, and that a short solution formula may not exist.

tional constraints as Collins proposed is valid in the 4variable case. The criteria are not satisfied for this example.

81

7.

Input: f , g and H as described above

TOWARDS AN ALGORITHM

In order to incorporate the results and ideas from the previous sections into an algorithm, two things are needed: a plan for checking whether or not the hypotheses of the various theorems are satisfied, and a plan for dealing with situations in which the hypotheses are not satisfied globally. An in-depth look at either of these two problems is outside the scope of this paper; especially the second, as it is related to the larger problem of how to cope with the situation in which the Brown-McCallum projection fails because a projection factor is nullified. In this paper, our goal is to provide a reasonable strategy without worrying about finding the best plan. The two previous examples are intended to serve as a guide in this. The hypotheses of Theorems 3.1 and 3.2 will typically only fail when certain polynomials or sets of polynomials of level4 less than r are zero. We will make the non-vanishing of these polynomials an assumption in our CAD construction and thus, in the usual way with partial CADs, not lift over any cells in which they fail. Solving the original input problem for the case in which these assumptions fail is a separate problem, and one that is simpler in the sense of being more constrained. This process is demonstrated in our two examples.

Output: formula A in the variables x1 , . . . , xr−1 and CAD D representing f = g = 0 ∧ H ∧ A 1. set R := resxr (f, g) 2. let ρi := resxr ,xr−1 (f, g, hi ) for all i ∈ 1, . . . , s; if any of the ρi are zero, exit returning F AIL 3. set E to the set of irreducible factors of resxr ,xr−1 ,xr−2 (f, g, ∂f /∂xr , ∂g/∂xr ) 4. set C to the set of all pairs of factors of ldcf xr (f ) and ldcfxr (g). 5. set A := 


p∈E

p 6= 0

∧ 


(p,q)∈C

p 6= 0 ∨ q 6= 0

6. construct CAD D in the following way (a) construct a partial CAD of r−1 for the set {R, ρ1 , . . . , ρs , hs+1 , . . . , ht } under the assumption A.


(b) lift into r-space only over sections of R, treating only common roots of f and g as sections in these new stacks (c) evaluate the original formula at each sample point

7.1 An algorithm

7. return D, A

We describe an algorithm BEQCCAD, which constructs a CAD for input with bi-equational constraints. The algorithm’s input is a triple (f, g, H) representing the formula f = g = 0 ∧ H, where f , g and the polynomials appearing in H are all in [x1, . . . , xr ]. Assume that f and g are relatively prime, square-free r-level polynomials, let h1 , . . . hs be the r and (r − 1)-level factors of polynomials appearing in H, and let hs+1 , . . . , ht be the lower-level factors of polynomials appearing in H. To simplify this presentation, let us suppose that resxr (f, g) does not vanish identically at any point in an (r − 1)-level cell over which we lift. (Since resxr (f, g) will be a product of polynomials in the projection factor set, this condition will be checked during CAD construction.) The CAD D computed by BEQCCAD represents the set defined by the input formula, except over regions in r−1 in which the hypotheses of Theorems 3.1 and 3.2 are not satisfied. We simply do not lift over cells in such regions. Figure 7.1 summarizes the steps performed by the algorithm BEQCCAD. The validity of this algorithm follows by Theorems 3.1 and 3.2. The CAD D represents the set defined by the input formula, except possibly over some lower dimensional regions in which A is not satisfied. As both examples demonstrate, it can be constructed much more quickly than the CAD we would construct from the input formula without taking advantage of any equational constraints in the input. As the second example demonstrates, it can even be constructed more quickly than the CAD we get by simply assuming the validity of the method of equational constraints as originally formulated by Collins.

Figure 2: Algorithm BEQCCAD

8. CONCLUSION We have presented two theorems that allow us to efficiently construct CAD representations of sets defined by formulas with bi-equational constraints, i.e. of the form f = g = 0 ∧ H. The algorithm we have derived based on these theorems will typically construct a CAD representing the input with some assumptions.The situations in which these assumptions fail to hold need to be handled as separate problems, but they are more constrained problems and thus, in a sense, simpler. The given examples illustrate this. Extending these results to problems with more than two equational constraints is a natural and important direction for future research.




Acknowledgements The first author’s work was supported in part by NSF grant number CCR-0306440.

9. REFERENCES [1] S. S. Abhyankar. Algebraic Geometry for Scientists and Engineers. American Math. Society, 1990. [2] D. S. Arnon. Topologically reliable display of algebraic curves. In Proceedings of SIGGRAPH, pages 219–227, 1983. [3] D. S. Arnon, G. E. Collins, and S. McCallum. Cylindrical algebraic decomposition I: The basic algorithm. SIAM Journal on Computing, 13(4):865–877, 1984. [4] James C. Beaumont, Russell J. Bradford, James H. Davenport, and Nalina Phisanbut. A poly-algorithmic

4

The level of a polynomial in the variables x1 , . . . , xr is the maximum i for which the polynomial’s degree in xi is positive.

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[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12] H. Hong. Quantifier elimination for formulas constrained by quadratic equations via slope resultants. Computer J., 36(5):440–449, 1993. [13] M. Marden. Geometry of Polynomials, 2nd edition. American Math. Society, 1966. [14] S. McCallum. An improved projection operation for cylindrical algebraic decomposition of three-dimensional space. Journal of Symbolic Computation, 5(1,2):141–161, 1988. [15] S. McCallum. An improved projection operator for cylindrical algebraic decomposition. In B. Caviness and J. Johnson, editors, Quantifier Elimination and Cylindrical Algebraic Decomposition, Texts and Monographs in Symbolic Computation. Springer-Verlag, Vienna, 1998. [16] S. McCallum. Factors of iterated resultants and discriminants. Journal of Symbolic Computation, 27:367–385, 1999. [17] S. McCallum. On projection in CAD-based quantifier elimination with equational constraint. In Sam Dooley, editor, Proc. International Symposium on Symbolic and Algebraic Computation, pages 145–149, 1999. [18] S. McCallum. On propagation of equational constraints in CAD-based quantifier elimination. In Bernard Mourrain, editor, Proc. International Symposium on Symbolic and Algebraic Computation, pages 223–230, 2001. [19] H. Whitney. Complex Analytic Varieties. Addison-Wesley, 1972.

approach to simplifying elementary functions. In Proc. International Symposium on Symbolic and Algebraic Computation, pages 27–34, 2004. C. W. Brown. Improved projection for cylindrical algebraic decomposition. Journal of Symbolic Computation, 32(5):447–465, November 2001. C. W. Brown. Simple CAD construction and its applications. Journal of Symbolic Computation, 31(5):521–547, May 2001. Christopher W. Brown. QEPCAD B: a program for computing with semi-algebraic sets using CADs. ACM SIGSAM Bulletin, 37(4):97–108, 2003. B.F. Caviness and J. R. Johnson, editors. Quantifier Elimination and Cylindrical Algebraic Decomposition. Texts and Monographs in Symbolic Computation. Springer-Verlag, 1998. G. E. Collins. Quantifier elimination for the elementary theory of real closed fields by cylindrical algebraic decomposition. In Lecture Notes In Computer Science, volume Vol. 33, pages 134–183. Springer-Verlag, Berlin, 1975. Reprinted in [8]. G. E. Collins. Quantifier elimination by cylindrical algebraic decomposition - 20 years of progress. In B. Caviness and J. Johnson, editors, Quantifier Elimination and Cylindrical Algebraic Decomposition, Texts and Monographs in Symbolic Computation. Springer-Verlag, 1998. G. E. Collins and H. Hong. Partial cylindrical algebraic decomposition for quantifier elimination. Journal of Symbolic Computation, 12(3):299–328, Sep 1991.

83

Hybrid Symbolic-Numeric Integration in Multiple Dimensions via Tensor-Product Series Orlando A. Carvajal, Frederick W. Chapman, Keith O. Geddes

∗

Symbolic Computation Group, School of Computer Science, University of Waterloo Waterloo, ON, N2L 3G1, Canada

ABSTRACT

Categories and Subject Descriptors

We present a new hybrid symbolic-numeric method for the fast and accurate evaluation of definite integrals in multiple dimensions. This method is well-suited for two classes of problems: (1) analytic integrands over general regions in two dimensions, and (2) families of analytic integrands with special algebraic structure over hyperrectangular regions in higher dimensions. The algebraic theory of multivariate interpolation via natural tensor product series was developed in the doctoral thesis by Chapman, who named this broad new scheme of bilinear series expansions ”Geddes series” in honour of his thesis supervisor. This paper describes an efficient adaptive algorithm for generating bilinear series of Geddes-Newton type and explores applications of this algorithm to multiple integration. We will present test results demonstrating that our new adaptive integration algorithm is effective both in high dimensions and with high accuracy. For example, our Maple implementation of the algorithm has successfully computed nontrivial integrals with hundreds of dimensions to 10-digit accuracy, each in under 3 minutes on a desktop computer. Current numerical multiple integration methods either become very slow or yield only low accuracy in high dimensions, due to the necessity to sample the integrand at a very large number of points. Our approach overcomes this difficulty by using a Geddes-Newton series with a modest number of terms to construct an accurate tensor-product approximation of the integrand. The partial separation of variables achieved in this way reduces the original integral to a manageable bilinear combination of integrals of essentially half the original dimension. We continue halving the dimensions recursively until obtaining one-dimensional integrals, which are then computed by standard numeric or symbolic techniques.

I.1.2 [Symbolic and Algebraic Manipulation]: Algebraic algorithms; G.4 [Mathematical Software]: Algorithm design and analysis

General Terms Algorithms

Keywords multiple integration, symbolic-numeric algorithms, approximation of functions, tensor products, bilinear series, splitting operator, Geddes-Newton series expansions, Geddes series scheme, deconstruction/approximation/reconstruction technique (DART)

1.

INTRODUCTION

The problem of approximating definite integrals is commonly known as quadrature (for single or multiple integrals) or cubature (for multiple integrals). Any iterated definite integral with variable limits of integration can be reduced to the following standard form via simple linear changes of variables: 1

I(f ) = 0

···

1 0

f (x1 , x2 , . . . , xd) dxd · · · dx2 dx1 .

The region [0, 1]d is known as the unit hypercube of dimension d or the unit d-cube. Multiple integration problems arise in various application areas, including atomic physics, quantum chemistry, statistical mechanics, and Bayesian statistics. The numerical evaluation of multiple integrals is computationally difficult, especially for larger dimensions, due to the size of the region over which the integrand must be sampled. Various methods have been developed for this problem, but no single method is found to be best for all cases. We propose a new method which we believe is superior to other methods for certain important classes of multiple integrals. We shall present an algorithm to approximate a multivariate function via natural tensor product series. There are two main advantages of this approximation. The first is that the approximation is represented using basis functions with only half as many variables as the original function. The second advantage is that because these basis functions belong to the same family as the original function (e.g., polynomial, rational, trigonometric, exponential, or various special functions), the interpolation series typically needs only a modest

∗This work was supported in part by NSERC of Canada Grant No. RGPIN8967-01 and in part by the MITACS NCE of Canada.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. ISSAC’05, July 24–27, 2005, Beijing, China. Copyright 2005 ACM 1-59593-095-7/05/0007 ...$5.00.

84

2.2

number of terms. We thus obtain a method to approximate a multiple integral in terms of a combination of integrals of half the original dimension. In contrast, other methods rely on subdividing the region to obtain the required accuracy. Our method exploits the possibility of symmetrizing the integrand. In two dimensions, this symmetrization is always possible while in higher dimensions it is non-trivial. Nonetheless, we are able to obtain the symmetry we require in higher dimensions for several important classes of integrands. Our method is able to compute multiple integrals efficiently and to high accuracy, in some cases where other methods require too much computation time.

2.

Globally adaptive methods iteratively divide the integration region into subregions until the desired accuracy is achieved. At each iteration, product formulas are used to estimate the value of the integral over the subregions. If the estimated error is not small enough, the subregion with the highest error is subdivided. The most popular implementations of these methods are ADAPT [7] and DCUHRE [1], the latter being an evolution of the former.

2.3

Monte Carlo Methods

The basic idea behind Monte Carlo methods is very simple: if we evaluate the integrand at n uniformly distributed random points in the region of integration, we can approximate the integral by multiplying the arithmetic mean of these n function values by the volume of the region [8]. The convergence rate of O(n−1/2 ) for such a method is at the same time its great advantage and sad drawback. Because the convergence is independent of the dimension, the method works better with integrals in higher dimensions than deterministic methods. At the same time, this convergence is very slow, and therefore the accuracy which can be achieved within a reasonable amount of computation time is very low.

EXISTING METHODS

In one dimension, there are various well-known methods for numerical integration. Among the most common methods are Clenshaw-Curtis, Newton-Cotes and Gaussian quadratures. Yet, in one dimension we find that some methods are better suited for certain families of problems than others; for example, some methods are better at handling singularities. In the Maple computer algebra system, a hybrid symbolicnumeric polyalgorithm is applied to compute the numerical value of a definite integral (see [5, 6]). In that approach, various symbolic-mode analytical techniques are applied, if necessary, to deal with infinite regions of integration and to deal with integrand singularities including derivative singularities which may slow the convergence of numerical quadrature methods. These analytical techniques include change of variables to transform an improper integral into a proper integral, or to eliminate singularities of the integrand, as well as approximating the integrand by a generalized series expansion near a non-removable singularity. At its base, the polyalgorithm makes a (hopefully intelligent) choice among several quadrature methods to compute the numerical result for the (possibly transformed) integration problem. Computing the numerical value of an integral in multiple dimensions is much harder than in one dimension. In multiple dimensions, the geometry of the region over which the integration takes place can be very complex. Even for very regular geometries, the number of points at which the integrand must be sampled grows exponentially with the dimension, assuming a numerical method which is a multidimensional generalization of one-dimensional quadrature formulas. This exponential growth is sometimes called the curse of dimensionality for definite integration. We now briefly describe various existing techniques for numerical multiple integration.

2.1

Globally Adaptive Methods

2.4

Quasi Monte Carlo Methods

Quasi Monte Carlo methods have received significant attention during the past decade [10, 11]. These methods aim to improve the convergence rate of the Monte Carlo method by using quasi-random numbers with specific distributions. A very comprehensive study of a general class of quasi Monte Carlo methods known as lattice rules is given by Sloan [12]. In general, lattice rules are a good choice only for smooth one-periodic integrands over a hypercube.

2.5

Dimensionality Reducing Expansions

He [9] presents analytical methods which reduce the dimension of the integral by one. This reduction is achieved by replacing the original region of integration with its boundary. The resulting dimensionality reducing expansions for multiple integration are derived using a multidimensional version of integration by parts, and thus require partial derivatives of the original integrand. This method is mostly used to create boundary type quadrature formulas (quadrature formulas which sample the integrand only on the boundary of the region) and in the evaluation of oscillatory integrals. The tensor product methods presented in this paper have a significant advantage over dimensionality reducing expansions: tensor product methods reduce the dimension geometrically by a factor of two, rather than arithmetically by only one dimension.

Quadrature Formulas

The most basic technique to approximate a definite integral uses a quadrature formula that computes the value of the integrand at various sample points in the region of integration. Product formulas extend the quadrature formulas used for one-dimensional integrals to higher dimensions. However, the number of sample points grows exponentially with the dimension. For some specific cases such as polynomial and trigonometric integrands, there are non-product formulas with the property that the number of sample points grows less than exponentially. Stroud [13] provides a comprehensive list of both product and non-product formulas for a wide variety of regions.

3.

TENSOR PRODUCT SERIES

We start by giving the definition of a tensor product and of the nonlinear splitting operator defined by Chapman [3] for generating natural tensor product series expansions.

3.1

Tensor Products

A tensor product is a finite sum of terms, where each term is a product of univariate functions: sn (x, y) =

n
i=1

85

gi (x) hi (y) .

Refining the terminology introduced by Chapman [3], we call the resulting series sn the Geddes-Newton series expansion1 of f to n terms with respect to the splitting points {(ai , bi )}n i=1 . In summary, the above algorithm applies the splitting operator at the i-th splitting point (ai , bi ) to the previous remainder ri−1 to generate the formula Υ(ai ,bi ) ri−1 for the i-th term of the Geddes-Newton series sn .

In mathematics, we find many bivariate functions and families of functions that can be expressed as tensor products: ex+y = ex ey cos(x + y) = cos(x) cos(y) − sin(x) sin(y) (x + y)2 = x2 + 2xy + y2 .

Example 1

The minimum number of terms among all the equivalent representations of the tensor product is called the rank of the tensor product. A tensor product is natural when the factors of each term can be derived from the original function by a finite linear combination of linear functionals.

3.2

The following function is defined on the unit square [0, 1]2 : 2 2

f (x, y) = ex

Given a bivariate function f and a point (a, b) where f (a, b) = 0, the splitting operator Υ(a,b) at the point (a, b) is defined by

s3 (x, y) =

3


ci gi (x) hi (y)

i=1

f (x, b) · f (a, y) . f (a, b)

where c1

The point (a, b) will be called a splitting point. Note that Υ(a,b) splits f into a rank-one tensor product. Two important properties of Υ(a,b) f (x, y) are:

= −0.884017, 2

g1 (x) = ex cos(x + 1), 2

h1 (y) = ey cos(y + 1);

• Υ(a,b)f (x, y) interpolates f (x, y) on the lines x = a and y = b . Thus Υ(a,b) f (a, y) ≡ f (a, y) and Υ(a,b) f (x, b) ≡ f (x, b) .

c2

= 0.794868, 2

g2 (x) = cos(x) + 0.477636 ex cos(x + 1),

• If there is a value x such that f (x, y) ≡ 0, it follows that Υ(a,b)f (x, y) ≡ 0 as well. Likewise for a y such that f (x, y) ≡ 0 .

2

h2 (y) = cos(y) + 0.477636 ey cos(y + 1); c3

The combination of these two properties is what allows us to generate a natural tensor product series approximation. This series is generated simply by iterating the splitting operator while varying the choice of splitting point.

3.3

cos(x + y) .

The Geddes-Newton series to three terms is as follows, where the splitting points are (a, a) for a = 1, 0, 0.616336 :

The Splitting Operator

Υ(a,b) f (x, y) =

y

= −9.83284, 2

g3 (x) = e0.379870 x cos(x + 0.616336) − 2

0.356576 ex cos(x + 1) − 0.623342 cos(x), 2

h3 (y) = e0.379870 y cos(y + 0.616336) −

Geddes-Newton Series Expansions

2

0.356576 ey cos(y + 1) − 0.623342 cos(y).

Not every bivariate function f can be expressed as a tensor product of finite rank; however, if an approximation is desired within a compact rectangle R = [a, b]×[c, d] where f is continuous, we can uniformly approximate f by a natural tensor product sn of rank n. We define the n-th remainder to be rn = f − sn and call the function |rn | the absolute error for the approximation sn of f on R. The following simplified algorithm generates a natural tensor-product series expansion sn of rank n of a given function f on R. (Here the uniform norm •∞ is with respect to the region R.)




1

0.8

0.6

0.4

1. Define the initial remainder by r0 := f and initialize the counter i := 1.

0.2

2. While the uniform error satisfies ri−1 ∞ > δ, iterate the following two steps:

0 0

(a) Choose a splitting point (ai , bi ) ∈ R such that |ri−1 (ai , bi )| = ri−1 ∞ .

0.2

0.4

0.6

0.8

1

Figure 1: Lines of Interpolation in Example 1

(b) Let ri := ri−1 − Υ(ai,bi ) ri−1 and i := i + 1.

1 Geddes-Newton series expansions are merely one kind of bilinear series in the general Geddes series scheme. The scheme includes numerous other classes as well, such as Geddes-Taylor series, Geddes-Fourier series, GeddesChebyshev series, and many kinds of Geddes-wavelet series.

3. After exiting the loop, let n := i − 1 and sn := f − rn . Return sn as the desired series approximation of f with uniform error rn ∞ ≤ δ over R.

86

unit square, it suffices to show that the n-th mean-value constant Mn := (∂/∂x)n (∂/∂y)n rn ∞ has an asymptotic growth rate satisfying Mn = o(n!2 ) as n → ∞. Note that even a rapid growth rate like Mn = O(nn ) still ensures uniform convergence. A rigorous upper bound on Mn can be obtained from upper bounds on the higher-order partial derivatives of f on [0, 1]2 and a positive lower bound on the smallest singular value of the nonsingular n × n matrix [f (ai , bj )]n i,j=1 . The elements of this matrix arise naturally as the values of f at the n2 points where the interpolatory grid lines x = ai and y = bj intersect. The proof of these results is beyond the scope of this paper and will appear separately.

1

0.8

0.6

0.4

0.2

0 0

0.2

0.4

0.6

0.8

1

4. Figure 2: Lines of Interpolation: 12 splitting points

INTEGRATION IN TWO DIMENSIONS

Now that we have a Geddes-Newton series to approximate a continuous function f within a compact rectangle R = [a, b] × [c, d], we can use it to calculate the definite integral:

As illustrated in Figure 1, the splitting points in Example 1 are all located on the diagonal y = x of the unit square. However, due to the interpolating properties of the splitting operator (see Section 3.2), the approximation s3 (x, y) agrees exactly with the given function f (x, y) at all points on the horizontal and vertical lines passing through the three splitting points. With this strong interpolating property, we find that even with only three splitting points used, the maximum error over the unit square for the approximation in Example 1 is f − s3 ∞ ≈ 3.0 × 10−3 . Using the strategies presented in the following sections for generating a Geddes-Newton series expansion, if a function is symmetric (i.e., f (x, y) ≡ f (y, x)) as in Example 1, the splitting points which are chosen in the unit square usually all lie on the diagonal. Moreover, we will always be able to symmetrize any bivariate function. If we request a series accurate to 15 digits for the function f (x, y) defined in Example 1, we find that only 12 terms are required (i.e., 12 splitting points) to yield an approximation satisfying f − s12 ∞ < 5.0 × 10−15 . The distribution of the 12 splitting points for this particular function is illustrated in Figure 2. The issue of convergence of a Geddes-Newton series expansion to the original function has similarities with the case of univariate polynomial interpolation via the Newton interpolation series. In both cases, the addition of a new term in the series corresponds to the addition of a new interpolation point. However, in the case of Geddes-Newton series the interpolation property holds not only at the new point but at all points on two lines passing through that point, as illustrated in Figure 2. Intuitively, for any function f which is continuous in the unit square U = [0, 1]2 , if the splitting points are dense along the diagonal then it is reasonable to expect that sn converges uniformly to f on U as n → ∞. This expectation is based on the fact that the series sn interpolates the function f on the lines x = ai and y = bi of the two-dimensional grid generated by the splitting points. The remainder rn therefore vanishes on the boundary of each cell in the resulting n × n grid. By choosing well distributed splitting points we can cover the whole region U with small grid cells, and thus make rn as close to zero as we wish inside these grid cells by the uniform continuity of rn . In order to prove that the Geddes-Newton series expansion converges uniformly to the original function f on the

I(f ) =

f (x, y) dx dy R

I(f ) ≈ In =

sn (x, y) dx dy . R

The main goal achieved when we approximate f (x, y) by sn (x, y) is a reduction in the dimension of the integrals. Specifically, by separating the variables x and y, we replace the calculation of one integral in two dimensions by 2n integrals each in one dimension: b

d

In =

sn (x, y) dy dx = a

c

n 
i=1

b

d

gi (x) dx a

 hi (y) dy .

c

The fact that we can compute one-dimensional integrals relatively efficiently, combined with the power of this particular interpolation scheme which results in n being of modest size, makes this technique very effective. The basic idea of the integration algorithm is now clear, at least for two dimensions. Assuming that the double integral has been transformed to be over the unit square [0, 1]2 , the following conditions will be assumed to hold: • f is continuous in the closed region [0, 1]2 , which implies that the maximum norm of the integrand in [0, 1]2 is finite: f ∞ < ∞ . • The one-dimensional integrals 1

1

f (x, a) dx 0

and

f (a, y) dy 0

can be computed (numerically or symbolically) for any value a ∈ [0, 1]. A simplistic implementation of the algorithm can lead to unnecessarily large running times. We shall explain how a smart implementation of the algorithm dramatically improves its efficiency.

4.1

Preparation of the Integral

First, let us show how to meet two requirements for the initial integral: the integrand must be symmetric and the integration limits must be constants.

87

an expensive two-dimensional sampling of the remainder rn over the whole unit square. This becomes even more expensive after each iteration because the complexity of the remainder rn grows quadratically in the number of iterations n. What can we do about this difficulty? We use the following observation: after some (typically small) number of iterations, convergence of the interpolation series becomes essentially monotonic. Once this point is reached, the norm of the remainder is almost always attained on the diagonal line y = x. This property is a consequence of the symmetry of the integrand. Therefore, after a certain number of iterations it is not necessary to sample on the whole unit square to estimate the norm. The important role of the diagonal leads us to break the approximation process into two phases named the confinement phase and the convergence phase.

We can apply a simple change of variables to convert any integral with non-constant limits to a new one with constant limits. (This can be extended to integrals in any number of dimensions). The change of variables x = a · (1 − s) + b · s y = c(x(s)) · (1 − t) + d(x(s)) · t transforms the integral as follows: b

1

d(x)

1

f (x, y) dy dx = a

0

c(x)

fˆ(s, t) dt ds .

0

Thus, we can limit our attention to integrals over the unit square. We say that a function is symmetric if f (x, y) ≡ f (y, x) and anti-symmetric if f (x, y) ≡ −f (y, x). We can express any bivariate function as a sum f = fS + fA of a symmetric part fS and an anti-symmetric part fA given by

The Confinement Phase

f (x, y) + f (y, x) f (x, y) − f (y, x) ; fA(x, y) = . 2 2 Additionally, we have that the integral of fA over a symmetric region such as [0, 1]2 is always 0. This conveniently gives us

Our objective in this phase is to confine the location (ai , bi ) of the maximum error ri−1 ∞ to the diagonal y = x. After many experiments and a few prototypes, we arrived at the following conclusions:

fS (x, y) =

[0,1]2

f (x, y) dy dx =

[0,1]2

• We should select splitting points (ai , bi ) on the diagonal y = x, unless the function becomes numerically zero on the diagonal. Only in this case do we select an off-diagonal splitting point.

fS (x, y) dy dx .

Example 2

• To preserve the symmetry of the remainder, off-diagonal splitting points must be chosen in symmetric pairs: selecting the point (a, b) as a splitting point implies that the next splitting point must be (b, a). Since a = b and ri (a, a) = ri (b, b) = 0, the sum of the two new terms is symmetric.

Consider the double integral 1 0

1−x

2 2

ex

y

cos(x + y) dy dx .

0

Applying the change of variables to transform into an integral over the unit square, and then applying symmetrization, yields the new problem: 1 0

1 0

• The criterion for deciding when the first phase is over will be based on the norm of the integrand. Achieving a uniform error which is 1/100th of the initial norm has proven to be a good threshold for switching to the convergence phase (i.e., ri ∞ ≤ f ∞ /100). The number of splitting points required to obtain such a threshold depends on the qualitative behaviour of f ; oscillatory functions take longer to complete phase one.

F (s, t) dt ds , where F (s, t) = fˆS (s, t) =

 2  2 2 2 2 2 1 cos(s + t − st) es (1−s) t (1 − s) + es (1−t) t (1 − t) . 2 The Geddes-Newton series for the new integrand F (s, t) was computed to three terms based on the splitting points (a, a) for a = 0, 1, 0.434450 . Then the series was integrated (applying one-dimensional quadratures) and this yielded the following estimate for the original double integral: 0.385433 . This result agrees with the correct value of the integral to 5 significant digits, which is excellent for a three-term approximation. By using more than three terms in the series approximation, more accuracy can be obtained.


4.2

• To avoid the quadratic growth of the remainder as a symbolic expression, we convert the original function to a discrete representation. With the sample values of the function stored in a matrix, the operations to calculate a new term and update the remainder become simple linear algebra. Regarding the discretization mentioned in the latter point, for the confinement phase we choose an initial grid of 25×25 sample points. Each time we choose a splitting point (from this grid), we reduce the number of sample points that are used to estimate ri ∞ , so we must monitor the process and possibly increase the size of the grid. This depends on the qualitative behaviour of f . The criterion we have adopted is that if the number of non-zero rows/columns falls below 15 while the f ∞ /100 threshold has not been reached, the grid is expanded by a factor of two in each dimension. The result of the confinement phase is a list of splitting points (in appropriate order) that were used to generate an approximation satisfying the above-specified threshold. The matrix used in the discretization is now discarded.

A Two-Phase Algorithm

We now describe an implementation of the integration algorithm. Some of the implementation details have been developed based on empirical evidence from experimentation. One important characteristic of our implementation is the division of the approximation process into two phases. Let us explain the primary reason for doing so. The algorithm for generating the series approximation is quite simple as presented in Section 3.3. The only step that can be expensive is finding the splitting point (ai , bi ) where the absolute error |ri−1 | attains its maximum norm ri−1 ∞ . The estimation of the norm in two dimensions would require

88

4.3

Symmetry Considerations

Symmetry of the remainder throughout the process is necessary to achieve optimal performance. We have defined a criterion for the selection of splitting points which preserves symmetry. There are even more benefits of this symmetry. The series resulting from our algorithm has the following algebraic form:

0.018 0.016 0.014 0.012 0.01 0.008 0.006

sn (x, y) =

0.004

i=1

0.002 0

n


0.2

0.4

0.6

0.8

  ci

i
j=1



ki,j f (x, bj )

i




li,j f (aj , y)

j=1

where ci , ki,j , li,j = 0 are real-valued coefficients, and (ai , bi ) is the splitting point used to generate the tensor product term in the i-th iteration. Although ai = bi does not always n hold for specific i, note that {ai }n i=1 = {bi }i=1 . We can represent the series using matrices and vectors as

1

Figure 3: A Typical Error Curve on the Diagonal

sn (x, y) = VT (x) · LT · P · D · L · V(y) where

The Convergence Phase

• V(x) is the column vector of dimension n whose elements are the univariate functions f (x, ai ).

During the convergence phase, the remainder typically exhibits the following behaviour. The remainder vanishes on the boundary of each grid cell (by the interpolation theory) and has constant sign inside each grid cell. These signs alternate in adjacent grid cells resulting in a checkerboard pattern. The maximum error over the whole region generally occurs in one of the grid cells located along the diagonal. The absolute error |ri (x, x)| on the diagonal oscillates as illustrated in Figure 3, and a reasonable estimate for the point of maximum error is obtained by sampling at the midpoint between adjacent splitting points along the diagonal. Based on the aforementioned properties, at each iteration in the convergence phase a new splitting point is chosen as follows: sample the remainder at each midpoint between adjacent splitting points along the diagonal, and select the point where the absolute error is largest. A minor addition is that if either or both of the points (0, 0) and (1, 1) were not chosen as splitting points during the confinement phase then they are added as candidates for the next splitting point. As discussed in Section 3.3, even though the splitting points are on the diagonal, the interpolation property holds on the vertical and horizontal lines through those points. Hence, the set of points where the remainder is zero will be dense in the unit square as long as the splitting points are dense on the diagonal. Furthermore, if the sequence of remainders {rn }∞ n=1 converges uniformly, the uniform limit will be continuous (since each sn and rn inherit the continuity of f ). The uniform limit must therefore vanish identically on the unit square (since it vanishes on a dense subset). In conclusion, if sn converges uniformly to something on [0, 1]2 as n → ∞, then this uniform limit must be the original function f. This is a direct consequence of the continuity of f, the density of the splitting points on the diagonal, and the interpolation properties of each sn . Experimentation has supported the effectiveness of this approach. The reduction in running times achieved by restricting the search to a one-dimensional region (i.e., the diagonal) is significant. At iteration n only O(n) evaluations of the remainder are computed in order to determine the next splitting point. The convergence phase ends when the estimated norm on the diagonal is less than the requested accuracy. At this point we proceed to integrate the series.

• D is an n×n diagonal matrix whose diagonal elements correspond to the coefficients ci = 1/ri−1 (ai , bi ). • P is an n × n permutation matrix that allows the coefficients ki,j to be obtained from the coefficients li,j via [ki,j ] = P·[li,j ]. The matrix P is symmetric and blockdiagonal. Each on-diagonal splitting point (a, a) generates a diagonal block of the form [1], and each pair of off-diagonal splitting points (a,b) and (b, a) generates 0 1 a diagonal block of the form . If there are 1 0 no off-diagonal splitting points, then P is the identity matrix. • L =[li,j ] is an n × n unit lower triangular matrix. This representation reduces the cost of handling what would otherwise become extremely complex expressions for rn and sn . The direct benefits are: • The cost of evaluating sn and rn is reduced from O(n2 ) to O(n) evaluations of the original function f . • The factors can be grouped to use only matrix-vector multiplications and one inner product, making the computation very efficient. • We only need to perform n one-dimensional integrations of cross-sections of the original function: f (x, ai ) for i = 1, 2, . . . , n. In the end, nearly all the processing time is spent evaluating the integrand, which is what we would hope for, and cannot avoid.

5.

INTEGRATION IN HIGH DIMENSIONS

For integrals in more than two dimensions we again wish to generate an approximation of the integrand by a tensor product series. The number of variables in the new functions to be integrated will thereby be cut in half, and applying the concept recursively will reduce the problem to some number of one-dimensional integration problems.

89

therefore the range of both s and t will be [0, 1]. Otherwise, c1 = c2 and only one of them will have

 range [0, 1], while

A major issue is how to guarantee the symmetry which is central to the method. As we have seen in previous sections, it is the symmetry of the integrand which allows the computation to be efficient. Carvajal’s master’s thesis [2] presents a novel approach which allows us to exploit our twodimensional approximation techniques in high-dimensional integration problems. Due to space limitations, we can only outline the main concepts of the new method here; further details are presented in the thesis.

5.1

the other will have range 0, min(cc1 ,c2 ) ⊂ [0, 1]. Step 2. We now compute an approximation sn (s, t) of the symmetric bivariate function v(s, t) in [0, 1]2 with our Geddes-Newton series approximation algorithm. Step 3. Next, we produce the reconstruction of f from v. Take the Geddes-Newton series expansion sn (s, t) of v(s, t) in [0, 1]2 and substitute for s and t using their defining equations above. This yields a multivariate series expansion Sn (x1 , . . . , xk ; xk+1 , . . . , xd) of f (x1 , . . . , xk ; xk+1 , . . . , xd) in the original d variables. This series approximation will be valid over the entire unit d-cube [0, 1]d . Step 4. Since we can separate the variables s and t in the Geddes-Newton series sn , we can separate the variables x1 , . . . , xk from xk+1 , . . . , xd in the multivariate series Sn . This reduces the d-dimensional integrals of the terms in Sn to integrals of dimension d/2. Finally, we apply the same technique recursively to each integral of dimension d/2 until we have only one-dimensional integration subproblems.


Description of DART

Our new method does not claim to handle all possible integrands in multiple dimensions. Rather, it is an approach that proves to be very effective for many common integrands, as described in Carvajal [2]. We call this new method of multivariate approximation and integration the deconstruction/approximation/reconstruction technique (DART). This method exploits the fact that high-dimensional integrals arising in applications frequently fit certain patterns. Multivariate functions constructed from a sum or a product of univariate functions are fairly common. We will see that we do not even need to have the original function be symmetric. The integration method has four steps: (1) Find a change of variables which converts an integrand f (x1 , x2 , . . . , xd ) in d > 2 variables into a symmetric bivariate function v(s, t). (2) Generate a Geddes-Newton series approximation sn (s, t) of v(s, t). (3) Substitute for s and t to transform sn (s, t) into a tensor product series Sn (x1 , x2 , . . . , xd ) in the d original variables. (4) Separate these d variables into two distinct groups and evaluate the resulting integrals in d/2 and d/2 variables by applying the method recursively. This recursion reduces the original d-dimensional integration problem to a collection of independent one-dimensional integrals. The following example illustrates the steps of the method.

The special form of integrand appearing in Example 3 has more general applicability than might be expected. Univariate functions of a linear combination of several variables, as in Example 3, are known as ridge functions. It has been proved that every continuous function on a compact subset of Euclidean space can be uniformly approximated by a finite sum of ridge functions [4]. Therefore, this special class of integrands can potentially be used to develop a very general method for multiple integration that would combine approximation by ridge functions and approximation by Geddes-Newton series expansions.

5.2

Example 3 (Ridge Functions)

f (x1 , x2 , . . . , xd) = u(g1 (x1 ) & g2 (x2 ) & · · · & gd (xd)),

Let us assume that a function f (x1 , x2 , . . . , xd ) in d = 2 k variables can be rewritten as

where & always denotes the same operator, either + or ×. Carvajal [2] gives a detailed presentation of the method for integrands fitting this pattern, for various template functions gi . He also discusses how to achieve an efficient implementation of the method.

f (x1 , x2 , . . . , xd ) = u(a1 x1 + a2 x2 + · · · + ad xd ) with ai > 0. We wish to calculate the integral of f over the unit d-cube [0, 1]d; thus, xi ∈ [0, 1]. Step 1. We start with a deconstruction of the original function into a symmetric bivariate function by making a change of variables. We can split the sum into two groups, each containing k variables, and let a1 x1 + a2 x2 + · · · + ak xk , s= c ak+1 xk+1 + ak+2 xk+2 + · · · + a2k x2k , t= c where c1 =

k
i=1

ai ,

c2 =

2k


ai ,

Other Integrand Patterns

DART can be applied to integrands having the following general form. Suppose that the integrand f can be expressed in terms of a univariate function u as

6.

COMPUTATIONAL RESULTS

Our integration algorithm was implemented in Maple 9. The linear algebra operations use the NAG/BLAS routines, which can be executed in both hardware floating point and in arbitrary precision software floating point in Maple. The tests were run on a computer with an Intel Pentium 4 processor at 2.02 GHz, with 0.98 GB of RAM, and running Microsoft Windows XP with Service Pack 1. We first present some results for integration in two dimensions. For this case, we consider the performance of the algorithm as the requested accuracy tolerance is decreased (i.e., the requested number of digits of accuracy is increased). Table 1 shows the results for the following double integral:

c = max (c1, c2 ) .

i=k+1

We obtain f (x1 , x2 , . . . , xd ) = u(a1 x1 + a2 x2 + · · · + a2k x2k ) = u(c · (s + t)) = v(s, t).

1 0

Note that the resulting function v(s, t) is symmetric. Since all xi ∈ [0, 1], in the best case we would have c = c1 = c2 and

1 0

sin(8π x(1 − x) y (1 − y)(x − y)2 ) dy dx .

Maple’s current numerical integration methods are not able

90

Tol 5 e-10 5 e-15 5 e-20

Time 0.3 3.7 5.3

5 e-25

7.2

5 e-30

14.2

Result 0.069551393139 0.06955139313890799 0.06955139313890799 01727 0.06955139313890799 0172712980 0.06955139313890799 017271297964487

Pts 15 21 26

RelErr 2.7 e-13 1.9 e-19 9.9 e-24

33

4.5 e-31

38

2.1 e-35

each integrand, Dim specifies the dimension d for that integrand. The dimension was chosen based on the criterion that for the given integrand, dimension 2 d would lead to a computation time of more than a few minutes. Time, Result and RelErr have the same meaning as in Table 1. The results of Table 2 show that our new method can accurately compute some integrals in very high dimensions, and can do so quite rapidly. In contrast, a state-of-theart method such as DCUHRE is successful only at modest dimensions: at most d = 15, and usually much less.

Table 1: A 2-D Integral at Varying Precisions Fcn F5 F6 F7 F16 F17 F18 F19 F22

Dim 32 16 512 256 64 16 128 32

Time 49.7 21.3 37.9 165.2 147.0 30.4 99.5 92.6

Result 3.1000000000 e01 -1.9250000000 e02 -1.8045810938 e-11 1.0000000000 e00 3.4940406596 e00 5.9973550251 e-02 4.9999927298 e-01 -5.6249710526 e01

7.

RelErr 1.9 e-13 1.6 e-13 3.1 e-10 2.0 e-14 2.8 e-12 1.5 e-11 6.4 e-13 9.8 e-12

Table 2: Integrals in High Dimensions to compute this integral beyond hardware floating point precision. In order to compute an accurate reference value, we used a specialized method based on a Taylor series expansion of sin(x) to obtain a result to 34 digits of accuracy. In Table 1, Tol is the requested accuracy tolerance, Time is the CPU time (seconds) used by our method, Result is the numerical value computed, Pts is the number of splitting points, and RelErr is the actual relative error in the computed result. The latter value is based on a comparison with the accurate reference value. Note that we display Result with a number of digits corresponding to Tol, but the computation uses 4 guard digits so the result may have additional accuracy. Indeed, there is room for the stopping criterion to be fine-tuned since RelErr is significantly smaller than Tol. A detailed discussion of the performance of the method is presented in [2]. In Table 2, we present some results for high-dimensional integration problems via DART. For this case, the requested accuracy tolerance is always 5×10−10 and the computations proceed in hardware floating point precision. The region of integration is the unit d-cube: xi ∈ [0, 1] for all i. The integrands cited in Table 2 are the following functions selected from the families of integrands considered in [2] :

F5 = F6 =

 2 d i=1 bi xi  3

F7 = cos

d i=1 bi xi



F16 = exp

d i=1

 d





xi

i=1



xi





d

F17 = ln 1 +

xi

−1 d i=1 2xi   i=1

F18 = 1 + F19 = sin 2 F22 =



π 4

d i=1

xi

d

xi

i=1  2 d cos

i=1

REFERENCES

[1] J. Bernsten, T. O. Espelid, and A. C. Genz. Algorithm 698: DCUHRE - An Adaptive Multidimensional Integration Routine for a Vector of Integrals. ACM Transactions on Mathematical Software, 17:452–456, 1991. [2] O. A. Carvajal. A New Hybrid Symbolic-Numeric Method for Multiple Integration Based on Tensor-Product Series Approximations. Master’s thesis, Univ of Waterloo, Waterloo, ON, Canada, 2004. [3] F. W. Chapman. Generalized Orthogonal Series for Natural Tensor Product Interpolation. PhD thesis, Univ of Waterloo, Waterloo, ON, Canada, 2003. [4] W. Cheney and W. Light. A Course in Approximation Theory. The Brooks/Cole Series in Advanced Mathematics. Brooks/Cole Publishing Co., Pacific Grove, California, 2000. [5] K.O. Geddes. Numerical Integration in a Symbolic Context. In B. W. Char, editor, Proc of SYMSAC’86, pages 185–191, New York, 1986. ACM Press. [6] K.O. Geddes and G. J. Fee. Hybrid Symbolic-Numeric Integration in Maple. In P. Wang, editor, Proc of ISAAC’92, pages 36–41, New York, 1992. ACM Press. [7] A.C. Genz and A. A. Malik. An Adaptive Algorithm for Numerical Integration over an N-Dimensional Rectangular Region. Journal of Computational and Applied Mathematics, 6:295–302, 1980. [8] J. M. Hammersley and D. C. Handscomb. Monte Carlo Methods. Methuen, 1964. [9] T. X. He. Dimensionality Reducing Expansion of Multivariate Integration. Birkha˝ user, 2001. [10] F. J. Hickernell. What Affects Accuracy of Quasi-Monte Carlo Quadrature? In H. Niederreiter and J. Spanier, editors, Monte Carlo and Quasi-Monte Carlo Methods, pages 16–55. Springer-Verlag, Berlin, 2000. [11] C. Lemieux and P. L’Ecuyer. On Selection Criteria for Lattice Rules and Other Quasi-Monte Carlo Point Sets. Mathematics and Computers in Simulation, 55(1-3):139–148, 2001. [12] I. H. Sloan and S. Joe. Lattice Methods for Multiple Integration. Oxford University Press, 1994. [13] A. H. Stroud. Approximate Calculation of Multiple Integrals. Prentice-Hall, 1971.

 xi .

F5 and F6 contain coefficients bi which were assigned nonzero integer values in the range −5 ≤ bi ≤ 5 (see [2] for details). In Table 2, Fcn denotes the integrand function and for

91

A BLAS Based C Library for Exact Linear Algebra on Integer Matrices Zhuliang Chen

Arne Storjohann

http://www.uwaterloo.ca/˜z4chen

http://www.uwaterloo.ca/˜astorjoh

School of Computer Science, U. Waterloo Waterloo, Ontario, N2L 3G1 Canada

ABSTRACT

Linear Algebra over

Algorithms for solving linear systems of equations over the integers are designed and implemented. The implementations are based on the highly optimized and portable ATLAS/BLAS library for numerical linear algebra and the GNU Multiple Precision library (GMP) for large integer arithmetic.



 

Nonsingular Solving over



Categories and Subject Descriptors



Word-Size Linear Algebra

I.1.2 [Symbolic and Algebraic Manipulation]: Algorithms; G.4 [Mathematical Software]: Algorithm design and analysis, Efficiency

 ATLAS/BLAS

General Terms







GMP Bignum

Algorithms Figure 1: Organization of IML

Keywords integer matrix, linear system solving

1.

lifting algorithm [5, 19] for nonsingular solving. Second, we design and report on an implementation of a new algorithm, based on ideas from [13, 14, 21], for certified solving. A feature of the certified solver is that lattice basis reduction may optionally be used to reduce the size of the particular solution. The implementations of these exact solvers are included in the recently released free library of C source code called IML — Integer Matrix Library1 . Below we discuss these two contributions in more detail, but first consider the organization of the IML library shown in Figure 1. The modules in the lowest level refer to two highly optimized and portable software libraries: the Automatically Tuned Linear Algebra Software library [29] for numerical linear algebra, and the GNU Multiple Precision library [16] for large integer arithmetic. Now consider the dashed box in Figure 1. The module WordSize Linear Algebra refers to the computation of matrix invariants (e.g., determinant, inverse, rank, left and right nullspace, row echelon form) over a small prime field. Section 2 discusses our implementation of this module. We use the standard representation for elements of p — nonnegative integers between 0 and p 1. In IML we always choose the moduli p small enough to allow direct use of the numerical BLAS routines for basic arithmetic operations on matrices and vectors. For example, by stipulating that n p 1 2 253 1 (the size of a double mantissa), the multiplication over p of two matrices with inner dimension n can be performed with single matrix multiplication over followed by reduction of entries modulo p. A more general purpose BLAS interface for matrix arithmetic over finite fields is already described

INTRODUCTION

The fundamental problem in exact linear algebra is to compute a particular solution to a system of linear equations. Linear solving can be divided into two cases. The first case – nonsingular solving – is to compute the unique solution to a nonsingular system Ax n n and b n 1. The second and more general b, where A case – certified solving – is to compute a solution with minimal denominator to a system Ax b, or to certify that the system has n m has arbitrary shape and rank. no solution, where A Nonsingular solving is the main building block of many recently proposed algorithms for other problems, including Diophantine solving [13, 20], the certified solving problem mentioned above [21], determinant [1, 10], Smith form [10, 23], and special cases of Hermite form [28]. Nonsingular solving is the main computational task driving the cost of all the algorithms in [1, 10, 13, 20, 21, 23, 28]. The two main contributions of this paper are as follows. First, we describe an efficient implementation of the well-know p-adic


 


 


 

 

  

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. ISSAC’05, July 24–27, 2005, Beijing, China. Copyright 2005 ACM 1-59593-095-7/05/0007 ...$5.00.



1 http://www.scg.uwaterloo.ca/˜z4chen/iml.html

92

in [8], where a wider class of fields (i.e., larger, nonprime) are handled efficiently by using various techniques (e.g., lazy reduction, blocking to preserve exactness, alternative field representations). See also [6, 9].


 


a certified solution of the compressed system from the single nonsingular system solution C1 1 C2  b , which is then easily extended to get a solution of the original system. The nonsingular system solution C1 1 C2  b is computed using the fast nonsingular solver discussed above. Asymptotically, the expected cost of our certified solver is the same as the algorithm in [21], but there are two important advantages in practice. First, the approach in [21] needs to solve multiple systems of the form ABy b, for a sequence of random dense matrices B. The algorithm here solves a single nonsingular system C1 y  C2  b with right hand side constant number of columns. It is considerably more efficient to solve one nonsingular system with right hand side k columns than to solve k different nonsingular systems with right hand side one column. Just consider that only one as opposed to multiple matrix inverses modulo p need be computed. Second, lattice reduction may be used to optionally reduce the bitlength of numerators in the solution vector. Our algorithm for solving the compressed system Cx b recovers at the same time an integer right kernel basis for C which has dimension k (e.g., k 10). This kernel can be used to reduce the solution size using lattice basis reduction, based on an idea described in [25, Page 381]. Although a reduction is not guaranteed, this method works well in practice. For example, consider a system Ax b with A of dimension 1000  2000 and all entries in A and b chosen randomly in the range 7  7. Without lattice reduction, a particular (in this case Diophantine) solution is computed in 52s and has entries with about 3097 decimal digits. With basis reduction the time increases to 89s but entries in the improved solution have only 311 decimal digits. The key to this approach is that lattice reduction is applied to a vector space of dimension k (in this example k 10). Applying the idea in [25, Page 381] directly would require computing and reducing the right kernel of dimension 1000 of the original matrix A; we expect this (since the system is random) to produce a solution vector with very small entries (i.e., single decimal digit) but with current techniques the lattice reduction of this large kernel is prohibitively expensive in terms of both time and space. For a matrix or vector A over , we denote by  A  the maximum magnitude of entries in A.


 

Nonsingular solving. The goal is to solve a system Ax b, n n is nonsingular and b n 1. A key feature of this where A problem
is expression swell. We expect the unique solution vector n 1 to require about as much space to write down as the A 1b entire input system. The three main computations in the variation of p-adic lifting that we use are: (i) a matrix inverse modulo p, log p O log n  log  A  ; (ii) O n matrix-vector products with entries of bitlength about log p; (iii) O n operations with integers of bitlength about n log p. Stage (iii) is accomplished with GMP. Our implementation reduces stages (i) and (ii) to Level 3 BLAS Gemm (wordsize matrix-matrix multiplication) and Level 2 BLAS Gemv (wordsize matrix-vector multiplication) by working over a well chosen residue number system. For the practitioner and implementor, the most important section of this paper is Section 3, where all the various optimization techniques used in our p-adic solver are described. The goal of our implementation is both performance and generality. For example, transposed systems At x bt are directly handled, the right hand side b may be multiple columns, and the integers in A and b may be arbitrarily large. However, if b is a single column and/or entries in A are all word-size (i.e., 32 or 64 bit) then the solver is optimized to take advantage. All timings we give in this paper use the same  environment:  Intel R Itanium R 2 @ 1.3GHz, 50Gb RAM, Intel R C++ compiler 8.0, GMP v. 4.1.3, ATLAS v. 3.6.0. Our code will solve a system of dimension 500, 1000 and 2  000 with single decimal digit entries in about 1.2 seconds, 7 seconds, and 42 seconds (see Table 2). While integers in these input systems were chosen randomly in the range 7  7, the numerators and denominators of entries in the solution vectors have 141, 1904, and 4110 decimal digits, respectively. Our implementation is designed for dense matrices, and doesn’t take advantage of any structure or sparseness that may be present in the input system.













2.

Certified solving. A reduction of Diophantine solving to non-


 

singular solving is given in [13]; an integer solution (if one exists) n m has arbitrary shape and rank of a system Ax b, where A profile, can be computed with high probability by combining a few random rational solution vectors. The ideas in [13] are taken up in [20] and extended in [21] to get an algorithm for certified solving: compute either a minimal denominator solution or prove that the system is inconsistent. Section 6 describes our new algorithm for the certified solving problem; the definition of certified solving is recalled at the start of the section. n m has full row rank. On the one Suppose for now that A hand, the approach of [21] is to solve multiple nonsingular sysm n is chosen randomly. tems of the form ABx b, where B On the other hand, the algorithm we describe here computes a sinm n  k , gle compression of the form C AB, for random B and then computes a certified solution of the compressed system Cx b. The matrix C can be written as C  C1  C2  where C1 is nonsingular and the column dimension k of C2 is a small constant. Thus, the compressed system is almost square. Section 4 sketches our (lengthy) proof that the compression will be successful with high probability even when k 25 (i.e., that the compressed system Cx b has the same minimal denominator as the original system Ax b). Section 5 gives a deterministic algorithm for computing


 


 

WORD-SIZE LINEAR ALGEBRA


  

Almost all vector space invariants of a given A p n m can be recovered by computing an echelon transform: a structured nonsingular U and a permutation P such that UPA is in echelon form. The following example has rank profile 1  4  5 .



U  



U2 dI3

PA

 

U1



      



R

1 





1



1   



A recursive reduction of echelon transform computation to matrix multiplication is described in [27, Section 2.1]. We have implemented an optimized version of the echelon transform algorithm  in IML. The functionality is also available in Maple R 10 as the command Modular[RowEchelonTransform]. All matrix multiplications arising in the algorithm have inner dimension at most  n 2 , and thus can be performed directly with Gemm provided that  n 2 p 1 2  p 1 253 1. The additive term p 1 arises because the algorithm takes advantage of an extra matrix addition operation allowed to be performed by Gemm. On the one hand, two n  n matrices can be multiplied with 2n3 n2 arithmetic operations. On the other hand, the computation of the inverse of A using the echelon transform also has cost


 





93









2n3  O n2 arithmetic operations. Of course, the real advantage of using a recursive reduction to matrix multiplication is that we can exploit Level 3 BLAS, which is typically orders of magnitude faster than an equivalent computation using Level 2. To evaluate our implementation in IML we use the ratio test: the time for an invariant computation (e.g., inverse or determinant) of an n  n matrix is divided by the time for computing the product of two matrices. Table 1 shows some timings for a call to Gemm A  A for comput-







Gemm A  A 0.01 s 0.06 s 0.46 s 3.58 s 11.56 s 0.89 m

Dimension 300 500 1000 2000 3000 5000







detA mod p Gemm A A

5 3 2.3 1.9 1.7 1.4

2 1.33 0.98 0.74 0.66 0.54



ing AA for different dimensions of A. In the same table, we show the ratios of the times to compute A 1 mod p and det A mod p with the time for the call to Gemm A  A . For dimension five thousand an inverse computation takes about one and a half times as long as a matrix multiplication, while the determinant can be computed in about half the time. Similar results are obtained for the computation of rank, nullspace, and reduced row echelon form. We conclude that the echelon transform gives a highly effective (in practice) reduction of matrix (vector space) invariant computation to matrix multiplication.

3.



NONSINGULAR SOLVING


 

 

(i)

B:



1

mod A





The second phase computes vectors ci such that mod A 1b  pi  1 c0  c1 p   ci pi for i 0  1   k 1, k to be determined.



(ii)



3.1

 




(iii) c : x:



  

k c 0  c1 p   ck 1 p k RatRecon c  N  D  p ;



1;

















Fast solution reconstruction

 





# radix conversion



 



The first part of phase (iii) is to compute the sum c c0  c1 p   ck 1 pk 1 . Initially we used a Horner scheme but discovered this was too slow. Instead, we implemented recursive algorithm for the radix conversion which incorporates large integer multiplication (see [12, Exercise 9.20]) to take better advantage of GMP. The next step is rational reconstruction. The goal is to compute a common denominator d such that d D and  mod dc  pk  N. The naive approach to accomplish this is to apply rational reconstruction on each entry of c independently and then set d to be the lcm of the n denominators. In practice, this is slow since each entry in the solution vector typically has denominator a large factor of d (in many cases equal to d.) In particular, GMP doesn’t currently offer an optimized function for rational number reconstruction. Our implementation3 is very sensitive to the size of the denominator.

Phase (iii) is accomplished using GMP. The optimized implementation of phase (iii) will be discussed later. First consider phases (i) and (ii). Our implementation reduces all of the matrix arithmetic in these stages to a number of calls to the highly efficient BLAS routines Gemm (matrix-matrix products) and Gemv (matrix-vector products). This is accomplished by working in a residue number system. On the one hand, in order to be able to apply Gemm and Gemv directly we need to work with moduli that are bounded in magnitude by B, where n B 1 2 253 1, or, equivalently, B 2t









Hadamard’s inequality bounds the numerators and denominators of entries in A 1b by N : nn 2  A  n 1  b  and D : nn 2  A  n , respectively. Thus, if k is chosen to satisfy pk 2ND we can re
n 1 cover A 1b using radix conversion and rational number reconstruction, see [12, Section 5.10].



















r : b; for i from 0 to k 1 do ci : mod B mod r p  p ; r : r Aci p od;

 










p ;



 

n n and a right hand side b Let a nonsingular matrix A n m be given. The first phase of p-adic lifting is to compute the inverse of A modulo p, where p det A.









Table 1: Ratio test for word-size inverse and determinant.









A 1 mod p Gemm A A



where t : log2 1  253 1 n . On the other hand, to maximize the amount of work in each BLAS call the moduli should be chosen as large as possible. For n 20  000 we have t 19  35, so for all practical purposes there will be sufficiently many distinct primes in the range 2t 1  2t . Thus, we may assume that all our moduli are about t bits in length. The obvious approach is to choose p to be a single t-bit prime such that p det A (such a prime can be found easily with a random choice.) Then stage (i) is accomplished with a single application of the echelon transform algorithm described in Section 2. Moreover, each computation of ci in stage (ii) can be accomplished with a n  A   ci  n  A  p 1 , the single call to Gemv.2 Since  Aci  computation of Aci in stage (ii) can be accomplished by working over a residue number system with about 1  log2 n  log2  A  t moduli. Each iteration of the loop in stage (ii) makes t bits of progress (the bitlength of p), so the number of calls to Gemv per t bits of progress is about 2  log2 n  log2  A  t. Experiments with the approach just described revealed that much more time is spend in phase (ii) than in phase (i). This is not unexpected since a single matrix-matrix multiplication using Level 3 BLAS is orders of magnitude faster than n matrix-vector multiplications using Level 2 BLAS. To better balance the cost between phases (i) and (ii) we modify the above approach by choosing p to be a multiple of t-bit primes: p p1 p2 pl , l to be determined. Then the computation of ci in phase (ii) requires l calls to Gemv. Computing the updated r in phase (ii) by first computing Aci requires a residue number system with about l  log2 n  log2  A  t primes. But we can optimize this by taking advantage of the fact that r Aci p is integral. Using GMP, compute the division with remainder of each entry in p. Then r Aci p r by p to obtain r pQ  R, where  R  is equal to Q  R Aci p. Since  R Aci p  1  n A , we can compute R Aci p directly in a residue number system with only about log2 n  log2  A  t moduli. This optimized approach requires only l  log2 n  log2  A  t calls to Gemv per lt bits of lifting progress, compared to the previous estimate of 2l  log2 n  log2  A  l t calls. Of course, as l increases the cost of phase (i) increases, since l inverses need to be computed. Therefore, we need to find a balance between these two concerns. Our experiments suggest the choice l 2 logn  log2  A  if entries in A are small and can fit into signed long. Otherwise, we choose l log2 n  log2  A  . In the next two subsections we discuss two more optimizations, aimed at minimizing the cost of phase (iii) and minimizing the number of lifting steps in phase (ii).







2 We will assume here that the right hand side has column dimension m 1. If m 1 then a single call to Gemm should be used. 3 We implemented the standard iterative algorithm based on the ex-



94

Dimension 100 500 1000 1500 2000 2500 3000

Instead, we use a trick that is implemented in Shoup’s NTL library [24]. Initialize d : 1, and for i 1  2   n reconstruct the ith entry of the solution vector as follows. Let e denote the ith entry in c. Compute e¯ : mod de  pk and then apply rational number reconstruction on e¯ to get n d.¯ Then n d d¯ is the ith entry in the solution vector. Update d to be d : d d¯ and proceed to the next i. At the end d will be the lcm of all the denominators. The advantage of this approach is that the product of all n denominators being reconstructed is equal to d, instead of being as large as d n . Thus, the net cost of all n calls to the rational number reconstruction routine is about the same as one call with a number having denominator d. We mention an anecdotal timing result. For a nonsingular system with dimension 3000 and single decimal digit entries, the new technique reconstructs the solution within five seconds. Applying rational reconstruction to each entry independently used twentyfive minutes. Thus, a three hundred times speedup is gained. The reason for this is that the GMP library provides highly optimized multiplication and division subroutines (i.e., for the computation of e¯ : mod de  pk ).





3.2







Digits 141 877 1905 2991 4110 5257 6430

Dimension 3500 4000 4500 5000 6000 7000 8000

Time 3.1 m 4.7 m 6.5 m 9.5 m 14.7 m 23.4 m 35.4 m

Digits 7620 8821 10040 11271 13761 16288 18849

Table 2: Timings to solve a nonsingular system Ax b, where b is a vector and entries of A and b are randomly chosen in the range 7  7. The Digits column is the number of decimal digits in entries of the solution vector.

Dimension 200 500 1000 2000



Output sensitive lifting



Time 0.2 s 1.2 s 6.6 s 19 s 42 s 1.3 m 2.2 m

IML 0.2 s 1.5 s 6.6 s 42 s

NTL 0.1 s 17 s 37 s 5.6 m

Maple 5.8 s 87 s 23 m 5 7 h

In the worst case, the number of lifting steps k needs to be chosen satisfying pk 2ND in order to reconstruct the solution correctly, where N nn 2  A  n 1  b  and D nn 2  A  n come from Hadamard’s bound. However, Hadamard’s bound is often a pessimistic [2] and we can take advantage of this by performing the lifting in an output sensitive manner.

Table 3: Comparison of timings to solve a nonsingular system Ax b using IML NonSingSolve, NTL solve1, and Maple Modular[LinIntSolve]. Entries in A and column vector b are randomly chosen between 7 and 7.

1. Initialize k0 to be zero and k to be a small positive integer (e.g., k 10).

point out that the time spent for the matrix inverse computations is almost the same as the time spent for the lifting iteration, due to our optimized choice of size of the lifting basis. A more detailed breakdown of the timing can be found in [4]. Table 3 compares our implementation with function solve1 in  NTL [24] v. 5.3.2 and Modular[LinIntSolve] in Maple R v. 9.5. NTL also makes full use of GMP for the large integer arithmetic, but except for the fast rational reconstruction technique described in Section 3.1, the implementation in NTL doesn’t incorporate the optimizations described in this section. The speedup of IML over NTL is obtained in large part because NTL is not using the ATLAS/BLAS library to perform the matrix-vector and matrix-matrix multiplications. The solver in Maple is actually BLAS based but uses a homomorphic imaging scheme instead of lifting. Giorgi [15] has implemented a BLAS based version of lifting for nonsingular solving in the LinBox5 library [7]. For input matrices with small entries (i.e., in the range 7  7) the timings obtained are very similar to those in Table 2. For matrices with larger entries the IML implementation using multiple primes for lifting gives an improvement (e.g., the solution to a 2000  2000 input matrix with 100 bit entries takes about twice as long in LinBox as with IML). Finally, we mention a technical optimization based on a feature of BLAS. If the right hand side b has column dimension m 1, the computation of r and ci in phase (ii) of the algorithm should make use of single calls to Gemm instead of m calls to Gemv. For example, Table 4 shows that solving a system with right hand side ten columns takes less than five times as long as solving a system with right hand side a single column.

 

 







2. Perform k k0 iterations in phase (ii) to compute the lifting coefficients ck0   ck 1 . 3. Compute c :

c 0  c1 p 

 

ck

1p

k 1.

4. Use the rational reconstruction scheme of the previous section to attempt to compute a common denominator d and pk 2 . y : mod dc  pk such that max d   y 







 





5. If the reconstruction succeeds and Ay db then return y d. Otherwise, assign k to k0 and increase k by a small positive integer4 and goto step 2. In our implementation we optimize the above scheme by making the following two changes. First, we merge steps 3 and 4. As soon as we compute the ith entry e of c using radix conversion, we perform the rational reconstruction on e before starting to compute the i  1 st entry of c. If the reconstruction of e fails, we avoid computing entries i  1   n of c. In practice, if k is not large enough a failure is reported quickly. Second, instead of assaying correctness of y in step 5 with an expensive matrix-vector multiplication (note that entries in y may have bitlength n times larger than entries in A) we verify the solution by checking the magnitude bound according to the following lemma, due to [3].





L EMMA 1. If d  b  n  A   y 

3.3

 pk 1  2 then Ay



bd.

Timings

Table 2 gives some timings to solve the nonsingular system Ax b for different dimensions of A, where b is a vector and entries of A and b are relatively small. Although not shown in the table, we

4.

LATTICE COMPRESSION

 n

Let A


 m  n with full row rank m be given, n 




m. Let B , with entries chosen uniformly and randomly from a finite set Λ 0  1   λ , where λ 2. We call the action of post-

tended euclidean scheme, see [12, Section 9.10]. 4 Our implementation uses k : k  10, which has not been optimized.

m  k




5 http://www.linalg.org

95



Dimension 500 2000 4000

m 1 1.2 s 42.0 s 4.7 m

m 10 4.9 s 3.4 m 22.3 m



Ratio 4.1 4.9 4.7





U 

Table 4: Comparison of timings to solve a nonsingular system Ax b with m 1 and m 10, where m is the column dimension of b and the entries of A and b are randomly chosen between 7 and 7.



U1 





N

N B


 


 

 


 

 








where 

















1s



1

..



.

sm



t1

(1)  

S2 S3




 





and only if all the diagonal entries in the Smith form of N B are equal to 1. B1 B2 B3 and introduce two indeterPartition B as B minants t1 and t2 such that the column dimensions of B1, B2 and B3 is m t1 , t1  t2 and k t2 respectively. There exist unimodular matrices U and V which separately apply row operations and column



























96













.

CERTIFIED SOLVING WITH KERNEL BASIS

 

    p Lemma 3 implies that C mod p has full row rank m over  for any prime p if and only if N B  mod p has full row rank   p . Since  N B  is equivalent to its Smith form, n over   p for any prime p if  N B  mod p has full row rank over 



In this section we present a deterministic algorithm to certified solve a linear system Cx b. We consider a special case of the A B problem: we assume that C may be decomposed as C n n  k where A n n is nonsingular. The algorithm here is applicable for arbitrary B, but is designed for the case when the column dimension k of B is small (e.g., k O 1 ). In Section 6 we show how to use the lattice compression technique of Section 4 to certified solve an input system with arbitrarily column dimension by reducing to the special case we describe here, with k 25. Before describing the
algorithm we recall the difference between a right nullspace (over ) and right kernel (over ). The matrix C is over the principal
ideal domain , but we may also consider C
n  k k to be over the field . On the
one hand, a matrix N is a right nullspace for C over if rank N k and CN 0. On n k k to be a right kernel for C the other hand, for a matrix K over the following additional condition must be satisfied: every integer vector in the right nullspace of C must be generated by an integer linear combination of columns of K. Computing a right kernel is a more subtle problem than computing a right nullspace. For example, scaling a nullspace by multiplying by the least common multiple of the denominators of entries will produce a nullspace with integer entries, but this is unlikely to be a kernel. The first step of our algorithm is to compute A 1 B b . Then n  k k for C over , the algorithm constructs a right kernel K

m n and B n l be given. Let N be L EMMA 3. Let A a right kernel for A. Then for any prime p, rank AB mod p rank N B  mod p rank N mod p over the finite field p.





I2



 

5.

 


 









 


 



since N L In m . Having set up the transform as above, we can see that a sufficient condition for all the diagonal entries of the Smith form of N B to equal one is that s1 s2  sm t1 1 1 and S2 S3 R It1  1 . The first step of our proof is to bound the sm t1 1 1 using the technique from probability that s1  [10, Section 6]. A problem of adapting the technique is that it can only provide a useful bound on the probability that s1   sm t1 1 are equal to one, without determining the value of sm t1 . However, if S2 S3 R It1  1 , then all the entries on the diagonal of the Smith form of N B are necessarily 1. So, the second step of our proof is to bound the probability that S2 S3 R It1  1 using the results from [21, Section 3]. In the final step, we derive the result of Theorem 2 by fixing the values of t1 and t2 and combining the previous two results.

Now we give the idea of our proof for Theorem 2. Refer to [4] for the complete proof. Let C AB. To bound the probability of A R C, an equivalent conversion is to assume that A R Im and to bound the probability that C R Im . Since A R Im if and only if for all primes p, A mod p has full row rank over p , our goal is to derive a lower bound on the probability that C mod p has full row rank over p for all primes p. n n m denote a right kernel for A. Then N Let N L In m (i.e., N has the Hermite row basis an identity matrix). Thus, for any prime p, N mod p has full column rank over p . From [21, Lemma 15] we obtain the following.

 


 



S2 S3  U2 B2 B3  and the t1  1  n submatrix U2 is in the nullspace for N. The first n m invariant factors are 1









m n be given, where rank A m and T HEOREM 2. Let A  n m k n m. Let B have entries uniformly and independently chosen at random from Λ 0  1   λ 1 . If λ max m  1  log2 n  log2  A  , then the probability that A R AB is at least 1 16 1 2 k 5. In particular, the probability is 1 2 if k 25.



 

S ..











1 




 





I3







B1 B2 B3






V 



U2

multiplication of A by B lattice compression: C AB where C has k more columns than rows. Let A R C denote that A and C have the same Hermite column basis (i.e., the set of all -linear combinations of columns of A is equal to the set of all -linear combinations of columns of C). In this section we sketch the result in [4] which gives a lower bound on the probability that A R C; the bound exponentially converges to one with respect to k. The lattice compression technique will be used by the algorithm in next section. The strongest form of lattice compression has k 0. For ex1) such that all ample, let V be a unimodular matrix (detV C but the first m columns of AV are zero: AV , where m m is necessarily nonsingular. If we take B as the subC matrix of V comprised of the first m columns, then C AB and A R C. However, the most efficient algorithm to compute such a V is too expensive by factor of m and, moreover, guarantees only the bound  B  m3m 2  3  A  3m (see [27, Proposition 8.10]) which causes this approach to be very inefficient. Alternatively, by choosn m  k to be a random matrix with k a small constant, ing B we have A R C with high probability.





operations on N B and transform the submatrix N B1 to Smith form. Such U and V can be partitioned using a conformal block decomposition as





 









n  k 1 such that Cy a minimal denominator solution y b,
1 n and a certificate vector z which proves the minimality of the denominator of y. For clarity, we first consider the computation of only K, then show how extend the method to compute y and z. For our size estimates in this section we assume that O n logn  log  C  . log  b  Finally, we recall a technique for optionally reducing the the solution using lattice reduction.








v









k 1 such that

x1  Nv has minimal denominator. Let







N

sx1











have Hermite form 

T

 

c e



 

(2)

s






Two rational matrices that are left equivalent necessarily have the
k 1 such that same denominator. Thus, our goal is to find a v the vector 



Constructing a right kernel K 

N

We will compute K to be the last k columns of the unique unimodular matrix of dimension n  k such that





sx1





sv 1 

1  or, equivalently, s


T

 

c e



sv 1 

1 s


s K A

B Ik



 










H 




(3) has minimal denominator. The equivalent presentation in (3), together with the choice v T 1c 1 s , reveals that the minimal denominator is s e. But since sT 1 is integral, we can choose sT 1 c mod se 1 s2 , giving the minimal denominator sov k 1 can s2x1 N v¯ s2, where v¯ sT 1 c mod se lution y 1 be computed by first computing sT mod se using similar method as used for the computation of H from sT¯ 1 mod s. The cost estimate remains the same as before.








where the matrix on the right hand side is in column Hermite form. Solving gives K StackMatrix A 1BH  H . We now show how to construct H efficiently from A 1B. Let s be the denominator of n  k k . A 1 B b and set N StackMatrix sA 1B  sIk
Then N is a basis for the right nullspace of C over . The algorithm of [17] can be used to compute the upper triangular row k k. The algorithm is easily modified to comHermite basis T k k that is lower pute, instead of T , a left equivalent matrix T¯ triangular with off-diagonal entries in each column reduced modulo the positive column leader (just an alternative definition of the Hermite row basis). This algorithm uses O nk 2 integer operations and takes advantage of the rows sIk to keep all intermediate entries during the computation of T and T¯ reduced modulo s. Then N T¯ 1 is also a right kernel of C over , so the last k rows of K and N T¯ 1 are right equivalent (i.e., H is the column Hermite form of sT¯ 1 ). The following code fragment shows how to compute H from T¯ , keeping all off-diagonal entries in H reduced modulo s.










 







 








 







 





q 






 





sA 1B sA sIk





1b

 



e

s







mod s

which shows that q is as desired. It remains to show how to compute u, row k  1 of a unimodular matrix effecting the transformation in (2). The transformation to Hermite form can be accomplished with a sequence of O nk unimodular transformations, each transformation on one or two rows (cf. [27, page 55]). Multiplying these transformations together would yield a unimodular matrix U UO nk U3U2U1 , but 2 this would be too expensive (O n k arithmetic operations) and the U produced might be dense, with O n2 nonzero entries. Instead, store all these transformations and then, working modulo s, apply them in reverse order to row k  1 of the identity matrix. As a result, u can be computed at the same time. Finally, compute z : qA 1. If k O 1 the nonsingular systems A 1 B b and qA 1 can be solved with O n3 log n  log  C  2 bit operations (see [20]). Additional O n operations with integers of bitlength O n logn  log  C  bits is required. This gives the following.













Once H has been recovered, compute K : NH 1 s . By Hadamard’s bound and Cramer’s rule, all of s and  N  are bounded in length by O n logn  log  C  bits. Assuming A 1B has already been computed, the dominant step is to compute T¯ at a cost of O nk 2 operations with integers bounded in absolute value by s.





























Constructing a minimal denominator solution y



















T HEOREM 4. Suppose log  b  O n logn  log  C  and k O 1 . Given a prime p such that p O log n  log  C  and p det A, the algorithm described above computes K, y and z with O n3 logn  log  C  2 bit operations.

Now we extend the algorithm just described to compute a minimal denominator solution y. Recall that N StackMatrix sA 1B  sIk is a basis for the right nullspace of C.
A particular solution of n  k 1, and the set of Cx b is x1 : StackMatrix A 1b  0
n 1 all rational solutions is x1  Nv  v . Our goal is to find a



Constructing a certificate z

H : sIk ; for i from 1 to k do for j from 1 to i 1 do for l from 1 to j do Hil : Hil T¯i j H jl mod s od od; for j from 1 to i do Hi j : Hi j T¯ii od od



 

1 n Finally, we show how to construct a certificate vector z such that zC is integral and zb has denominator s e. Our approach 1 n such that qA 1B is integral and qA 1 b is to construct a q has denominator s e, and then set z : qA 1. Let u be equal to row k  1 of a unimodular matrix that transforms the first matrix in (2) to it’s Hermite form. Let q be the vector comprised of the first n q entries of u reduced modulo s, so that u mod s. Recall that N StackMatrix A 1B  sIk and x1 StackMatrix A 1b  0 .  Then

















Incorporating lattice reduction Let y be a minimal denominator solution for Cx b and K be a right kernel for C. If y and K are computed using the algorithm




97



Dimension n m 500  1000 1000  2000 3000  6000 6000  8000

supporting Theorem 4 the bitlength of entries will be O n logn  log  A  . We can try to find a minimal denominator solution y¯ with improved bitlength using the following approach described in [25, Page 381]. ¯ Use lattice basis reduction [18] to compute a reduced kernel K. k so that the vector d y y¯ : d y y  Ku is size Then compute u ¯ In all our experiments with reduced with respect to the vectors in K. random matrices, this produces a minimal denominator solution y¯ which has numerators with bitlength about a factor of k smaller than those in y. The main cost of the above approach is to reduce the lattice K of dimension k in in n  k -dimensional space. This lattice has a very special shape since k is so small compared to n  k (e.g., k O 1 ). Moreover, the lattice is very skew since the norms of column vectors in K are large (i.e., O n logn  log  A  ). Using directly the LLL algorithm [18] would be too expensive. Instead, we use the modification in [26] that works in three stages: (1) compute K T K k k; (2) compute, modulo an integer M O kn  K  , a unimodk k; (3) set K ¯ KU mod M. Step (2) dominates ular matrix U the cost with O k 4n logn  logA arithmetic operations involving integers bounded in length by O kn logn  log  A  bits. The algorithm in [22], which is less sensitive to  K  , may also be well suited for this task.

















 



6.


 



k 10 20 30 40 50









 





1.





 m











1 n with

qA

0   0




1 m and qb

0.











Timings



In the first case, y is a solution with minimal denominator and z serves as a certificate for the minimality of the denominator of y. The idea of minimal denominator certificate is a generalization of the integer version of Farkas’ lemma in [11]. In the second case, vector q certifies that the system is inconsistent and has no rational solution. The idea for certifying inconsistency is due to [14]. Refer to [21] for explanations in detail. There are three stages to our certified solver. The first is to either prove that the input system Ax b is inconsistent or to reduce to an equivalent system which has full row rank. Our implementation of this first stage is accomplished using a similar approach as in [21, CertifiedSolver, Page 506]. The description of this phase is omitted here since it is so similar. Henceforth we will assume that the input n m has full row rank n. matrix A The second stage is to compress the system Ax b with A n m using lattice compression as described in Section 4. This give us an almost square system Cx b (e.g., C AB with B

 




The certified solver described in Section 5 and above was implemented in IML directly as described using GMP. All nonsingular solving uses the ATLAS/BLAS based algorithm described in Section 3. For the lattice compression phase we chose by default λ 2 and k 10 (the user has the option of choosing a larger k). Using this choice, the lattice compression C AB succeeded in all our experiments, and hence only a single nonsingular system C1 1 C2 b with right hand side eleven columns needed to be solved. Table 4 shows that the time for solving this system is only about five times as long as the time for solving a system with right hand side one column. In Table 5 we give timings to compute a minimal denominator solutions to the system Ax b with entries in A and b randomly chosen. The table shows that the solution size can be approximately reduced by a factor of ten using lattice basis reduction. For a randomly chosen input system, we always observed this correspondence between the solution reduction and the kernel basis dimension. Table 6 considers the effect of increasing the value of k on the quality of the solution and the running time. Most of the increase in time is due to the lattice reduction phase. Finally, we remark that an optimized BLAS based algorithm for diophantine solving has been implemented in LinBox [7]. See [15] for a description and detailed comparison with IML.





 



n 25

6.1




Time 39 s 3.9 m 15 m 42 m 1.6 h



zb and y have the same denominator.

q

Digits 254 128 86 65 53

chosen randomly). The matrix multiplication AB is accomplished by reducing to Gemm by using a residue number system. Note that if m is not too big with respect to n (e.g., m n  25) then we can choose B Im in which case C A. The third stage computes a certified solution y¯  z to the compressed system Cx b using the algorithm described in the previous section. Using GMP, we assay if By¯  z is a certified solution to the original system Ax b by checking that zA is over , and that zb and By¯ have the same denominator.

y z , where

m  1 with Ay b, y

1  n with zA
 1  m, and z

2. (“no solution”, q), where




n m and b n 1 be given. We first recall the properLet A ties of minimal denominator. Let d y denote the denominator of a solution vector y to the system Ax b, where b can be a vector or a matrix. The set of denominators of all the solutions to the system Ax b generates an ideal I of . Let d A  b denote the generator of the ideal I. Then d A  b divides all the elements in I and hence is the minimal denominator. Our algorithms for certified linear system solving has the same functionality as the algorithms in [21], which take as input A and b and return as output one of the following:




 

Table 6: Timings to compute a size-reduced minimal denominator solution to a system Ax b for different k, where k is the column dimension of the right kernel of the compressed system. 500 1000 and b 500 1 are randomly chosen Entries in A from 210 to 210. The column Solution Digits is the number of decimal digits of the largest entry in the solution numerator.

CERTIFIED LINEAR SYSTEM SOLVING


 

Reduced Solution Digits Time 146 16 s 311 89 s 1102 33 m 2347 4.8 h

Table 5: Timings to compute a minimal denominator solution n m and b n 1. Entries of to the system Ax b, where A A and b are randomly chosen between 7 and 7. The column Digits is the number of solution decimal digits.





 



Typical Solution Digits Time 1445 9s 3097 52 s 10995 22 m 23451 3.3 h





98



7.

REFERENCES

[15] P. Giorgi. Arithmetic and algorithmic in exact linear algebra for the LinBox library. PhD thesis, Ecole normale superieure de Lyon, LIP, Lyon, France, December 2004. [16] T. Granlund. The GNU multiple precision arithmetic library, 2004. Edition 4.1.4. http://www.swox.com/gmp. [17] C. S. Iliopoulos. Worst-case complexity bounds on algorithms for computing the canonical structure of finite abelian groups and the Hermite and Smith normal forms of an integer matrix. SIAM Journal of Computing, 18(4):658–669, 1989. [18] A. K. Lenstra, H. W. Lenstra, and L. Lov´asz. Factoring polynomials with rational coefficients. Math. Ann., 261:515–534, 1982. [19] R. T. Moenck and J. H. Carter. Approximate algorithms to derive exact solutions to systems of linear equations. In Proc. EUROSAM ’79, volume 72 of Lecture Notes in Compute Science, pages 65–72, Berlin-Heidelberg-New York, 1979. Springer-Verlag. [20] T. Mulders and A. Storjohann. Diophantine linear system solving. In S. Dooley, editor, Proc. Int’l. Symp. on Symbolic and Algebraic Computation: ISSAC ’99, pages 281–288. ACM Press, New York, 1999. [21] T. Mulders and A. Storjohann. Certified dense linear system solving. Journal of Symbolic Computation, 37(4):485–510, 2004. [22] P. Nguyen and D. Stehl´e. Floating-point LLL revisited. In Proceedings of Eurocrypt ’05, 2005. [23] D. Saunders and Z. Wan. Smith normal form of dense integer matrices, fast algorithms into practice. In J. Gutierrez, editor, Proc. Int’l. Symp. on Symbolic and Algebraic Computation: ISSAC ’04, pages 274–281. ACM Press, New York, 2004. [24] V. Shoup. NTL: A Library for Doing Number Theory, 2005. http://www.shoup.net/ntl/. [25] C. C. Sims. Computation with finitely presented groups, volume 48 of Encyclopedia of mathematics and its applications. Cambridge University Press, 1984. [26] A. Storjohann. Faster algorithms for integer lattice basis reduction. Technical Report 249, Departement Informatik, ETH Z¨urich, July 1996. [27] A. Storjohann. Algorithms for Matrix Canonical Forms. PhD thesis, Swiss Federal Institute of Technology, ETH–Zurich, 2000. [28] U. Vollmer. A note on the Hermite basis computation of large integer matrices. In R. Sendra, editor, Proc. Int’l. Symp. on Symbolic and Algebraic Computation: ISSAC ’03, pages 255–257. ACM Press, New York, 2003. [29] R. C. Whaley, A. Petitet, and J. J. Dongarra. Automated empirical optimization of software and the atlas project. Parallel Computing, 27(1-2), 2001.

[1] J. Abbott, M. Bronstein, and T. Mulders. Fast deterministic computation of determinants of dense matrices. In S. Dooley, editor, Proc. Int’l. Symp. on Symbolic and Algebraic Computation: ISSAC ’99, pages 197–204. ACM Press, New York, 1999. [2] J. Abbott and T. Mulders. How tight is Hadamard’s bound? Experimental Mathematics, 10(3):331–336, 2001. [3] S. Cabay. Exact solution of linear systems. In Proc. Second Symp. on Symbolic and Algebraic Manipulation, pages 248—253, 1971. [4] Z. Chen. A BLAS based C library for exact linear algebra on integer matrices. Master’s thesis, School of Computer Science, University of Waterloo, 2005. [5] J. D. Dixon. Exact solution of linear equations using p-adic expansions. Numer. Math., 40:137–141, 1982. [6] J.-G. Dumas. Efficient dot product over word-size finite fields. CoRR, cs.SC/0404008, 2004. [7] J.-G. Dumas, T. Gautier, M. Giesbrecht, P. Giorgi, B. Hovinen, E. Kaltofen, B. D. Saunders, W. J. Turner, and V. G. LinBox: A generic library for exact linear algebra. In A. J. Cohen and N. Gao, X.-S. andl Takayama, editors, Proc. First Internat. Congress Math. Software ICMS 2002, Beijing, China, pages 40–50, Singapore, 2002. World Scientific. [8] J. G. Dumas, T. Gautier, and C. Pernet. Finite field linear algebra subroutines. In Proc. Int’l. Symp. on Symbolic and Algebraic Computation: ISSAC ’02, pages 63–74. ACM Press, New York, 2002. [9] J. G. Dumas, P. Giorgi, and C. Pernet. Finite field linear algebra package. In J. Gutierrez, editor, Proc. Int’l. Symp. on Symbolic and Algebraic Computation: ISSAC ’04, pages 119–126. ACM Press, New York, 2004. [10] W. Eberly, M. Giesbrecht, and G. Villard. Computing the determinant and Smith form of an integer matrix. In Proc. 31st Ann. IEEE Symp. Foundations of Computer Science, pages 675–685, 2000. [11] J. Edmonds and R. Giles. A min-max relation for submodular functions on graphs. Annals of Discrete Mathematics, 1:185–204, 1977. [12] J. von zur Gathen and J. Gerhard. Modern Computer Algebra. Cambridge University Press, 2 edition, 2003. [13] M. Giesbrecht. Efficient parallel solution of sparse systems of linear diophantine equations. In M. Hitz and E. Kaltofen, editors, Second Int’l Symp. on Parallel Symbolic Computation: PASCO ’97, pages 1–10. ACM Press, New York, 1997. [14] M. Giesbrecht, A. Lobo, and B. D. Saunders. Certifying inconsistency of sparse linear systems. In O. Gloor, editor, Proc. Int’l. Symp. on Symbolic and Algebraic Computation: ISSAC ’98, pages 113—119. ACM Press, New York, 1998.

99

Structure and Asymptotic Expansion of Multiple Harmonic Sums C. Costermans

J.Y. Enjalbert

Universite´ Lille 2 1, Place Deliot, ´ 59024 Lille, France

Universite´ Lille 2 1, Place Deliot, ´ 59024 Lille, France

[email protected] Hoang Ngoc Minh

[email protected] M. Petitot

Universite´ Lille 2 1, Place Deliot, ´ 59024 Lille, France

Universite´ Lille 1 59655 Villeneuve d’Ascq, France

[email protected]

[email protected] ABSTRACT

with the convention Hs (N ) ≡ 1 when s is the empty composition. It can be proved that the limit

We prove that the algebra of multiple harmonic sums is isomorphic to a shuffle algebra. So the multiple harmonic sums {Hs }, indexed by the compositions s = (s1 , · · · , sr ), are Rlinearly independent as real functions defined over N. We deduce then the algorithm to obtain the asymptotic expansion of multiple harmonic sums.

ζ(s) =

G.2.1 [Discrete Mathematics]: Combinatorics—combinatorial algorithms, generating functions

General Terms Algorithms, Languages

Keywords

Ha (N ) · Hb (N ) = Ha,b (N ) + Hb,a (N ) + Ha+b (N ).

Polylogarithms, multiple harmonic sums, Lyndon words, polyzˆetas

INTRODUCTION N X 1 n=1

ns

(s ∈ N>0 , N ∈ N>0 )

,

(1)

can be generalized to any composition s of length r ≥ 0, i.e. a sequence of positive integers s = (s1 , . . . , sr ) by putting Hs (N ) =

X N ≥n1 >···>nr >0

1 n1 s 1 . . . n r s r

(4)

So, the vector-space HR is closed under product. The main result of this article is to prove that in HR , the functions {Hs }s are linearly independent. As a consequence, HR appears to
be isomorphic to some shuffle algebra noted RhY i, . The structure of this algebra is well known and Hoffman showed that it is freely generated by Lyndon words on the alphabet Y . Let HR0 be the R-algebra generated by the functions Hs when s describes the set of all convergent compositions. We show that

Let N>0 be the set of positive integers. The harmonic sums Hs (N ) =

(3)

exists if and only if the composition s is empty or if s1 6= 1. In this case, we will say that s is a convergent composition. The values ζ(s) are called ”Multiple Zeta Values” (MZV). Harmonic sums and MZV arise in high-energy particule physics [3] and in analysis of algorithms [5]. We consider the R-vector space HR generated by the Hs , seen as functions from N to R. The theory of quasisymmetric functions shows that the {Hs (N )}s satisfy shuffle relations. In particular, the product of two harmonic functions is a sum of harmonic functions : for all a, b ∈ N>0 , we have

Categories and Subject Descriptors

1.

lim Hs (N )

N → ∞

HR ' HR0 [H1 ]

(2)

(5)

i.e. that any harmonic function can be decomposed P uniquely −1 in a univariate polynomial, on the sums H1 (N ) = N . n=1 n This decomposition is obtained thanks to a variant of Taylor expansion for univariate polynomials, by
defining a derivation D in the shuffle algebra RhY i, . In fact, this decomposition provides an asymptotic expansion, up to order 0, of Hs (N ) as N → ∞. We can then deduce an asymptotic expansion, up to any order, by using the second form of the Euler-Mac Laurin summation formula. Our result of linear independance of the functions Hs lies on the C-linear independance of the polylogarithm functions

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. ISSAC’05, July 24–27, 2005, Beijing, China. Copyright 2005 ACM 1-59593-095-7/05/0007 ...$5.00.

100

This means that every polynomial in RhY i, for shuffle product, can be decomposed uniquely in a linear combination of shuffle products of Lyndon words.

(z is a complex number so that |z| < 1) X

Lis (z) =

n1

n1 >···>nl

z s , s1 · · · nl l n >0 1

(6)

2.2

first proved in [9], then resumed in [13, 14]. In this article, harmonic functions are seen as Taylor coefficients X 1 Lis (z) = Hs (N )z N . 1−z N =0 ∞

(7)

l

Mu Mv = Mu

2.3

How to shuffle?

p=

(p|w) ∈ R.

(p|w)w,

v) + yj (u

v 0 ) + yi+j (u0

3.

=

Hw (N )

=

X

0

1 n1s1 . . . nsrr

for 0 ≤ N < r.

If |w| = 0, we put H (N ) = 1, for any N ≥ 0. Lemma 1. Let w = ys1 . . . ysr ∈ Y ∗ . The sum Hw (N ) is convergent, when N → ∞, if and only if s1 > 1. In this case, this limit is nothing but the polyzta (or MZV [15]) ζ(w), and thus, by an Abel’s theorem lim Hw (N ) = ζ(w) = lim Liw (z).

N →∞

z→1

(15)

Then, the word w is said convergent. A polynomial of RhY i is said convergent when it is a linear combination of convergent words.

(11)

Lemma 2. For w = ys w0 , we have

Hoffman generalized Radford theorem in the following way ([12, 10]). One has ) = (C0 [y1 ],

Hw (N )

N ≥n1 >...>nr >0

∗

) ' (R[Lyndon(Y )],

Harmonic sums

and

A nonempty word w is called a Lyndon word if it is strictly smaller (for lexicographic order) than any of its proper right factors, i.e. w < v for any factorization w = uv with u 6=  and v 6= . Let Lyndon(Y ) denotes the set of Lyndon words over Y . Let

(RhY i,

(14)

Definition 1. Let w = ys1 . . . ysr ∈ Y ∗ . Then, for N ≥ r ≥ 1, the harmonic sum Hw (N ) is defined as

 < w for w ∈ Y \  yi u < yj v if i > j or if i = j and u < v.

Theorem 1

z n1 · · · nsrr ,

HARMONIC SUMS ALGEBRA

3.1

(10)

C0 = R ⊕ (RhY i \ y1 RhY i).

ns 1 n1 >···>nr >0 1

In fact, this result can be improved by replacing C by any algebra of analytic functions defined over C − {0, 1}, for example C[z, 1/z, 1/(z − 1)], and the proof lies on an explicit evaluation of the monodromy group of Liw .

(9)

v 0 ),

X

Theorem 2 ([9]). The functions Liw , for w ∈ Y ∗ , are C-linearly independent.

with i, j ∈ N>0 and u0 , v 0 ∈ Y ∗ . This product is also extended to RhY i by linearity. Provided with this shuffle product, RhY i becomes
an associative and commutative R-algebra noted RhY i, . We can totally order Y by putting yi < yj if i > j. So y1 is the biggest letter of the alphabet Y . Lexicographic order is then recursively defined on words w ∈ Y ∗ by 

(13)

for r > 0 and by Li (z) = 1, for the empty word .

The concatenation product for words is extended to polynomials by linearity. The shuffle product of two words u = yi u0 and v = yj v 0 in Y ∗ is recursively defined by  w = w, for all w ∈ Y ∗ and v = yi (u0

(u, v ∈ Y ∗ ).

Polylogarithms

Liw (z) =

w∈Y ∗

u

v,

Let us associate to any word w = ys1 · · · ysr the polylogarithm Liw (z) defined for |z| < 1 by

We consider the non commutative alphabet Y = {yn | n ∈ N>0 }. As usual, the set of all words over Y is denoted Y ∗ and the empty word is denoted . The length of the word w is denoted |w| and the word w resulting from the concatenation of two words u and v is the word w = uv. Let R be a commutative ring containing Q. A polynomial p ∈ RhY i is a linear combination of words, with coefficients in R. The coefficient of the word w in polynomial p is noted (p|w) and therefore X

(12)

which is a formal series on the letters tn ∈ T , with coefficients in N. For the empty word, we define M = 1. Thanks to Gessel works, it is known that the product of quasi monomial functions is a sum of quasi monomial functions. Moreover, we have the following identity between formal series

BACKGROUND

2.1

sl , tsn11 · · · tn l

n1 >···>nl >0

where ξ1 , . . . , ξl are roots of unity.

2.

X

Mw =

By using the combinatorics developed by M. Bigotte in [2], it is possible to consider a similar generalization for the harmonic sums related to coloured MZV X ξ1 n1 . . . ξl nl l , (8) ζ(ξs11 ,...,ξ ,...,sl ) = n1 s 1 . . . n l s l n >···>n >0 1

Quasi monomial functions

Let T = {tn | n ∈ N>0 } a countable commutative set. To each word w = ys1 . . . ysl ∈ Y ∗ , we associate the quasi monomial function

Hw (N ) = ).

N X Hw0 (k − 1) k=1

101

ks

Corollary 1. For w = ys w0 , we have ζ(w)

=

X Hw0 (k − 1)

ls

k≥1

Hw (N + 1) − Hw (N )

=

Proof. Proposition 2 gives P as an algebra morphism and by Theorem 2, P is the expected isomorphism. ,

(16)

(N + 1)−s Hw0 (N )

Thanks to the relations existing between Liw (1 − z) and Liw (z) [8], we can precise the asymptotic behaviour of Liw in the neighbourhood of 1. For example,

(17)

1 − Li4 (t) + log(t) Li3 (t) − log(t)2 Li2 (t) 2 1 2 + log(t)3 Li1 (t) + ζ(2)2 . 6 5 So, we find, by formula (14), the expansion of Li2,1,1 (1 − ε) and, by dividing it by ε, we find the one of P2,1,1 (1 − ε) :

([10]). For any words u and v, we have

Lemma 3

Hu

v (N )

Li2,1,1 (1 − t)

= Hu (N ) Hv (N ).

Proof. The harmonic sum Hw (N ) can be obtained by specialization of the quasi-mononial function Mw at ti = 1/i if 1 ≤ i ≤ N and ti = 0 if i > N . By (13), Hw (N ) satisfies the expected result.

3.2

1 1 1 2 ζ(2)2 + log3 ε − log2 ε + log ε 5 ε 6 2 ε ε ε − 1+ log3 ε − log2 ε + log ε + O (ε) 12 8 8 From this, we can also deduce the expansion of the Taylor coefficients of P(z) (see [4]). P2,1,1 (1 − ε)

Generating series

Definition 2 ([6]). For any word w ∈ Y ∗ , let Pw be the ordinary generating series of {Hw (N )}N ≥0 : Pw (z) =

X

Hw (N )z N , with P (z) =

N ≥0

1 . 1−z

3.3

Liw (z) = (1 − z)Pw (z). N

Proposition 3. The map H : u 7→ Hu is an isomorphism from (RhY i, ) to the algebra HR .

Proof. Since Pw (z) = N ≥0 Hw (N )z , it is known that the series expansion of (1 − z)Pw (z) is given by

Since Lyndon(Y ) generate freely the shuffle algebra then

[Hw (N ) − Hw (N − 1)]z N .

Corollary 3. Any harmonic sum in HR can be decomposed, uniquely, as a polynomial on the series Hl , for l ∈ Lyndon(Y ), i.e. HR ' R[Hl ; l ∈ Lyndon(Y )].

N ≥1 0

But, by (17), for w = ys1 w , Hw (N ) − Hw (N − 1) = N −s1 Hw0 (N − 1),

Lemma 4. Any l ∈ Lyndon(Y ) is convergent if and only if l 6= y1 . Any convergent polynomial can be decomposed uniquely as shuffle of convergent Lyndon words.

so (1 − z)Pw (z) = Hw (0) +

X Hw0 (N − 1) N ≥1

N s1

z N = Liw (z).

Proof. By definition, a Lyndon word l is strictly smaller (for lexicographic order) than any of its proper right factors. So, if l = y1 u, with u ∈ Y ∗ , we have y1 u < u which is impossible (remind that y1 is the greatest letter of Y ). Thus, the only Lyndon word beginning by y1 is y1 itself, and our first statement is proved. The second one is based on the remark : if y1 appears as a ) in the Radford decomposition of w, then this factor (for word begins by y1 . Since a convergent polynomial contains convergent terms, which do not begin by y1 , the statement is proved.

Definition 3. The Hadamard product is a bilinear function from C[[z]] × C[[z]] to C[[z]] defined, for all integers n and m,by (

n

z z Thus,

P∞ n=0

an z n

m

=

P∞ n=0

z n if n = m, 0 if n 6= m.

bn z n =

P∞ n=0

an bn z n .

Proposition 4. Every harmonic sum Hw ∈ HR can be decomposed in a unique way in a univariate polynomial in H1 , with coefficients in the convergent harmonic sums. This can also be expressed as follows :

Proposition 2. For u, v ∈ Y ∗ , one has Pu (z) Pv (z) = Pu

v (z).

Proof. By Lemma 3,

X

N ≥0

Hu (N )z N

X

Hv (N )z N

=

N ≥0

X

HR ' HR0 [H1 ], Hu (N )Hv (N )z N

where HR0 is defined as the R-algebra generated by the functions Hw , for all convergent words w ∈ C0 .

N ≥0

=

Algebra HR

From Corollary 2, we deduce then

P

X

=

Definition 4. The algebra HR of harmonic sums is defined as the R-vector space HR = spanR (Hw | w ∈ Y ∗ ), equipped with the ordinary product.

Proposition 1 ([6]). For any word w ∈ Y ∗ and for any complex number z satisfying |z| < 1, one has

(1 − z)Pw (z) = Hw (0) +

=

X

Hu

v (N )z

N

.

Example – The Radford decomposition gives, in Lyndon basis, y1 y4 y2 = y1 y4 y2 − y5 y2 − y4 y1 y2 − y4 y2 y1 − y4 y3 . Thus, H1,4,2 = H1 H4,2 − H5,2 − H4,1,2 − H4,2,1 − H4,3 .  By Proposition 3, we deduce ker H = {0}. In other words,

N ≥0

Corollary 2 ([6]). Extented by linearity, the map ) to the P : u 7→ Pu is an isomorphism from (RhY i, Hadamard algebra of {Pw }w∈Y ∗ .

Proposition 5. The harmonic sums Hw , for w ∈ Y ∗ are R-linearly independent.

102

4.

ASYMPTOTIC EXPANSIONS

So we deduce the asympotic expansion up to order q + 2, which will appear to be very useful afterwards :

We are going to construct a recursive algorithm to find the asymptotic expansion of Hw . For that, considering any real
sequence {sn }n∈N , we will define ASq sn as the asymptotic expansion up to order q of sn , i.e. so that

N X k=2

log(k − 1) kq

sn − ASq (sn ) = O(n−q ).

4.1

with K =

n

Bn

n≥0

z exp(−z) z z = = , n! exp(z) − 1 1 − exp(−z)

x exp(tx) = exp(x) − 1

Z

n

Bn (t)

n≥0

1 qB2 log(N ) log(N ) + − 2N q qN q 2 N q+1

P+∞ k=2

+

m X

N

f (x)dx + M

n=M

x . n!

N −1

1

log(x) dx = (x + 1)q +

1 = (2m + 1)!

B2m+1 (x − [x])f

(2m+1)

(x)dx.

M

q−1 X Bk 1

k Nk



+O

1 Nq



4.2

X

j=1



∞

X 2j − 1

ck (w) = + (q)2j−1

i=1 ∞

2j−2

k=0

where K =

k

P+∞

(2j − 2 − k)!(q)k

k=2 log(k − 1)k

∞ X

−q

X (−y1 )i Di i=0

1 + iN q+2j−1+i i!N 2j+q−1+i

(19)

(20)

i!

Dk (w).

Since Dk w = 0 as soon as k > |w|, this formula can be summered as follows :

X (−1)i (2j − k − 2)i  i=0

y1 k . k!

|w|−k

B2j (−1)2j (2j)! N q+2j−1

ck (w)

Proposition 6. Let w ∈ Y ∗ , a word of length |w|. Then the polynomials ck (w) are given by

X 1 1 log N + − (q − 1)iN i 2N q 2iN q+i i=1

log N

|w| X

In particular, Dw = 0 when w is convergent and D(y1 w) = w for each word w ∈ Y ∗ . We can prove that D is a derivation . for the shuffle product In the following sequence, all products and powers will be . carried out with the shuffle product

X log N 1 − q−1+i (1 − q)N q−1 (1 − q)iN i=1

× −(q)2j−1

log(N − 1) , 2N q ! 2j−2 X 2j − 1 (2j − 2 − k)!(q)k k x2j−k−1 (x + 1)q+k k=0 log(x) . (q)2j−1 (x + 1)q+2j−1

(Dp|w) = (p|y1 w).

∞

+

  q−2 1 X1 1 1 − j q − 1 j=1 j N j 2

We want to calculate the convergent polynomials ck (w) ∈ C0 . For that, let D : RhY i → RhY i be the linear application defined, for each p ∈ RhY i and for each word w ∈ Y ∗ by the duality

log(k − 1) kq

i=q−1 ∞

log(N − 1) log(N − 1) + (1 − q)N q−1 q−1   N 1 log 1−q 2

k=0

Lemma 6. One has, for any integer q ≥ 2,

−



1 N q+2

Taylor algorithm

w=

Proof. With the function f (x) = x−r , the summation (18) between M = 1 and N gives the expected results.

∞ X



+O

By Theorem 1, any w ∈ Y ∗ can be expressed as follows,

1 (r − 1)N r−1 !   q−1 X Bk−r+1 k − 1 1 1 , + O k − r + 1 r − 1 Nk Nq

with r ≥ 2.

=K+

1 N q+1

where (s)k = Γ(s + k)/Γ(s) for k ∈ N. We just need to insert the previous terms in the summation (18), expand log(N − 1), and make m → ∞.

k=r

k=2

=

ζ(r) −

= −

N X

f (2j−1) (x)

−

k=1

Hr (N )

=

N

log N + γ −

=

f (1) + f (N − 1) 2 (18)

([1]). One has

Lemma 5 H1 (N )

Z

+

f (M ) + f (N ) 2

 B2j  (2j−1) f (N ) − f (2j−1) (M ) + R2m (2j)!

j=1

where Rm

Z f (n) =



log(k − 1)k−q .

We need the second form of Euler-Maclaurin summation [11] given by, for all integers q, M , N with N > M , N X

B2 q 2 − 4q − 3 − 2 2q 2 − 2

Proof. Let q > 0 f (x) = log(x)(x + 1)−q . We use the Euler-Maclaurin summation (18) from M = 1 to N − 1, which leads us to calculate each term involved in this sum :

and {Bn (·)}n∈N the Bernoulli polynomials defined by X

+ +

Let {Bn }n∈N be the set of Bernoulli numbers obtained in the expansion of the following series X

K+



Euler Mac-Laurin formula

log(N ) 1 − (1 − q)N q−1 (q − 1)2 N q−1

=

ck (w) = e−y1 D Dk (w)

,

with the convention exp(−y1 D) = making D and y1 commute.

.

103

P

i≥0

(−y1 )i Di /i!, i.e. by

Proof. For a polynomial p ∈ R[X] of degree l, the Taylor expansion is finite, and given by p(x) = p(y) + Dp(y)(x − y) + · · · +

In a second time, if w 6∈ Lyndon(Y ), as before, we use Radford decomposition and the table of the asymptotic expansion for the Lyndon words. Example – Let l = y4 y2 ∈ Lyndon(Y ). By Lemma 2,

Dl p(y) (x − y)l . l!

ck (w) = Dk (w) − D Dk (w)y1 + · · · +

c0

=

i=0

=

Dl k D (w)(−y1 )l . l!

But H2 (i − 1) = ζ(2) − H4,2 (N )

c1

=

i=0

ck

=

(−y1 )i Di (y1 y4 y2 ) = y1 y4 y2 − y1 i! i

ζ(4, 2) − ζ(2)

+

1 2

y4 y 2

H4,2 (N )

(−y1 ) D (y4 y2 ) = y4 y2 i!

∞ X i=N +1

0 for k ≥ 2

Algorithm for asymptotic expansions

) + O(N

).

H1,4,2 (N )

+

(21)

• We assume all expansions for Lyndon words of length lower or equal to L are stored. We then consider a Lyndon word of length L + 1, w = ys u. We know from Lemma 2 that the expansion of Hw is linked to the one of Hu by

i=N +1

ζ(4, 2)

=

ζ(4, 1, 2)

=

ζ(5, 2)

=

ζ(4, 2, 1)

=

ζ(4, 3)

=

H1,4,2 (N )

= +



AEq−s+1 (Hu (i − 1)) . is

+ −




. If u ∈ Lyndon(Y ) then AEq−s+1 Hu (i − 1) is assumed to be stored.

+

. If u 6∈ Lyndon(Y ), with the Radford decomposition, we write u as finite sum of terms t = ··· lr , where c ∈ Q, li ∈ Lyndon, with c l1 AEq−s+1 (Ht (i − 1)) = c

1 3

ζ(2) +

1 ζ(2) + 3 N3 2 5

N5



+O

1 2

ζ(2) +

1 4

N4



1 . N6

log(N )ζ(4, 2) − ζ(4, 1, 2) + γζ(4, 2) ζ(5, 2) − ζ(4, 2, 1) − ζ(4, 3) 1 ζ(4, 2) 1 ζ(4, 2) 1 ζ(2) − + 2 N 12 N 2 9 N3   1 1 1 − 24 ζ(2) − 16 + 120 ζ(4, 2) 1 + O N4 N5

32 ζ(2)3 105 5 5 3 ζ(7) + ζ(2)ζ(5) − ζ(2)2 ζ(3) 8 2 2 4 −11ζ(7) + 5ζ(2)ζ(5) + ζ(2)2 ζ(3) 5 221 11 7 − ζ(7) + ζ(2)ζ(5) + ζ(2)2 ζ(3) 16 2 5 17ζ(7) − 10ζ(2)ζ(5), ζ(3)2 −

So, we deduce the reduced form of the previous expansion

So, there are two possibilities

r Y

 

∞

Thanks to the table giving the relations between MZV up to weight 161 [7],we have the following identities

• If w = ys , then AEq (Hw (N )) = ASq (Hw (N )), an expansion which is already known by Lemma 5, and so can be stored.

AEq (Hw (N )) = ζ(w) − ASq

= − +

Lemma 2 and Lemma 3 give us the following algorithm. We use the notation AEq (Hw (N )) for the asymptotic expansion of Hw (N ) up to order q. In a first time, we store the table of the asymptotic expansions, for w ∈ Lyndon(Y ). For this, we proceed by recurrence on the length of w.

 X ∞

∞

 Example – Let l = y1 y4 y2 ∈ / Lyndon(Y ). The Radford decomposition of l is given by l = y1 y4 y2 −y5 y2 −y4 y1 y2 − y4 y2 y1 − y4 y3 . Using our algorithm, we find :

We now are going to use both previous tools (Euler MacLaurin formula and Taylor algorithm) to get an asymptotic expansion of Hw up to order q, in the scale of functions {N −β logα N, α ∈ N, β ∈ N}. This means that we are looking for a polynomial p ∈ R[X, Y ] verifying Hw (N ) = p(log N, N

X 1 1 + 4 i i5 i=N +1

X 1 1 + O 7 i6 i i=N +1

ζ(4, 2) −

= −

−q

∞ X



1 , so i3

Expanding the sums in N , we finally find

i

−1



i=N +1

So we get y1 y4 y2 = c0 +c1 y1 = −y4 y1 y2 −y4 y2 y1 −y4 y3 − y5 y2 + y4 y2 y1 . Note that, in this case, Taylor algorithm gives directly the Radford decomposition. 

4.3

1 1 1 − +O i 2 i2

=

−y4 y1 y2 − y4 y2 y1 − y4 y3 − y5 y2 2 X

H2 (i − 1) , i4

i=N +1

Example – Let w = y1 y4 y2 . Note that D(w) = y4 y2 and so that Dk (w) = 0, for k ≥ 2. By using Proposition 6, 3 X

∞ X

H4,2 (N ) = ζ(4, 2) −

So, taking x = 0, y = y1 , p = Dk (w), we find

+

32 32 ζ(2)3 ) − γζ(2)3 105 105 7 γζ(3)2 − 3ζ(2)ζ(5) − ζ(2)2 ζ(3) 10 32 115 1 ζ(3)2 − 105 ζ(2)3 ζ(7) + 16 2 N 32 1 ζ(3)2 − 105 ζ(2)3 1 ζ(2) + 12 N2 9 N3 1 1 1 4 − 24 ζ(2) − 16 + 120 ζ(3)2 − 1575 ζ(2)3 log(N )(ζ(3)2 −

 O

1 N5



N4






1

This table is in agreement with the Zagier’s dimension conjecture [15] and is available at http://www.lifl.fr/~petitot/publis/MZV.

AEq−s+1 Hlp (i − 1) .

p=1

104

4.4 More examples − ln(N ) − 1 − γ N 1 ln(N ) + 12 γ + 14 2 N2     1 1 5 1 1 − γ− − ln(N ) + O 6 36 6 N3 N4

H2,1 (N ) =

ζ (3) +

+ +

H3,1 (N ) = + +

H2,1,1 (N )

ζ (2, 1) N 1 1 γ + ζ (2, 1) + 12 ln(N ) + 34 2 2 N2   19 1 1 1 − − ζ (2, 1) − γ − 1/3 ln(N ) 36 6 3 N3 ζ (2, 2, 1) −

H2,2,1 (N ) = + +



+



− 1 ln(N ) − 14 − 12 γ ζ (3, 1) + 2 N2 1 1 1 ln(N ) + 2 γ + 6 2 N3     ln(N ) 7 1 1 1 − ln (N ) + O − γ− 4 48 4 N4 N5

=

−1 −

ζ (2, 1, 1) +

1 2

− ln(N )γ − γ −

+

+

+ + + +

ln(N ) −

1 8

+

=

1 1 ln(N ) +O 12 N4

+

2

ln (N ) N5

− − − + − +



H5,1 (N ) H4,1 (N ) =

ζ (4, 1) + 1 2

− 31 γ −

1 3

ln (N ) − N3

+

γ+

=

1 2

+

1 2

H4,2 (N )

= +

H3,2 (N )

= +

1 2

ζ (2) + 1 ζ (2) + 2 N2 N3   − 14 ζ (2) − 38 1 +O N4 N5

ζ (3, 2) −

1 3

= − − + +

1 8 1 4 1 4 1 2

−

− 14 γ −

1 4

1 ln (N ) − 16 4 N   ln(N ) + 12 γ + 1/10 ln(N ) +O N5 N6

1 + 12 ζ (2) 1 ζ (2) 4 + 3 N3 N4   −2/5 − 1/3 ζ (2) 1 +O N5 N6

ζ (4, 2) −

1 1 1 ζ (2) − ln(N )2 − ln (N ) 6 6 9

=

ζ (4, 1, 1) +

−

1 1 1 1 1 − ln(N )γ − γ − γ 2 27 3 9 6 N3

+

O





1 1 (ln (N ))2 − ln(N ) 4 4 1 1 1 − ln (N ) γ − γ + ζ (2) 2 4 4   1 1 1 2 γ + − ζ (2) + γ N2 4 6 1 1 1 γ 2 − + (ln(N ))2 + ln(N ) 9 4 6   2  ln (N ) 1 ln(N )γ + O N3 N4

ζ (3, 1, 1) +

1 3 1 γ − ζ (3) 6 3 1 1 ζ (2) γ + ζ (2) ln(N ) + ζ (2) 2 2 1 ln(N )3 − (ln (N ))2 − 1 − ln(N ) 2 1 2 (ln(N )) γ − ln(N )γ − γ − ln(N )γ 2 2   1 1 1 1 γ2 + − ζ (2) γ + γ3 + γ2 N 4 12 8 1 1 1 ζ (2) + ln(N )3 − ln (N ) − 12 8 16 1 1 2 ln(N ) γ + ln(N )γ − γ 4 8 1 1 ζ (2) ln(N ) + ln(N )γ 2 + ln(N )2 4 8   3  ln (N ) 1 ζ (3) +O N2 N3



H4,1,1 (N )



H3,1,1 (N )



ζ (5, 1) +

1 9

ln (N ) + 18 N4     3 1 1 1 1 − − γ − ln(N ) + O 20 3 3 N5 N6

+

1 2 1 6 1 2 1 2 1 8 1 4 1 4 1 6

−

γ 2 + 12 ζ (2) N ln2 (N ) − ln (N )

1 2



ln(N ) N5

ζ (2, 1, 1, 1) +

+

1 4



O

1 N4



H2,1,1,1 (N )

N ln2 (N ) + 21 ln(N )γ N2  − 14 ζ(2) + 14 γ + 14 γ 2 5 ln(N ) + − N2 36 29 5 1 1 − γ − ln(N )γ − ln2 (N ) 216 36 6 12  1 1 2 1 1 1 ζ(2) − γ γ− + 12 12 N3 12 96 1 4



1 61 1 γ+ + ln(N ) 24 288 24

−

H3,2,1 (N )

ln2 (N ) N4



1 ζ (2, 1) + 2 N2

1 4 γ+ 3 9

ζ (3, 2, 1) −

+

1 1 1 + ζ (2, 1) + ln(N ) 2 3 N3

−

1 3 1 ζ (2, 1) − ln(N ) +O 4 8 N4

 

105



=



− 

17 3 − γ 32 8 ln(N ) N5





H3,1,2 (N ) =

ζ (3, 1, 2) +

−

1 1 ζ (2) γ + ζ (3) 2 2

1 1 1 ζ (2) ln (N ) − ζ (2) + ζ (2, 1) 2 4 2

−



+





1 +O N3





ln(N ) N4

ζ (3, 1, 1, 1) +

− − − −



+

−

H4,3 (N ) =

=

H4,2,1 (N )



= − + − + + − + + −

+ +

1 28

ζ (2, 1, 1, 1, 1) +

−1−γ−

1 ζ (2, 1) + 3 N3

=

ζ (4, 2, 1) −

+

1 1 5 + ln(N ) + 16 4 N4

−

2/5 ln(N ) −

1 ζ (2, 1) 3 

H4,1,2 (N )

1 2 γ 2

1 4 1 1 4 ln (N ) − ln2 (N ) − γ 24 2 24 1 1 1 ζ (4) − ζ (3) − ln2 (N )γ 8 3 2 1 1 1 ln(N )γ 2 − ζ (3) ln (N ) − ζ (3) γ 2 3 3 1 1 1 1 ζ (2) + ζ (2) γ − γ 3 − ln3 (N ) 2 2 6 6 1 1 2 3 ζ (2) ln (N ) − ln (N ) γ 4 6 1 1 2 2 ln(N ) γ − ln(N )3 γ 4 6 1 1 ζ (2) ln(N )γ + ζ (2) γ 2 2 4 1 1 ζ (2) ln (N ) − ln(N ) − ζ (2, 2) 2 4   4  ln (N ) 1 ln(N )γ +O N N2

−







1 1 γ + ζ (2, 1) 4 2

− 2/5γ − 1 +O N5

ζ (4, 1, 2) +

+

1 1 1 ζ (3) − ζ (2) γ + ζ (2, 1) 3 3 3

+

O

ln(N ) N4



53 100 ln(N ) N6



1 1 ζ (2) ln(N ) − ζ (2) 3 9

=







1 N3



H4,1,1,1 (N ) = + − − −

5.

1 3 1 ζ (3) − γ 9 18 1 1 1 1 ζ (2) γ − ln2 (N )γ − ln(N )γ − γ 6 6 9 27 1 1 1 3 ln N − (ln (N ))2 − ln (N ) 18 18 27 1 1 1 1 + ζ (2) ln(N ) + ζ (2) − ln(N )γ 2 81 6 18 6   3  ln (N ) 1 2 1 γ +O 18 N3 N3

ζ (4, 1, 1, 1) +

−

ACKNOWLEDGMENTS

Thanks to Boutet de Monvel, Cartier, Jacob and Waldschmidt for useful discussions.

6. =

N6

+



ζ (2, 1, 1) 1 + ζ (2, 1, 1) N 2 1 2 3 3 7 1 γ + γ + ln(N ) + + (ln (N ))2 4 4 4 8 4    2 ln (N ) 1 1 1 ln(N )γ − ζ (2) + O 2 4 N2 N3

+

H6,1 (N )

1 6

− 13 ζ (3) + 1/10 1 ζ (3) 1 ζ (3) + + 3 N3 2 N4 N5   + 16 ζ (3) 1 +O N7 N8

ζ (2, 2, 1, 1) −

+

H2,1,1,1,1 (N )

ζ (4, 3) −

−

−

ζ (5, 1, 1) +

−



H2,2,1,1 (N )

1 1 ln(N )γ − γ 4 16 1 2 1 1 1 ln (N ) − ln(N ) − − γ2 8 16 64 8   2  ln (N ) 1 1 ζ (2) +O 8 N4 N5

H5,1,1 (N ) =

1 ln(N )2 γ 4 1 1 1 1 ln(N )γ − γ − ln(N )γ 2 − γ 2 4 8 4 8 1 1 1 (ln(N ))3 − ln(N )2 − ln (N ) 12 8 8 1 1 1 1 + ζ (2) ln (N ) + ζ (2) + ζ (2) γ 16 4 8 4   3  ln (N ) 1 1 3 1 γ − ζ (3) + O 12 6 N2 N3

=

5 − 12

+



H3,1,1,1 (N )

1 ζ (2) + 5 1 ζ (2) 2 + 4 N4 N5   5 23 − 12 ζ (2) 1 84 + + O N6 N7 N8

ζ (5, 2) −

=

1 N2

1 1 1 1 ζ (2) + ζ (2) γ − ζ (2, 1) − 6 2 2 3

1 1 ζ (3) + ζ (2) ln (N ) 2 2

−

1

H5,2 (N )

REFERENCES

[1] F. Bergeron, G. Labelle, P. Leroux.– Combinatorial Species and Tree-like Structures, Encyclopedia of Mathematics and its Applications, Vol. 67, Cambridge University Press (1998). [2] M. Bigotte.– Etude symbolique et algorithmique des polylogarithmes et des nombres Euler-Zagier colors, Ph.D., Lille, (2000).

1 − 51 ln(N ) − 25 − 1/5γ N5 1 1 ln(N ) + 12 + 12 γ 2 N6     ln(N ) 1 13 1 1 − γ− − ln (N ) + O 2 84 2 N7 N8

ζ (6, 1) +

106

[3] J. Bl¨ umlein.– Mathematical Structure of Anomalous Dimensions and QCD Wilson Coefficients in Higher Order, Nuclear Physics B (Proc Suppl.), 135, pp 225-231, (2004). [4] C. Costermans, J.Y. Enjalbert, Hoang Ngoc Minh.– Algorithmic and combinatoric aspects of multiple harmonic sums, in the proceedings of AofA, Barcelone, 6-10 June, (2005). [5] P. Flajolet, G. Labelle, L. Laforest, B. Salvy.– Hypergeometrics and the Cost Structure of Quadtrees, Random Structures and Algorithms, Vol. 7, No.2, pp 117-144, (1995). [6] Hoang Ngoc Minh.– Finite polyzetas, Poly-Bernoulli numbers, identities of polyzetas and noncommutative rational power series, proc. of 4th Int. Conf. on Words, pp. 232-250, September, 10-13 Turku, Finland, (2003). [7] Hoang Ngoc Minh & M. Petitot.– Lyndon words, polylogarithmic functions and the Riemann ζ function, Discrete Math., 217, pp. 273-292, (2000). [8] Hoang Ngoc Minh, M. Petitot & J. van der Hoeven.– L’algbre des polylogarithmes par les sries gnratrices, SFCA’99, Barcelone, (1999).

[9] Hoang Ngoc Minh, M. Petitot & J. van der Hoeven.– Shuffle algebra and polylogarithms, Discrete Mathematics, 225, pp 217-230, (2000). [10] M. Hoffman.– The algebra of multiple harmonic series, Jour. of Alg., August (1997). [11] A. Ivi´c.– The Riemann zeta function, J. Wiley, New York, (1985). [12] C. Reutenauer.– Free Lie Algebras, Lon. Math. Soc. Mono., New Series-7, Oxford Sc. Pub., (1993). [13] V.N. Sorokin.– On the linear independence of the values of generalized polylogarithms, Math. Sb., 192:8, pp. 139–154, (2001); English transl, Sb. Math. 192, pp 1225–1239, (2001). [14] E.A. Ulanskii.– Identities for generalized polylogarithms, Mat. Zametki, 73, pp. 613–624; English transl, Math. Notes, 73, pp 571–581, (2003). [15] D. Zagier.– Values of zeta functions and their applications, First European congress of Mathematics, Vol.2, Birkhuser, Basel, 1994, pp. 497-512.

107

Lifting Techniques for Triangular Decompositions Xavier Dahan

´ LIX, Ecole polytechnique 91128 Palaiseau, France

Marc Moreno Maza

ORCCA, University of Western Ontario (UWO) London, Ontario, Canada [email protected]

´ Eric Schost

´ LIX, Ecole polytechnique 91128 Palaiseau, France

[email protected] [email protected] Yuzhen Xie Wenyuan Wu ORCCA, UWO

ORCCA, UWO

[email protected]

[email protected]

ABSTRACT

Let us introduce the notation used below. If k is a perfect field (e.g., Q or a finite field), a triangular set is a family T1 (X1 ), T2 (X1 , X2 ), . . . , Tn (X1 , . . . , Xn ) in k[X1 , . . . , Xn ] which forms a reduced Gr¨ obner basis for the lexicographic order Xn > · · · > X1 and generates a radical ideal (so Ti is monic in Xi ). The notation T 1 , . . . , T s denotes a family of s triangular sets, with T i = T1i , . . . , Tni . Then, any 0dimensional variety V can be represented by such a family,
such that I(V ) = ∩i≤s T i  holds, and where T i  and T i   are coprime ideals for i = i ; we call it a triangular decomposition of V . This decomposition is not unique: the different possibilities are obtained by suitably recombining the triangular sets describing the irreducible components of V . In this paper, we consider 0-dimensional varieties defined over Q . Let thus F = F1 , . . . , Fn be a polynomial system in Z[X1, . . . , Xn ]. Since we have in mind to apply Hensel lifting techniques, we will only consider the simple roots of F , that is, those where the Jacobian determinant J of F does not vanish. We write Z(F ) for this set of points; by the Jacobian criterion [10, Ch. 16], Z(F ) is finite, even though the whole zero-set of F , written V (F ), may have higher dimension. Let us assume that we have at hand an oracle that, for any prime p, outputs a triangular decomposition of Z(F mod p). Then for a prime p, a rough sketch of an Hensel lifting algorithm could be: (1) Compute a triangular decomposition t1 , . . . , ts of Z(F mod p), and (2) Lift these triangular sets over Q. However, without more precautions, this algorithm may fail to produce a correct answer. Indeed, extra factorizations or recombinations can occur modulo p. Thus, we have no guarantee that there exist triangular sets T 1 , . . . , T s defined over Q , that describe Z(F ), and with t1 , . . . , ts as modular images. Furthermore, if we assume no control over the modular resolution process, there is little hope of obtaining a quantification of primes p of “bad” reduction. Consider for instance the variety V ⊂ C 2 defined by the polynomials 326X1 −10X26 +51X25 +17X24 +306X22 +102X2 + 34 and X27 +6X24 +2X23 +12. For the order X2 > X1 , the only possible description of V by triangular sets with rational coefficients corresponds to its irreducible decomposition, that is, T 1 : ( X1 −1, X23 +6 ) and T 2 : ( X12 +2, X22 +X1 ). Now, the following triangular sets describe the zeros of (F mod 7), which are not the reduction modulo 7 of T 1 and T 2 ;

We present lifting techniques for triangular decompositions of zero-dimensional varieties, that extend the range of the previous methods. We discuss complexity aspects, and report on a preliminary implementation. Our theoretical results are comforted by these experiments. Categories and Subject Descriptors: I.I.2 [Computing Methodologies]: Symbolic and Algebraic Manipulation – Algebraic Algorithms General Terms: Algorithms, experimentation, theory. Keywords: Polynomial systems, triangular sets, Hensel lifting.

1.

INTRODUCTION

Modular methods for computing polynomial GCDs and solving linear algebra problems have been well-developed for several decades, see [12] and the references therein. Without these methods, the range of problems accessible to symbolic computations would be dramatically limited. Such methods, in particular Hensel lifting, also apply to solving polynomial systems. Standard applications are the resolution of systems over Q after specialization at a prime, and over the rational function field k(Y1 , . . . , Ym ) after specialization at a point (y1 , . . . , ym ). These methods have already been put to use for Gr¨ obner bases [26, 1] and primitive element representations, starting from [13], and refined notably in [14]. Triangular decompositions are well-suited to many practical problems: see some examples in [3, 11, 24]. In addition, these techniques are commonly used in differential algebra [4, 15]. Triangular decompositions of polynomial systems can be obtained by various algorithms [16, 18, 21] but none of them uses modular computations, restricting their practical efficiency. Our goal in this paper is to discuss such techniques, extending the preliminary results of [24].

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. ISSAC’05, July 24–27, 2005, Beijing, China. Copyright 2005 ACM 1-59593-095-7/05/0007 ...$5.00.

1 t

108

X22 + 6X2 X12 + 2X2 + X1 X13 + 6X12 + 5X1 + 2

and

2 t

X2 + 6 , X1 + 6

A lifting algorithm should discard t1 and t2 , and replace 1 2 them by the better choice t : ( X1 + 6, X23 + 6 ) and t : 2 2 1 ( X1 + 2, X2 + X1 ), which are the reduction of T and T 2 modulo 7. In [24], this difficulty was bypassed by restricting to equiprojectable varieties, i.e. varieties defined by a single triangular set, where no such ambiguity occurs. However, as this example shows, this assumption discards simple cases. Our main concern is to lift this limitation, thus extending these techniques to handle triangular decompositions. Our answer consists in using a canonical decomposition of a 0-dimensional variety V , its equiprojectable decomposition, described as follows. Consider the map π : V ⊂ A n (k) → A n−1 (k) that forgets the last coordinate. To x in V , we associate N (x) = #π −1 (π(x)), that is, the number of points lying in the same π-fiber as x. Then, we split V into the disjoint union V1 ∪ · · · ∪ Vd , where for all i = 1, . . . , d, Vi equals N −1 (i), i.e., the set of points x ∈ V where N (x) = i. This splitting process is applied recursively to all V1 , . . . , Vd , taking into account the fibers of the successive projections A n (k) → A i (k), for i = n − 1, . . . , 1. In the end, we obtain a family of pairwise disjoint, equiprojectable varieties, whose reunion equals V , which form the equiprojectable decomposition of V . As requested, each of them is representable by a triangular set with coefficients in the definition field of V . Looking back at the example, both Z(F ) and Z(F mod 7) are described on the leftmost picture below (forgetting the actual coordinates of the points). Representing Z(F ) by 1 2 T 1 and T 2 , as well as Z(F mod 7) by t and t amounts to grouping the points as on the central picture; this is the equiprojectable decomposition. The rightmost picture shows the description of Z(F mod 7) by t1 and t2 . 11 00 11 00 1 0 0 1 0 1 1 0 0 1 0 1

11 00 11 00 1 0 0 1 0 1

1 0 0 1 0 1

00 11 00 11 00 11 11 00 0 11 1 00 1 0 0 1

1 0 0 1 0 1

1 0 0 1 0 1

00 11 00 11 00 11 11 00 0 11 1 00 1 0 0 1

height of a 0-dimensional variety V defined over Q: the former denotes its number of points, and the later estimates its arithmetic complexity; see [17] and references therein for its definition. Let then T 1 , . . . , T s be the triangular sets that describe the equiprojectable decomposition of Z = Z(F ). In [9], it is proved that all coefficients in T 1 , . . . , T s have height in O(nO(1) (deg Z + ht Z)2 ). However, better estimates are available, through the introduction of an alternative representation denoted by N 1 , . . . , N s . For i ≤ s, N i = N1i , . . . , Nni is obtained as follows. Let D1i = 1 and N1i = T1i . For 2 ≤  ≤ n and 1 ≤ i ≤ s, define Di =

1 0 0 1 0 1

i Ni = Di Ti mod (T1i , . . . , T−1 ).

• For any C ∈ N, let Γ(C) be the sets of primes in [C + 1, . . . , 2C]. We assume the existence of an oracle O1 which, for any C ∈ N, outputs a random prime in Γ(C), with the uniform distribution. • We assume the existence of an oracle O2 , which, given a system F and a prime p, outputs the representation of the equiprojectable decomposition of Z(F mod p) by means of triangular sets. We give in Section 2 an algorithm to convert any triangular decomposition of Z(F mod p) to the equiprojectable one; its complexity analysis is subject of current research.

1 0 0 1 0 1

00 11 00 11 00 11 11 00 0 11 1 00 1 0 0 1

a a h

h

• For F as in Theorem 1, we write F = (n, d, h), F = ndn (h+11 log(n+3)) and F = 5( F +1) log(2 F +1). The input system is given by a straight-line program of size L, with constants of height at most hL .

b

h

• C ∈ N is such that for any ring R, any d ≥ 1 and monic t ∈ R[X] of degree d, all operations (+, −, ×) in R[X]/t can be computed in Cd log d log log d operations in R [12, Ch. 8,9]. Then all operations (+, −, ×) modulo a triangular set T in n variables can be done in quasi-linear complexity in Cn and deg V (T ).

Theorem 1. Let F1 , . . . , Fn have degree ≤ d and height ≤ h. Let T 1 , . . . , T s be the triangular description of the equiprojectable decomposition of Z(F ). There exists A ∈ N − {0}, with ht A ≤ (n, d, h), and, for n ≥ 2,

a(n, d, h) = 2n d

and

It is proved in [9] that all coefficients in N 1 , . . . , N s have height in O(nO(1) (deg Z +ht Z)). Since T 1 , . . . , T s are easily recovered from N 1 , . . . , N s , our algorithm will compute the latter, their height bounds being the better. Theorem 2 below states our main result regarding lifting techniques for triangular decompositions; in what follows, we say that an algorithm has a quasi-linear complexity in terms of some parameters if its complexity is linear in all of these parameters, up to polylogarithmic factors. We need the following assumptions:

The above algorithm sketch is thus improved by applying lifting only after computing the equiprojectable decomposition of the modular output. Theorem 1 shows how to control the primes of bad reductions for the equiprojectable decomposition, thus overcoming the limitation that we pointed out previously. In what follows, the height of x ∈ Z is defined as ht x = log |x|; the height of f ∈ Z[X1, . . . , Xn ] is the maximum of the heights of its coefficients; that of p/q ∈ Q, with gcd(p, q) = 1, is max(ht p, ht q).

a

2 2n+1

∂Tji ∂Xj

1≤j≤−1

11 00 11 00 1 0 0 1 0 1

Y

Theorem 2. Let ε > 0. There exists an algorithm which, given F , satisfying

(3h + 7 log(n + 1) + 5n log d + 10),

a

4

and with the following property. If a prime p does not divide A, then p cancels none of the denominators of the coefficients of T 1 , . . . , T s , and these triangular sets reduced mod p define the equiprojectable decomposition of Z(F mod p).

F

b

+2 ε

F

+1<

h

1 exp (2 2

F

+ 1),

computes N 1 , . . . , N s defined above. The algorithm uses two calls to O1 with C = 4 F + 2 F /ε, two calls to O2 with p in [C + 1, . . . , 2C], and its bit complexity is quasi-linear in L, hL , d, log h, Cn , deg Z, (deg Z + ht Z), | log ε|. The algorithm is probabilistic, with success probability ≥ 1 − ε.

a

Thus, the set of bad primes is finite and we have an explicit control on its size. Since we have to avoid some “discriminant locus”, it is natural, and probably unavoidable, that the bound should involve the square of the B´ezout number. A second question is the coefficient size of the output. In what follows, we write deg V and ht V for the degree and

b

To illustrate these estimates, suppose e.g. that we have n = 10, d = 4, h = 100, hence potentially 1048576 solutions; to ensure a success probability of 99%, the primes should

109

B(i, di , B(di+1 , . . . , dn , V )). All B(di , . . . , dn , V ) are (i − 1)equiprojectable, only finitely many of them are not empty, and the non-empty ones form a partition of V . The equiprojectable decomposition of V is its partition into the family of all non-empty B(d2 , . . . , dn , V ). All these sets being equiprojectable, they are defined by triangular sets. Note that we have not proved yet that the B(d2 , . . . , dn , V ) are defined over the same field as V . This will come as a by-product of the algorithms of this section. To do so, we introduce now the notions of critical pair and solvable pair. Critical and solvable pairs. Let T = T  be two triangular sets. The least integer  such that T = T is called the level of the pair T, T  . If  = 1 we let K = k, otherwise we define K = k[X1 , . . . , X−1 ]/T1 , . . . , T−1 . Since a triangular set generates a radical ideal, the residue class ring K is a direct product of fields. Therefore, every pair of univariate polynomials with coefficients in K has a GCD in the sense of [22]. The pair T, T  is critical if T and T are not relatively prime in K [X ]. If T, T  is not critical, it is certified if U, U  ∈ K [X ] such that U T + U  T = 1 are known. The pair T, T  is solvable if it is not critical and if for all  < j ≤ n we have degXj Tj = degXj Tj . Introducing the notion of a certified solvable pair is motivated by efficiency considerations. Indeed, during the splitting step, solvable pairs are discovered. Then, during the merging step, the B´ezout coefficients U, U  of these solvable pairs will be needed for Chinese Remaindering. Solvable families. We extend the notion of solvability from a pair to a family of triangular sets. A family of triangular sets is solvable (resp. certified solvable) at level  if every pair {T, T  } of elements of is solvable (resp. certified solvable) of level . The following proposition shows how to recombine such families. When this is the case, we say that all T in divide S. In what follows, we write V ( ) for ∪T ∈T V (T ).

have only about 20 decimal digits, hence can be generated without difficulty. Thus, even for such “large” systems, our results are quite manageable. Besides, computing the polynomials N i instead of T i enables us to benefit from their improved height bounds. In the sequel, we use the following notation. For n ∈ N, for 1 ≤ j ≤ i ≤ n and any field k, we denote πji : A i (k) → A j (k) the map (x1 , . . . , xi ) → (x1 , . . . , xj ). The cardinality of a finite set G is written #G.

2.

SPLIT-AND-MERGE ALGORITHM

We start by reviewing the notion of equiprojectable decomposition of a 0-dimensional variety V , introduced in [8]. Then, in preparation for the modular algorithm of Section 4, we present an algorithm for computing this decomposition, given an arbitrary triangular decomposition of V . We call it Split-and-Merge, after its two phases: the splitting of what we call critical pairs (which is achieved by GCD computations) and the merging of what we call solvable families (which is performed by Chinese remaindering). The complexity analysis of the Split-and-Merge algorithm is work in progress [6]. From our preliminary study reported in [7], we believe that suitable improvements of the Split-and-Merge algorithm can run in quasi-linear time in the degree of V . Let k be a perfect field and k one of its algebraic closures. Following [2], we first define the notion of equiprojectability. Equiprojectable variety. Let V ⊂ A n (k) be a 0-dimensional variety over k. For 1 ≤ i ≤ n, the variety V is equiprojectable on πin (V ) if all fibers of the restriction πin : V → πin (V ) have the same cardinality. Then, for 1 ≤ i ≤ n, V is i-equiprojectable if it is equiprojectable on all πjn (V ), i ≤ j ≤ n. Thus, any 0-dimensional variety is n-equiprojectable. Finally, V is equiprojectable if it is 1-equiprojectable. It is the case if and only if its defining ideal is generated by a triangular set T1 , . . . , Tn with coefficients in k. In this case, k n being perfect, all fibers of the projection πin (V ) → πi−1 (V ) share the same cardinality, which is the degree of Ti in Xi . The variety V can be decomposed as the disjoint union of equiprojectable ones, in possibly several ways. Any such decomposition amounts to represent V as the disjoint union of the zeros of some triangular sets. The equiprojectable decomposition is a canonical way of doing so, defined by combinatorial means.

T

T

T

T

Proposition 1. If is certified solvable at level , one can compute a triangular set S such that V (S) = V ( ), using only multiplications in K [X ].

T

T

Proof. First, we assume that consists of the pair {T, T  }. We construct S as follows. We set Si = Ti for 1 ≤ i <  and S = T T . Let  < i ≤ n. For computing Si , we see Ti and Ti in K [X ][X+1 , . . . , Xi ]. We apply Chinese remaindering to the coefficients in Ti and Ti of each monomial in X+1 , . . . , Xi occurring in Ti or Ti : since the B´ezout coefficients U, U  for T , T are known, this can be done using multiplications in K [X ] only. It follows from the Chinese Remaindering Theorem that the ideal S is equal to T  ∩ T  ; for i > , the equality degXi Ti = degXi Ti shows that S is monic in Xi , as requested. Assume that consists of s > 2 triangular sets T 1 , . . . , T s . First, we apply the case s = 2 to T 1 , T 2 , obtaining a triangular set T 1,2 . Observe that every pair T 1,2 , T j , for 3 ≤ j ≤ s, is solvable but not certified solvable: we obtain the requested B´ezout coefficient by updating the known ones. Let us fix 3 ≤ j ≤ s. Given A1 , A2 , B1 , Bj , C2 , Cj ∈ K [X ] such that A1 T1 + A2 T2 = B1 T1 + Bj Tj = C2 T2 + Cj Tj = 1 hold in K [X ], we let α = B1 C2 mod Tj and β = A1 Cj T1 + A2 Bj T2 mod T1 T2 . Then, αT1,2 + βTj = 1 in K [X ], as requested. Proceeding by induction ends the proof. Splitting critical pairs. Let now V be a 0-dimensional variety over k. Proposition 3 below encapsulates the first

Equiprojectable decomposition. Let first W be a 0dimensional variety in A i (k), for some 1 ≤ i ≤ n. For x in A i−1 (k), we define the preimage i µ(x, W ) = (πi−1 )−1 (x) ∩ W ;

for any d ≥ 1, we can then define n

T

T

o

i (x), W ) = d . A(d, W ) = x ∈ W | #µ(πi−1

Thus, x is in A(d, W ) if W contains exactly d points x such i i (x) = πi−1 (x ) holds. Only finitely many of the that πi−1 A(d, W ) are not empty and the non-empty ones form a partition of W . Let 1 ≤ i ≤ n. Writing W = πin (V ), we define B(i, d, V ) = {x ∈ V | πin (x) ∈ A(d, W )} . Thus, B(i, d, V ) is the preimage of A(d, W ) in V , so these sets form a partition of V . If V is i-equiprojectable, then all B(i, d, V ) are (i − 1)-equiprojectable. We then define inductively B(V ) = V , and, for 1 < i ≤ n, B(di , . . . , dn , V ) =



110

same set of points, and all of which satisfy Pκ−1 . First, we partition κ using the equivalence relation T ≡ T  if and  . Assumption Pκ shows only if T1 , . . . , Tκ−1 = T1 , . . . , Tκ−1 that each equivalence class is certified and solvable of level κ. We then let (κ) be the family of triangular sets obtained by applying Proposition 1 to each equivalence class.

part of the Split-and-Merge algorithm: given any triangular decomposition of V , it outputs another one, without critical pairs. We first describe the basic splitting step.

T

T

T

S

be a triangular decomposition of Proposition 2. Let V which contains critical pairs. Then one can compute a triangular decomposition Split( ) of V which has cardinality larger than that of .

T

T

S

Lemma 1. Let S =  S  in (κ) . The pair {S, S  } is noncritical, certified, of level  < κ.

T



of level  and let Proof. Let T, T be a critical pair of G be a GCD of T , T in K [X ]. First, assume that G is monic, in the sense of [22]; let Q and Q be the quotients of T and T by G in K [X ]. We define the sets A B A B

= = = =

T

T

Proof. Let T, T  ∈ , which respectively divide S and S  . Due to assumption Pκ , there exists 0 ≤  ≤ κ such that  T1 , . . . , T−1 = T1 , . . . , T−1 and (T1 , . . . , T ) and (T1 , . . . , T ) have no common zero. Then,  < κ, since T ≡ T  . Thus, T1 , . . . , T = S1 , . . . , S and T1 , . . . , T = S1 , . . . , S . Since {T, T  } is certified of level  < κ, {S, S  } is also.

T1 , . . . , T−1 , G, T+1 , . . . , Tn , T1 , . . . , T−1 , Q, T+1 , . . . , Tn ,  T1 , . . . , T−1 , G, T+1 , . . . , Tn ,   T1 , . . . , T−1 , Q , T+1 , . . . , Tn .

S

We partition some more, into the classes of the equivalence relation S ≡ S  if and only if degXκ Sκ = degXκ Sκ . (κ) (κ) Let 1 , . . . , δ be the equivalence classes, indexed by the common degree in Xκ ; we define Mergeκ ( κ ) as the data of all these equivalence classes.

S

We let Split( ) = {A, B, A , B  }, excluding the triangular sets defining the empty set. Since the pair T, T  is critical, V (A) and V (B) are non-empty. Since T and T are not associate in K [X ], at least Q or Q is not constant. Thus, Split( ) has cardinality at least 3. Since T  and T   are radical, if Q ∈ K , G and Q are coprime in K [X ], so V (T ) is the disjoint union of V (A) and V (B). The same property holds for A and B  . Thus, the proposition is proved. Assume now that T , T have no monic GCD in K [X ]. Then, there exist triangular sets C 1 , . . . , C s , D1 . . . Ds such that V (T ) is the disjoint union of V (C 1 ), . . . , V (C s ), V (T  ) is the disjoint union of V (D1 ), . . . , V (Ds ), at least one pair C i , Dj is critical and Ci , Dj admits a monic GCD in K [X ]. These triangular sets are obtained by the algorithms of [22] when computing a GCD of T , T in K [X ]. Then the results of the monic case prove the existence of Split( ).

T

T

T

T

T

T

T

T

Proposition 4. V (

T

T

T

(κ) d )

= B(κ, d, V (

T S



T S

T

κ ))

for all d.

S

(κ)

T

S

S

S

S



The main merging algorithm. We can now give the main algorithm. We start from a triangular decomposition of V without critical pairs, and where every pair is certified, so it satisfies Pn . Let us initially define n = { }; note that n is a set of families of triangular sets. Then, for 1 ≤ κ ≤ n, assuming κ is defined, we write κ−1 = ∪U(κ) ∈Tκ Mergeκ ( (κ) ). Lemma 2 shows that this process is well-defined; note that each κ is a set of families of triangular sets as well. Let be a family of triangular sets in 0 . Then satisfies P0 , so by the remarks make previously, consists in a single triangular set. Proposition 4 then shows that the triangular sets in 0 form the equiprojectable components of V .

T

T

T

S

Proof. We know that V ( κ ) is the union of the V ( d ). (κ) Besides, both families {V ( d )} and {B(κ, d, V ( ))} form a partition of V ( κ ). Thus, it suffices to prove that for x (κ) in V ( κ ), x ∈ V ( d ) implies that πκn (x) ∈ A(d, W ), with n W = πκ (V ( κ )). First, for S in (κ) , write WS = πκn (S). Then Lemma 1 shows that the WS form a partition of W , κ and that their images πκ−1 (WS ) are pairwise disjoint. (κ) Let now x ∈ V ( d ) and y = πκn (x). There exists a unique S ∈ (κ) such that x ∈ V (S). The definition of (κ) shows that there are exactly d points y  in WS such d κ κ that πκ−1 (y) = πκ−1 (y  ). On the other hand, for any y ∈  κ (y) = WS
, with S = S, the above remark shows that πκ−1 κ (y  ). Thus, there are exactly d points y  in W such that πκ−1 κ κ πκ−1 (y) = πκ−1 (y  ); this concludes the proof.

T

T

satisfies Pκ−1 .



Thus, we can now suppose that we have a triangular decomposition of V without critical pairs, and where every pair is certified, such as the one computed in Proposition 3. We describe the second part of the Split-and-Merge algorithm: merging solvable families in a suitable order, to obtain the equiprojectable decomposition of V . For 0 ≤ κ ≤ n, we say that satisfies property Pκ if for all T, T  ∈ the pair {T, T } is certified, has level  ≤ κ and for all κ < i ≤ n satisfies degXi Ti = degXi Ti . Observe that if P0 ( ) holds, then contains only one triangular set, and that the input family satisfies Pn .

T

(κ) d

(κ)



T

S

S

Proof. Write 0 = , and define a sequence i by i+1 = Split( i ), if i contains critical pairs, and i+1 = i otherwise. Testing the presence of critical pairs is done by GCD computations, which yields the B´ezout coefficients in case of coprimality. Let D be the number of irreducible components of V . Any family i has cardinality at most D, so the sequence i becomes stationary after at most D steps.

T

T

Proof. Let S = S  in d , and let T, T  be as in the proof of Lemma 1; we now prove the degree estimate. For κ < i ≤ n, we have degXi Ti = degXi Si and degXi Ti = degXi Si ; assumption Pκ shows that degXi Si = degXi Si for κ < i ≤ n. Since degXκ Sκ = degXκ Sκ = d, the lemma is proved.

T

T

S

Lemma 2. Each family

Proposition 3. Let be a triangular decomposition of V . One can compute a triangular decomposition  of V with no critical pairs, and where each pair of triangular sets is certified.

T



(κ)

T

T

U

T

U

The basic merging algorithm. Let 1 ≤ κ ≤ n. We now define the procedure Mergeκ , which takes as input a family κ of triangular sets which satisfies Pκ , and outputs several families of triangular sets, whose reunion defines the

T

T

111

T

T

T

T

T U

U

3.

Proof. We prove on  = n + 1, . . . , 2 that for all d , . . . , dn , B(d , . . . , dn , Z) equals B(d , . . . , dn , Z); taking  = 2 gives the lemma. Since B(X) = X for any variety X, this property holds for  = n + 1. Assuming it for B(d+1 , . . . , dn , Z), we prove it for B(d , . . . , dn , Z). Let B = B(d+1 , . . . , dn , Z), n (B); Lemma 4 implies that reB = πn (B) and B−1 = π−1 duction modulo p is one-to-one on both B and B−1 . For y in B−1 and z in B−1 , we define

PROOF OF THEOREM 1

In this section, we consider the simple solutions Z(F ) of a system F = F1 , . . . , Fn in Z[X1, . . . , Xn ], that is, those where the Jacobian determinant J of F does not vanish. We prove that for all primes p but a finite number, the equiprojectable decomposition of Z(F ) reduces modulo p to that of Z(F mod p). These results require to control the cardinality of the “specialization” of a variety at p. Such questions are easy to formulate using primitive elements and associated representations, which we now define as a preamble. Primitive element descriptions. Let W ⊂ C  be a 0dimensional variety defined over Q. Let ∆ be a linear form in Z[X1, . . . , X ]. Its minimal polynomial is the minimal polynomial µ ∈ Q[T ] of the multiplication endomorphism by ∆ in Q[W ]; it is the squarefree part of Πx∈W (T − ∆(x)). Then ∆ is a primitive element for W if the map x → ∆(x) is one-to-one on W . In this case, µ has degree deg W and Q[W ] is isomorphic to the residue class ring Q[T ]/µ. Writing wi ∈ Q[T ] for the image of Xi , we deduce that µ(T ) = 0 and Xi = wi (T ), 1 ≤ i ≤ , form a parametrization of the points in W . We will use quantitative estimates on the size of the coefficients in this representation, in terms of the degree and height of W . The following result is [5, Th. 2]; using the coefficient χ leads to sharp height bound, as is the case for the polynomials N i defined in the introduction.

 µ(y) = (π−1 )−1 (y) ∩ B

and

 µ(z) = (π−1 )−1 (z) ∩ B .

We first prove that µ(y) and µ(y) have the same cardinality for all y in B−1 . To this effect, observe the equalities X

X

#µ(y) = #B , y∈B−1

#µ(z) = #B . z∈B−1

Let now y in B−1 . Since µ(y) ⊂ µ(y), injectivity of the reduction mod p on B implies that #µ(y) ≤ #µ(y). Thus, X

#B =

#µ(y) ≤

y∈B−1

X

#µ(y). y∈B−1

Injectivity of the reduction mod p on B−1 implies that X

X

#µ(y) = y∈B−1

#µ(z) = #B . z∈B−1

This sum equals #B . Thus, all inequalities are equalities, giving our claim.  (x)); define similarly ν(z) For x in B , write ν(x) = µ(π−1 for z in B . By the previous point, ν(x) and ν(x) have the same cardinality. Recalling from Section 2 that for d ∈ N, we have defined A(d, B ) as the set {x ∈ B | #ν(x) = d}, and A(d, B ) as the set {z ∈ B | #ν(z) = d}, one can see A(d, B ) = A(d, B ). To conclude, recall that by definition {x ∈ Z | πn (x) ∈ A(d, πn (B(d+1 , . . . , dn , Z)))} = B(d, d+1 , . . . , dn , Z). By the induction assumption, this equals {x ∈ Z | πn (x) ∈ A(d, B )}, and we have proved that this equals {x ∈ Z | πn (x) ∈ A(d, B )}. By definition, this is B(d, d , . . . , dn , Z), which is what we wanted.

Lemma 3. Let h∆ be an upper bound of the height of ∆, and H∆ = ht W + (deg W )h∆ + (deg W ) log( + 2) + ( + 1) log deg W . There exist χ, v1 , . . . , v in Z[T ], such that χ, χ , v1 , . . . , v have height at most H∆ , µn equals χ divided by its leading coefficient, and wi = vi /χ mod χ for all i. Geometric considerations. Let now Z = Z(F ). For 1 ≤ i ≤ n, let ∆i be a linear form in Z[X1, . . . , Xi ] which is a primitive element for πin (Z), let µi ∈ Q[T ] be its minimal polynomial, and let w1 , . . . , wn ∈ Q[T ] be the parametrization of Z associated to ∆n . Let finally p a prime. We first introduce assumptions on p (denoted by H1 , H2 , H3 ), that yield the conclusion of Theorem 1 in a series of lemmas; we then give quantitative estimates for these assumptions. H1 . The prime p divides no coefficients in µn , w1 , . . . , wn and µn remains squarefree modulo p.



Lemma 6. Let T 1 , . . . , T s be the triangular sets that describe the equiprojectable decomposition of Z. Then p cancels no denominator in the coefficients of T 1 , . . . , T s , and the reduction of these triangular sets modulo p defines the equiprojectable decomposition of Z.

Let Fq be a finite extension of Fp such that (µn mod p) splits in Fq , let Qq be the corresponding unramified extension of Qp [20] and Zq its ring of integers; then, µn splits in Qq , and has all its roots in Zq; thus, Z lies in Zn q . Note that p divides no coefficient in µ1 , . . . , µn : the roots of µi are the values of ∆i on πin (Z), so they are in Zq, hence the coefficients of µi are in Zq ∩ Q = Zp. The map Zq → Fq of reduction modulo p extends to maps a ∈ Ziq → a ∈ Fiq for all i. Given A ⊂ Ziq, A is the set {a | a ∈ A}. The same notation is used for the reduction of polynomials modulo p. H2 . All polynomials µi are squarefree.

Proof. For i ≤ s, let Zi = Z(T i ). By Lemma 5, Z1 , . . . , Zs are the equiprojectable components of Z. For i ≤ s, Zi is described by a triangular set ti with coefficients in Fp . The coefficients of T i are rational functions of the points in Zi , given by interpolation formulas [9, §3]. With these formulas, Lemma 4 shows that all denominators are nonzero modulo p. The coefficients of ti are obtained using the same formulas, using the coordinates of the points in Zi . Thus, ti = T i mod p.



H3 . The Jacobian determinant of F vanishes nowhere on Z.

Lemma 4. For i ≤ n, #πin (Z) equals #πin (Z).

Lemma 7. The set Z equals Z(F ).

Proof. The inequality #πin (Z) ≤ #πin (Z) is obvious. By assumption H2 , all values taken by ∆i on πin (Z) are distinct, so #πin (Z) ≥ deg µi = #πin (Z).

Proof. First, we prove that F vanishes on Z. Indeed, all Fi belong to the ideal generated by I = (µn , X1 − w1 , . . . , Xn − obner basis, so any wn ) in Q[T, X1 , . . . , Xn ]. Now, I is a Gr¨ Fi can be written in terms of I. Since p divides no denominator and no leading term in I, the division equality



Lemma 5. For all d2 , . . . , dn , B(d2 , . . . , dn , Z) equals B(d2 , . . . , dn , Z).

112

of all, we describe the required subroutines, freely using the notation of Theorem 2, and that preceding it. We do not give details of the complexity estimates for lack of space; they are similar to those of [24].

specializes modulo p, and F vanishes on Z, as requested. Let then Z  = Z(F ). By Assumption H3 , Z ⊂ Z  , so it suffices to prove that #Z  ≤ #Z. Let Fr be a finite extension of Fp that contains the coordinates of all these points and let Qr be the corresponding unramified extension of Qp . By Hensel’s lemma, all points in Z  lift to pairwise distinct  simple roots of F in Qn r . Thus, #Z ≤ #Z = #Z.

• EquiprojDecomposition takes as input a polynomial system F and outputs the equiprojectable decomposition of Z(F ), encoded by triangular sets. This routine is called here for systems defined over finite fields. For the experiments in the next section, we applied the triangularization algorithm of [21], followed by the Splitand-Merge algorithm of Section 2, modulo a prime. Studying the complexity of this task is left to the forthcoming [7]; hence, we consider this subroutine as an oracle here, which is called O2 in Theorem 2.



Quantitative estimates. By Lemmas 6 and 7, assumptions H1 , H2 and H3 imply Theorem 1. Thus, it suffices to give quantitative estimates for these assumptions. To this effect, we let D and H be upper bounds on the degrees and heights of the varieties πin (Z), h∆ be an upper bound of the height of ∆1 , . . . , ∆n , and H∆ = H + Dh∆ + D log(n + 2) + (n + 1) log D. Lemma 8. There exists a in N − {0} such that if p does not divide a, H1 and H2 hold. Moreover a verifies:

• Lift applies the Hensel lifting algorithm of [24], but this time to a family of triangular sets, defined first modulo a prime p1 , to triangular sets defined modulo κ the successive powers p21 . From [24], one easily sees that the κth lifting P step has a bit complexity quasilinear in (L, hL , Cn , i≤s deg V (T i ), 2κ , log p1 ), i.e. in (L, hL , Cn , deg Z, 2κ , log p1 ).

ht a ≤ n ((2D − 1)H∆ + (2D − 1) log(2D − 1)) . Proof. Fix i in 1, . . . , n, and let χ, χ , v1 , . . . , vi the polynomials associated to πin (Z) and ∆i in Lemma 3; all of them have integer coefficients of height at most H∆ . Let now ai be the resultant of χ and χ ; by Hadamard’s bound, ht ai ≤ (2D − 1)H∆ + (2D − 1) log(2D − 1). Suppose that p does not divide ai . Then, χ keeps the same degree and remains squarefree modulo p. Furthermore, p divides no coefficient in any wj , since all denominators in 1/χ mod χ divide ai . Thus, assumption H1 holds. Repeating this argument for all projections πin (Z), and taking a = a1 · · · an gives assumption H2 .

• Convert computes the polynomials N i starting from the polynomials T i . Only multiplications modulo triangular sets are needed to perform this operation, so its complexity is negligible before that of Lift. • RationalReconstruction does the following. Let a = p/q ∈ Q, and m ∈ N with gcd(q, m) = 1. If ht m ≥ 2ht a + 1, given a mod m, this routine outputs a. If ht m < 2ht a + 1, the output may be undefined, or differ from a. We extend this notation to the reconstruction of all coefficients of a family of triangular sets. Using the fast Euclidean algorithm [12, Ch 5,11], its complexity is negligible before that of Lift.



Lemma 9. There exists a in N − {0} such that if p does not divide aa , H1 , H2 and H3 hold, and with ht a ≤ 2Dn(dH∆ + h + log d + (d + 1)D log(n + 1)). Proof. Let χ, v1 , . . . , vn be associated to ∆n as in Lemma 3, let J h be the homogenization of J w.r.t. a new variable, and let a ∈ Z be the resultant of J h (χ , v1 , . . . , vn ) and χ; then, a = 0 by the definition of Z. The Jacobian determinant J has coefficients of height at most n(h + log d + (d + 1) log(n + 1)); estimating the height of the determinant of the Sylvester matrix of J h (χ , v1 , . . . , vn ) and χ yields the bound on ht a . Suppose now that p does not divide aa . Then the degree of χ does not drop modulo p, and thus no root of χ cancels J h (χ , v1 , . . . , vn ). In other words, all points described by χ(T ) = 0 and χ (T )Xi = vi (T ), 1 ≤ i ≤ n, are simple for F . This set of points equals Z, giving H3 .

• We do not consider the cost of prime number generation. We see them as input here; formally, in Theorem 2, this is handled by calls to oracle O1 . Computing a triangular decomposition by lifting techniques Input: The system F , primes p1 , p2 Output: The polynomials N 1 , . . . , N s .



T 1,0 , . . . , T s,0← EquiprojDecomposition(Z(F mod p1 ))
u1 , . . . , us ← EquiprojDecomposition(Z(F mod p2 ))

1 m , . . . , ms ← Convert(u1 , . . . , us ) κ←1 while not(Stop) do κ T 1,κ , . . . , T s,κ ← Lift(T 1,κ−1 , . . . , T s,κ−1 ) mod p21 1,κ s,κ 1,κ s,κ N ,...,N ← Convert(T , . . . , T ) N 1,κ, . . . , N s,κ ← RationalReconstruction(N 1,κ , . . . , N s,κ ) 1,κ s,κ 1 s
Stop ← {m , . . . , m } Equals {N , . . . , N } mod p2 κ← κ+1 end while return N 1,κ−1 , . . . , N s,κ−1

In view of Lemma 9, we prove Theorem 1 with A = aa . By [23, Lemma 2.1], all ∆i can be taken of height at most h∆ = n(log n + 2 log D) ≤ n(log n + 2n log d). Using the arithmetic B´ezout bound of [17], we get after simplifications that all H∆ are bounded by ndn (h+3 log(n+1)+2n log d+3). The previous lemmas then give the upper bounds below, which finish proving Theorem 1 after a few simplifications. ht a ht a

4.

≤ ≤

Q

2nd2n (h + 3 log(n + 1) + 2n log d + 7) 2n2 d2n+1 (2h + 4 log(n + 1) + 3n log d + 3).

PROOF OF THEOREM 2

Q

Q

We now give the details of our lifting algorithm: given a polynomial system F , it outputs a triangular representation of its set of simple solutions Z = Z(F ), by means of the polynomials N 1 , . . . , N s defined in the introduction. First

Q

Q

Q

We still use the notation and assumption of Theorem 2. From [9, Th. 1], all coefficients of N 1 , . . . , N s have height

113

Sys 1 2 3 4 5 6 7 8 9 10 11 12 13 14

in nO(1) (deg Z + ht Z), which can explicitly be bounded by F . For p1 ≤ exp (2 F + 1), define

h

h

d = d(p ) =



1



log2

h

2 F +1 log p1

h

d(p1 )



.

Then, p21 has height at least 2 F + 1. In view of the prerequisites for rational reconstruction, (p1 ) bounds the number of lifting steps. From an intrinsic viewpoint, at the last lifting step, 2κ is in O(nO(1) (deg Z + ht Z)). Suppose that p1 does not divide the integer A of Theorem 1. Then, Hensel lifting computes approximations κ T 1,κ , . . . , T s,κ = T 1 , . . . , T s modulo p21 . At the κth lifting step, let N 1,κ , . . . , N s,κ be the outputκ of Convert applied to T 1,κ , . . . , T s,κ , computed modulo p21 ; let N 1,κ , . . . , N s,κ be the same polynomials after rational number reconstruction, if possible. By construction, they have rational coefficients of height at most 2κ−1 log p1 . Supposing that p2 does not divide the integer A of Theorem 1, failure occurs only if for some κ in 0, . . . , − 1, and some j in 1, . . . , s, N j,κ and N j differ, but coincide modulo p2 . For this to happen, p2 must divide some non-zero number of height at most F + 2κ−1 log p1 + 1. Taking all κ into account, this shows that for any prime p1 , there exists a non-zero integer Bp1 such that ht Bp1 ≤ ( F + 1) + 2d log p1 , and if p2 does not divide Bp1 , the lifting algorithm succeeds. One checks that the above bound can be simplified into ht Bp1 ≤ F . Let C ∈ N be such that

d

Q

Q

Name Cyclohexane Fee 1 fabfaux geneig eco6 Weispfenning-94 Issac97 dessin-2 eco7 Methan61 Reimer-4 Uteshev-Bikker gametwo5 chemkin Sys 1 2 3 4 5 6 7 8 9 10 11 12 13 14

h

d

b



C=

a

4

F

b

+2 ε



F

,

so that C ≤

F

+ 1);

Sys 1 2 3 4 5 6 7 8 9 10 11 12 13 14

let Γ be the set of pairs of primes in [C + 1, . . . , 2C]2 and γ be the number of primes in C + 1, . . . , 2C; note that γ ≥ C/(2 log C) and that #Γ = γ 2 . The upper bound on C shows that all primes p less than 2C satisfy the requested inequality log p ≤ 2 F + 1. We can then estimate how many choices of (p1 , p2 ) in Γ lead to failure. There are at most F /log C primes p1 in C + 1, . . . , 2C which divide the integer A of Theorem 1, discriminating at most γ F /log C pairs (p1 , p2 ). For any other value of p1 , there are at most ( F + F )/log C choices of p2 which divide A and Bp1 . This discriminates at most γ( F + F )/log C pairs (p1 , p2 ). Thus the number of choices in Γ leading to failure is at most γ(2 F + F )/log C. The lower bound on γ shows that if (p1 , p2 ) is chosen randomly with uniform probability in Γ, the probability that it leads to failure is at most

h

a

a

a

b

a

b

a

b

a

a

b

a

b

b

a

γ(2 F + F ) γ(2 F + F ) 2 F+ F 4 = = ≤ #Γ log C γ 2 log C γ log C

p1 4423 24499 2671 116663 105761 7433 1549 358079 387799 450367 55313 7841 159223 850088191

d 7 8 7 10 10 7 6 11 11 11 9 7 10 18

a 2 4 5 5 3 3 5 7 4 6 2 5 -

Ca 15 70 110 162 40 31 102 711 89 362 19 125 -

F

b

+2 C

F

∆p 1 3 8 5 12 16 66 47 1515 2292 3507 4879 ∞ -

Ep 0.3 1 0.4 1 1.5 1.5 0.4 9 9 6 1 2 -

Lift 2 9 6 5 6 11 4 232 35 82 9 22 -

Total 7 20 22 18 35 43 133 427 2873 4686 5569 8796 -

Mem. 5 6 7 6 6 7 8 13 11 25 38 63 fail

Output size 243 4157 5855 4757 2555 3282 4653 122902 9916 50476 2621 12870 -

Table 3: Results from our modular algorithm Sys 1 2 3 4 5 6 7 8 9 10 11 12 13 14

,

which is at most ε, as requested. To estimate the complexity of this algorithm, note that since we double the precision at each lifting step, the cost of the last lifting step dominates. From the previous discussion, the number of bit operations cost at the last step is quasi-linear in (L, hL , Cn , deg Z, 2κ , log p1 ). The previous estimates show that at this step, 2κ is in O(nO(1) (deg Z + ht Z)), whereas log p1 is quasi-linear in | log ε|, log h, d, log n. Putting all these estimates ends the proof of Theorem 2.

5.

h 4395 24464 2647 116587 105718 7392 1511 358048 387754 450313 55246 7813 159192 850088102

h 3 2 13 2 0 0 2 7 0 16 1 3 8 11

Table 2: Data for the modular algorithm

h

1 exp (2 2

d 4 4 3 3 3 5 2 2 3 2 5 3 4 3

Table 1: Features of the polynomial systems

Q

d

h

n 3 4 3 6 6 3 4 10 7 10 4 4 5 13

Triang. 0.4 2 512 2.5 5 3000 1593 ∞ -

Mem. 4 6 275 4 5 250 fail fail 18 fail fail fail fail

Size 169 1680 6250 743 3134 2695 55592 -

gsolve 0.2 504 1041 9 4950 1050 ∞ -

Mem. 3 18 34 fail 5 66 31 error fail fail fail fail fail

Size 239 34375 27624 2236 34932 31115 -

Table 4: Results from Triangularize and gsolve

compared with two other Maple solvers, Triangularize, from the RegularChains library, and gsolve, from the Groebner library. Remark that the triangular decompositions modulo a prime, that are needed in our algorithm, are performed by Triangularize. This function is a generic code:

EXPERIMENTATION

We realized a first Maple 9.5 implementation of our modular algorithm on top of the RegularChains library [19]. Tests on benchmark systems [25] reveal its strong features,

114

Acknowledgment

essentially the same code is used over Z and modulo a prime. Thus, Triangularize is not optimized for modular computations. Our computations are done on a 2799 MHz Pentium 4. For the time being our implementation handles square systems that generate radical ideals. We compare our algorithm called TriangularizeModular with gsolve and Triangularize; For each benchmark system, Table 1 lists the numbers n, d, h and Table 2 lists the prime p1 , the a priori and actual number of lifting steps ( and a) and the maximal height of the output coefficients (Ca ). Table 3 gives the time of one call to Triangularize modulo p1 (∆p ), the equiprojectable decomposition (Ep ), and the lifting (Lift.) in seconds — the first two steps correspond to the “oracle calls” O2 mentioned in Theorem 2, which will be studied in [6]. Table 3 gives also the total time, the total memory usage and output size for TriangularizeModular, whereas Table 4 gives that data for Triangularize and gsolve. The maximum time is set up to 10800 seconds; we set the probability of success to be at least 90%. TriangularizeModular solves 12 of the 14 test systems before the timeout, while Triangularize succeeds with 7 and gsolve with 6. Among most of the problems which gsolve can solve, TriangularizeModular shows less time consumed, less memory usage, and smaller output size. Noticeably, quite a few of the large systems can be solved by TriangularizeModular with time extension: system 13 is solved in 18745 seconds. Another interesting system is Pinchon-1 (from the FRISCO project), for which n = 29, d = 16, h = 20, = 1409536095e + 29, which we solve in 64109 seconds. Both Triangularize and gsolve fail these problems due to memory allocation failure. Our modular method demonstrates its efficiency in reducing the size of the intermediate computations, whence its ability to solve difficult problems. We observed that for every test system, for which Ep can be computed, the Hensel lifting always succeeds, i.e. the equiprojectable decomposition over Q can be reconstructed from Ep . In addition, TriangularizeModular failed chemkin at the ∆p phase rather than at the lifting stage. Furthermore, the time consumed in the equiprojectable decomposition and the Hensel lifting is rather insignificant comparing with that in triangular decomposition modulo a prime. For every tested example the Hensel lifting achieves its final goal in less steps than the theoretical bound. In addition, the primes derived from our theoretical bounds are of quite moderate size, even on large examples.

h

The authors are thankful to Fran¸cois Lemaire (Universit´e de Lille 1, France) for his support with the RegularChains library. Merci, Fran¸cois !

7. REFERENCES [1] E. A. Arnold. Modular algorithms for computing Gr¨ obner bases. J. Symb. Comp., 35(4):403–419, 2003. [2] P. Aubry and A. Valibouze. Using Galois ideals for computing relative resolvents. J. Symb. Comp., 30(6):635–651, 2000. [3] F. Boulier, L. Denis-Vidal, T. Henin, and F. Lemaire. L´ episme. In ICPSS, pages 23–27. University of Paris 6, France, 2004. [4] F. Boulier and F. Lemaire. Computing canonical representatives of regular differential ideals. In ISSAC 2000, pages 37–46. ACM Press, 2000. [5] X. Dahan. Borne de hauteur (polynomiale) sur les coefficients d’une repr´ esentation triangulaire d’une vari´ et´ e z´ ero-dimensionnelle pr´ esentant des sym´ etries. Master’s thesis, ´ Ecole Polytechnique, 2003. ´ Schost, W. Wu, and Y. Xie. [6] X. Dahan, M. Moreno Maza, E. The complexity of the Split-and-Merge algorithm. In preparation. ´ Schost, W. Wu, and Y. Xie. On [7] X. Dahan, M. Moreno Maza, E. the complexity of the D5 principle. Preprint. ´ Schost, W. Wu, and Y. Xie. [8] X. Dahan, M. Moreno Maza, E. Equiprojectable decompositions of zero-dimensional varieties. In ICPSS, pages 69–71. University of Paris 6, France, 2004. ´ Schost. Sharp estimates for triangular sets. In [9] X. Dahan and E. ISSAC 04, pages 103–110. ACM Press, 2004. [10] D. Eisenbud. Commutative algebra, volume 150 of GTM. Springer-Verlag, 1995. [11] M.V. Foursov and M. Moreno Maza. On computer-assisted classification of coupled integrable equations. J. Symb. Comp., 33:647–660, 2002. [12] J. von zur Gathen and J. Gerhard. Modern Computer Algebra. Cambridge University Press, 1999. [13] M. Giusti, J. Heintz, J. E. Morais, and L. M. Pardo. When polynomial equation systems can be solved fast? In AAECC-11, pages 205–231. Springer, 1995. [14] M. Giusti, G. Lecerf, and B. Salvy. A Gr¨ obner free alternative for polynomial system solving. J. Complexity, 17(1):154–211, 2001. ´ Hubert. Notes on triangular sets and [15] E. triangulation-decomposition algorithms. In Symbolic and Numerical Scientific Computations, volume 2630 of LNCS, pages 1–39. Springer, 2003. [16] M. Kalkbrener. A generalized euclidean algorithm for computing triangular representations of algebraic varieties. J. Symb. Comp., 15:143–167, 1993. [17] T. Krick, L. M. Pardo, and M. Sombra. Sharp estimates for the arithmetic Nullstellensatz. Duke Math. J., 109(3):521–598, 2001. [18] D. Lazard. Solving zero-dimensional algebraic systems. J. Symb. Comp., 13:117–133, 1992. [19] F. Lemaire, M. Moreno Maza, and Y. Xie. The RegularChains library. In Maple 10, Maplesoft, Canada. To appear. [20] P. J. McCarthy. Algebraic extensions of fields. Dover, New York, 1991. [21] M. Moreno Maza. On triangular decompositions of algebraic varieties. Technical Report 4/99, NAG, UK, Presented at the MEGA-2000 Conference, Bath, UK. http://www.csd.uwo.ca/∼moreno. [22] M. Moreno Maza and R. Rioboo. Polynomial gcd computations over towers of algebraic extensions. In Proc. AAECC-11, pages 365–382. Springer, 1995. [23] F. Rouillier. Solving zero-dimensional systems through the rational univariate representation. AAECC, 9:433–461, 1999. ´ Schost. Complexity results for triangular sets. J. Symb. [24] E. Comp., 36(3-4):555–594, 2003. [25] The symbolicdata project, 2000–2002. http://www.SymbolicData.org. [26] W. Trinks. On improving approximate results of Buchberger’s algorithm by Newton’s method. In EUROCAL 85, volume 203 of LNCS, pages 608–611. Springer, 1985.

d

h

6.

CONCLUSIONS

We have presented a modular algorithm for triangular decompositions of 0-dimensional varieties over Q and have demonstrated the feasibility of Hensel lifting in computing triangular decompositions of non-equiprojectable varieties. Experiments show the capacity of this approach to improve the practical efficiency of triangular decomposition. By far, the bottleneck is the modular triangularization phase. This is quite encouraging, since it is the part for which we relied on generic, non-optimized code. The next step is to extend these techniques to specialize variables as well during the modular phase, following the approach initiated in [13] for primitive element representations, and treat systems of positive dimension.

115

Computing the Multiplicity Structure in Solving Polynomial Systems Barry H. Dayton

Zhonggang Zeng∗

Department of Mathematics Northeastern Illinois University Chicago, IL 60625

Department of Mathematics Northeastern Illinois University Chicago, IL 60625

[email protected]

[email protected]

ABSTRACT

are even more subtle, yet more essential in defining the nature of the system and the zero:

This paper presents algorithms for computing the multiplicity structure of a zero to a polynomial system. The zero can be exact or approximate with the system being intrinsic or empirical. As an application, the dual space theory and methodology are utilized to analyze deflation methods in solving polynomial systems, to establish tighter deflation bound, and to derive special case algorithms.

• The multiplicity m = 12.

• The Hilbert function {1, 2, 3, 2, 2, 1, 1, 0, · · · } that is a partition of the multiplicity 12. • The dual space D(0,0) (I) of the ideal is 12-dimensional with basis below grouped by the differential orders and counted by the Hilbert function:

Categories and Subject Descriptors

1

3

2

2

Here, the differentiation operator

Algorithms, Theory

∂j1 ···js ≡ ∂xj1 ···xjs ≡ 1

Keywords

s

1 ∂ j1 +···+js j1 ! · · · js ! ∂xj1 · · · ∂xsjs 1

.

(2)

The functionals in (1) vanish on the entire ideal I at the zero (0, 0), and form its multiplicity structure.

polynomial ideal, dual space, multiplicity

• The breadth β(0,0) (I) = 2 and the depth δ(0,0) (I) = 6.

INTRODUCTION

In this paper we present several algorithms for computing the multiplicity structure of a polynomial system, namely the dual space of the zero-dimensional ideal at a zero. Using approximate rank-revealing, those algorithms allow the systems and the zeros to be given approximately. Solving polynomial equations is one of the fundamental problems in computational mathematics with a wealth of applications. Multiple zeros, in particular, present a challenge in computation while possessing a rich spectrum of structural invariances, as shown in the following example. Example 1. Consider multiple zero (x1 , x2 ) = (0, 0) of a simple polynomial ideal I = hx31 , x21 x2 + x42 i. The multiplicity 12 may not be obvious. Other structural invariances ∗

(1)

1

1

z }| { z }| { z }| { ∂13 , ∂04 − ∂21 , ∂05 − ∂22 , ∂06 − ∂23 .

General Terms

1.

2

}| { z }| { z}|{ z }| { z ∂00 , ∂10 , ∂01 , ∂20 , ∂11 , ∂02 , ∂12 , ∂03 ,

G.1.5 [Mathematics of Computing]: Roots of Nonlinear Equations – systems of equations; I.1.2 [Symbolic and Algebraic Manipulations]: Algebraic Algorithms

The multiplicity at a zero to a polynomial ideal has been an important topic in algebraic geometry since the days of Newton and Leibniz [5]. In theory, the multiplicity can be computed via symbolic methods if the exact zero is known. Using Singular package [7] this can often be done in practice. However, computer algebra systems such as Maple and Mathematica still output multiplicities with inconsistent accuracy (see §7). When a zero is known approximately or the polynomial system is inexact, computing the multiplicity structure may be beyond the scope of symbolic computation as narrowly defined. Computing multiplicity structures has been studied extensively. The duality approach is originated by Macaulay [11, 15] in 1916, along with an algorithm (see §5) that appears to be largely unknown in modern era. This approach is then phrased in terms of dual spaces by Gr¨ obner in 1939 and elaborated recently by Marinari, M¨ oller and Mora in [13] with a symbolic algorithm. Stetter [20, 22] and Thallinger [24] propose a modified approach with an implementation. Formulation of multiplicity structure as matrix eigenproblems has been studied in [1, 14, 21]. For computing multiplicity only, a numerical algorithm in [8] utilizes Zeuthen’s rule, an eigenvalue approach is introduced in [12], along with a homotopy algorithm in [18] providing an upper bound.

Supported by NSF grant DMS-0412003

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. ISSAC’05, July 24–27, 2005, Beijing, China Copyright 2005 ACM 1-59593-095-7/05/0007...$5.00.

116

where dim(·) denotes the dimension of a vector space. Using (5), we introduce the breadth and the depth of dual space Dxˆ (I) denoted by βxˆ (I) and δxˆ (I) respectively as

Those approaches possess various strengths. However, as evidenced by the frequent inaccuracies of advanced CAS packages, finding practical, robust and accurate algorithms for computing the complete multiplicity structure is apparently still in an early stage of development, especially in the presence of data error and zero approximation. In this paper, we give an expository account of multiplicity structure via duality. The main algorithm MultStructure is then presented for computing the multiplicity structure. Finally, as an application of the duality analysis we investigate existing regularization strategies for computing the multiple zeros, along with our modifications and special case algorithms.

2.

βxˆ (I) = H(1)

δxˆ (I) = max{ α | H(α) > 0 }.

Depth and breadth play significant roles in˘zero-finding (§6). ¯ In contrast to Example 1, the system x31 , x42 also has zero (0, 0) of multiplicity 12 with a different Hilbert function {1, 2, 3, 3, 2, 1, 0, · · · } and a dual space spanned by 1

2

3

3

2

1

z}|{ z }| { z }| { z }| { z }| { z}|{ ∂00 , ∂10 , ∂01 , ∂20 , ∂11 , ∂02 , ∂21 , ∂12 , ∂03 , ∂13 , ∂22 , ∂23

System {x32 , x2 −x23 , x3 −x21 } at origin is again 12-fold with Hilbert function {1, · · · , 1, 0, · · · } and a dual space basis

NOTATION AND PRELIMINARIES

Throughout this paper, N ≡ { 0, 1, 2, · · · } and C is the complex field. Matrices are denoted by capital letters A, B, etc, while O and I represent zero matrices and identities respectively. For any matrix A, N ( A ), rank ( A ) and nullity ( A ) are the nullspace, rank and nullity of A, respectively. Vectors are columns denoted by boldface lower case letters u, v, etc. For any matrix or vector, (·)> denotes the transpose while (·)H the conjugate transpose. Symbols 0 and e1 represent zero vectors and [1, 0, · · · , 0]> respectively. We use C[x] ≡ C[x1 , · · · , xs ] to denote the ring of polynomials with complex coefficients in variables x = (x1 , · · · , xs ). For every index array j = [j1 , · · · , js ] ∈ Ns , the monomial xj = xj11 · · · xjss , and (x − y)j = (x1 − y1 )j1 · · · (xs − ys )js . The operator ∂j is as in (2) with order |j| ≡ j1 + · · · + js . Consider a system of t polynomials {f1 , · · · , ft } in s ˆ where t ≥ s. Polynomials variables with an isolated zero x f1 , · · · , ft ∈ C[x] generate an ideal I = hf1 , · · · , ft i. As ˆ∈C in (2), we define a functional at x

1

1

1

1

}| { z }| { z }| { z }| { z ∂000 , ∂100 , ∂200 + ∂001 , · · · , ∂600 + ∂401 + ∂202 + ∂210 + ∂003 + ∂011 , 1

z }| { · · · , ∂11,00 + ∂901 + ∂702 + · · · + ∂312 + ∂105 + ∂320 + ∂113 + ∂121 .

The last example is a breadth-one case that is of special interest. Dual spaces in this case can be computed via a simple recursive algorithm in §6.2.

3. THE HILBERT FUNCTION It can be assumed without loss of generality that the sysˆ = 0. There tem {f1 , . . . , ft } has a zero at the origin x ˆ are various constructions of the local ring at the point x (e.g. see [2, §4.2]), the easiest is C[[x]]/hf1 , · · · , ft i where C[[x]] = C[[x1 , · · · , xs ]] denotes the ring of formal power ˆ = 0, and I = hf1 , · · · , ft i is the ideal genseries at x erated by f1 , · · · , ft in C[[x]]. These local rings contain the information needed to calculate the multiplicity structure. `The intersection multiplicity of the zero is given by ‹ ´ dimC C [[x]] I (see [5] or [2, Chap.4, Prop 2.11]) and can be decoded from the associated graded ring L α α+1 GrM (C[[x]]/I) = ∞ = C[x]/In(I) α=0 M /M

∂j [ˆ x] : C[x] → C, where ∂j [ˆ x](p) = (∂j p)(ˆ x) for p ∈ C[x]. ˆ is a linear comGenerally, a (differential) functional at x ˆ that vanish on I bination of ∂j [ˆ x]’s. All functionals at x ˆ form a vector space Dxˆ (I) called the dual space of I at x ˛ n o X ˛ Dxˆ (I) ≡ c = (3) cj ∂j [ˆ x] ˛ c(f ) = 0, for all f ∈ I j∈Ns

where M = hx1 , . . . , xs i is the maximal ideal at the origin and In(I) is the ideal of initial forms relative to a local degree ordering [6, §5.5]. This ring is a standard graded algebra. The Hilbert function is then defined as [19, 6, 2].

where cj ∈ C for all j ∈ Ns .

ˆ of an Definition 1. [11, 22] The multiplicity of zero x ideal I ⊂ C[x] is m if the dual space Dxˆ (I) is of dimension ˆ. m, while Dxˆ (I) itself defines the multiplicity structure of x

H(α) = dimC M α /M α+1 , α ∈ N.

This definition is a generalization from the univariate case, where x ˆ is an m-fold root of p ∈ C[x] if Dxˆ (I) is spanned by ∂0 [ˆ x], ∂1 [ˆ x], · · · , ∂m−1 [ˆ x]. Notice that Dxˆ (I) consists of functionals vanishing not only on polynomials f1 , · · · , ft but also on the entire ideal I they generate. In other words, let ˆ . Then c ∈ Dxˆ (I) if and only if c be a functional at x c(p fi ) = 0 for all p ∈ C[x] and 1 ≤ i ≤ t.

and

(6)

The lemma below summarizes the relevant properties of the Hilbert function. Parts (i),(ii) follow from [6], while (iii),(iv) can be derived from properties of the Hilbert function discovered by Macaulay and enumerated in [19, Thm. 2.2]. Lemma 1. Let H : N −→ N be the Hilbert function defined in (6). Then ` ´ ` ´ P (i) dimC C [[x]]/I = dimC C [x]/In(I) = ∞ α=0 H(α).

(4)

For α ∈ N, Dxα ˆ (I) with ˆ (I) consists of functionals in Dx differential orders bounded by α. The Hilbert function, to be defined in §3, has a convenient property associated with the multiplicity structure (see Theorem 2): 8 ” “ 0 > > < H(0) = dim “Dxˆ (I) ” ≡ 1 “ ” α−1 α (5) (I) (I) − dim D H(α) = dim D ˆ x ˆ x > > : for α ∈ {1, 2, · · · },

ˆ is isolated if and only if H(α) = 0 for (ii) The zero x sufficiently large α ∈ N. ` ´ (iii) If H(1) = β, then H(α) ≤ β+α−1 for all α ∈ N. β−1 (iv) If H(α) ≤ 1 for some α then H(σ + 1) ≤ H(σ) ≤ 1 for all σ ≥ α.

117

THE MULTIPLICITY MATRICES

If an entry of Sα is at the intersection of the row and ˆ )k fi ` and ∂j respectively, column indexed by (x − x then ´ ˆ )j fi (see Example this entry is the value of ∂j [ˆ x] (x − x 2). With this arrangement, Sα is the upper-left mα × nα submatrix of subsequent multiplicity matrices Sσ , for σ ≥ α, as illustrated in Example 2.

Based on (5), the multiplicity structure of the the polyˆ nomial ideal I = hf1 , · · · , ft i at x can computed as dimensions and bases for the vector spaces Dxα for ˆ (I) α = 0, 1, · · · until “ ” “ ” dim Dxα = dim Dxα+1 (I) (7) ˆ (I) ˆ

cj ∂j [ˆ x] ∈ Fxα is

if and only if it vanishes on I = hf1 , · · · , ft i:

−1 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 f1 f2 x 1 f1 x 1 f2 x 2 f1 x 2 f2 x21 f1 x21 f2 x 1 x 2 f1 x 1 x 2 f2 x22 f1 x22 f2

0 0 0 0 0 0 0 0 0 0 0 1

0 0 0 0 0 1 0 0 0 0 −1 −1 0 0 0 1 0 0 0 0 −1 −1 1 1 0 0 0 0 1 0 −1 −1 1 1 0 0 0 0 1 0 0 0 1 1 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0

|j| = 1 z }| { ∂10 ∂01

that form a linear system of homogeneous equations. This observation leads to the following definition/theorem which formulates the multiplicity matrices.

1 1 0 0 0 0 0 0 0 0 0 0

z ∂20

Since c is linear, (8) is equivalent to ” “ ˆ )k fi = 0 for k ∈ Ns , i ∈ {1, · · · , t} c (x − x

|j| = 0 z }| { ∂00

(8)

multiplicity matrices &

c(p fi ) = 0 for p ∈ C[x1 , · · · , xs ] and i ∈ {1, · · · , t}.

0 0 1

|j|≤α

0 0 1

P

0 1 0

. A functional c =

0 1 −1

«

1 0 0

α+s α

bases for nullspaces (transposed as row vectors)

in

Dxα ˆ (I)

„

|j| = 2 }| { ∂11 ∂02

with nα =

z ∂30

which is isomorphic to the space of complex vectors o n ˆ ˜ ˛˛ Cnα ≡ c = cj : |j| ≤ α ˛ cj ∈ C, j ∈ Ns

0 1 0 0 −1 −1 0 0 0 0 0 0

∂21

|j| = 3 }| ∂12

{ ∂03

at the smallest α, say α = σ. When (7) occurs, the corresponding Hilbert function satisfies H(σ + 1) = 0, and consequently H(α) = 0 for all α > σ by Lemma 1–(iv). Thus Dxˆ (I) = Dxσˆ (I). The dual subspace Dxα ˆ (I) is a subspace of ˛ o nX ˛ α Fxˆ ≡ cj ∂j [ˆ x] ˛ cj ∈ C, j ∈ Ns , |j| ≤ α ˛ n o ˛ ≡ span ∂j [ˆ x] ˛ j ∈ Ns , |j| ≤ α

0 0 0

Example 2: Consider system {x1 −x2 +x21 , x1 −x2 +x22 } at ˆ = (0, 0). The following array shows the expansion of x multiplicity matrices from S1 to S2 , then to S3 , with rows and columns labeled by xk fi and ∂j respectively. The table beneath the multiplicity matrices shows the bases for the nullspaces as row vectors using the same column indices.

0 0 −1 −1 1 1 0 0 0 0 0 0

4.

The nullspaces can be converted to bases of dual subspaces using the indices in the table:

which corresponds to a homogeneous linear system Sα c = nα 0 „ for c ∈ « C . Here Sα„, an m «α × nα matrix with mα = α−1+s α+s t and nα = , is called the α-th order α−1 α

0 D(0,0) (I) = span { ∂00 } 1 D(0,0) (I) = span { ∂00 , ∂10 + ∂01 } 2 D(0,0) (I) = span { ∂00 , ∂10 + ∂01 , −∂10 + ∂20 + ∂11 + ∂02 }

multiplicity matrix. Consequently, the Hilbert function (5)

The multiplicity matrices are constructed in a way similar to the convolution matrices as in depend on ˛ and ¯ ˘ [25, §5.3], the ordering of index set Iα ≡ j ∈ Ns ˛ |j| ≤ α . For an obvious ordering ≺ of I1 , we can arrange S1 =

6 6 4

f1 (ˆ x) . . . ft (ˆ x)

J(ˆ x)

3 7 7 5

≡

2

0 6 . 6 . 4 . 0

J(ˆ x)

3 7 7 5

{z |k| = 2

}

S3

S2

{z } |k| = 1 |

It is also easy to verify that nullity ( S3 ) = nullity ( S2 ) = 3. Therefore, the Hilbert function H(N) = {1, 1, 1, 0, · · · }. The multiplicity equals 3. The dual space D(0,0) (I) = 2 D(0,0) (I) with breadth β(0,0) (I) = H(1) = 1 and depth δ(0,0) (I) = max{α | H(α) > 0} = 2. The complete multiplicity structure is obtained. 2 The Hilbert function can be calculated from another perspective. The multiplicity matrix Sα is expanded from Sα−1 in the manner of the following matrix partition

H(α) = nullity ( Sα ) − nullity ( Sα−1 ) , α = 1, 2, · · · , (10) ˆ ˜> with S0 ≡ f1 (ˆ x), · · · , ft (ˆ x) = Ot×1

2

S1

| {z } |k| = 0|

N ( S0 ) N ( S1 ) N ( S2 ) N ( S3 )

ˆ be an isolated zeroPof f1 , · · · , ft ∈ Theorem 1. Let x x] is C[x1 , · · · , xs ]. For α ∈ N, a functional |j|≤α cj ∂j [ˆ α if and only if the coefficient vector c = in ˆ Dxˆ (hf1 , ·˜· · , ftni) cj : |j| ≤ α ∈ C α satisfies “ ” P ˆ )k fi (ˆ x) = 0, |j|≤α cj ∂j (x − x (9) s k ∈ N , |k| < α, i ∈ {1, 2, · · · , s},

Sα =

(11)

»

Sα−1 O

∗ ∗

–

=

»

˛ – ˛ Aα ˛˛ Bα

with Aα =

»

Sα−1 O

–

. (12)

Let Nα be a matrix whose rows form a basis for the left nullspace of Aα . For the ideal hf1 , . . . , ft i, let Inα (I) denote the degree α part of In(I) as in [6]. The following lemma is a result of standard matrix theory.

where J(ˆ x) is the Jacobian of the system {f1 , . . . , ft } at ˆ . Generally, we index rows of Sα by (x − x ˆ )k fi for x (k, i) ∈ Iα−1 × {1, · · · , t} with ordering (k, i) ≺ (k0 , i0 ) if k ≺ k0 in Iα−1 or k = k0 but i < i0 . The columns are indexed by the differential functionals ∂j for j ∈ Iα .

Lemma 2. With the assumptions of Theorem 1 and the partition (12) along with Nα as given above, formula (10)

118

is equivalent to H(α) =

„

α−1+s α−1

«

where (·)† denotes the Moore-Penrose inverse. From a random u0 , this iteration virtually guarantees convergence to an approxi-nulvector u, while {ςk } converges to the distance ς between A and the nearest rank-deficient matrix. ˆ= finding an approxi-nulvector u, we form matrix A » After H – kAk∞ u ˆ the resulting sequence . Applying (13) on A, A

− rank ( Nα Bα ) , α ∈

N. Moreover, rank ( Nα Bα ) = dimC Inα (I).

We now assert that the notion of multiplicity and Hilbert function in §2 agree with those in §3. The second statement of the following theorem is well known in the folklore and implicit in [11, 13, 15], but we have not been able to find a clear and explicit statement with a proof in the literature. This theorem can be derived from Lemmas 1 and 2.

{ˆ uk } converges to an approxi-nulvector v of A orthogonal to u, while the scalar sequence {ˆ ςk } converges to the distance between A and the nearest matrix with nullity 2. This process can be recursively continued by stacking kAk∞ vH on ˆ and applying (13) on the new stacked matrix. top of A We now describe the numerical procedure that carries out the shaded step in Algorithm MultStructure. The nullspace ˆNθ ( S0 ) = ˜span { [1] }. Assume orthonormal basis Y = y1 , · · · , yµ for» Nθ (– Sα−1 ) is»obtained. –

Theorem 2. Under the assumption of Theorem 1, the Hilbert function defined in (6) satisfies (5). Consequently, ˆ is identical to the the intersection multiplicity P∞ of the zero x arithmetic multiplicity α=0 H(α) and the dual multiplicity given in Definition 1.

5.

COMPUTING THE MULTIPLICITY

Sα = where

∗ find N ( Sα ) by expanding N ( Sα−1 )

Rα−1 O

is

Sα−1 O » – F1 F2

– » – F T ZH , G Sα

= QH α−1

» – O . F

=

2

TY H 4Sα−1 O

Let

32 » Rα−1 O Qα−1 5 4 F ˆ O G O » – » – ˆ ˆ R = F2 be Q G O

3 F1 ˆ R5 O

= Qα

»

Rα O

–3 F1 F 2 ˜5 G

a QR

–

(14)

ˆ into Qα . with proper accumulation of Qα−1 and Q Notice that (14) implies N ( Rα ) equals ” \ “ \ N ( Sα ) N Z H = N ( Sα ) Nθ ( Sα−1 )⊥ .

Therefore Nθ ( Rα ) consists of approxi-nullvectors of Sα that are approximately orthogonal to those of Sα−1 . The procedure below produces the approxi-nullspace Nθ ( Rα ).

This algorithm turns out to be essentially equivalent to Macaulay’s procedure of 1916 for finding inverse arrays of dialytic arrays [11, 15], except that Macaulay’s algorithm requires construction of dialytic arrays with full row rank. This requirement is difficult and costly to implement with approximate systems or the zeros. Implementation of MultStructure is straightforward for symbolic computation when the system and zero are exact and properly represented. Applying this multiplicityfinding procedure on approximate zeros and/or inexact systems requires the notion and algorithms of numerical rankrevealing at the shaded step in Algorithm MultStructure. The approxi-rank of a matrix A is defined as the minimum rank of matrices within a threshold θ: min

»

decomposition. Then 2 – » Rα−1 T ZH = Qα 4 O Sα O

∗ if nullity ( Sα ) = nullity ( Sα−1 ) then δ = α − 1, H(α) = 0, break the loop otherwise, get H(α) by (10), end if end for • convert N ( Sδ ) to Dxˆ (I) P Output: multiplicity m = α H(α), the Hil. func. H, Dxˆ (I) basis, depth δxˆ (I), breadth βxˆ (I) = H(1)

kA−Bk2 ≤θ

= Qα−1

available, where Qα−1 is unitary, Rα−1 is square uppertriangular and T is a diagonal scaling matrix. Embedding yi ’s into Cnα by appending zeros at the bottom to form zi for i = 1, · · · , µ, we obtain Z = ˆ ˜ z1 , · · · , zµ whose columns form a subset of an orthonormal basis for Nθ ( Sα ). Notice the matrix partitions

Algorithm: MultStructure ˆ ∈ Cs Input: system {f1 , · · · , ft } and zero x • initialize S0 , N ( S0 ) = span { [1] }, H(0) = 1 • for α = 1, 2, · · · do ∗ expand Sα−1 to Sα ∗ embed N ( Sα−1 ) into N ( Sα )

rankθ ( A ) =

TY H Sα−1

Also assume QR decomposition

The multiplicity and its structure can be computed using symbolic, symbolic-numeric or floating point computation based on Theorem 2 and Theorem 1, depending on the representation of the polynomial system and the zero. The main algorithm can be outlined in the following pseudo-code.

• •

let A = Rα for i = 1, 2, · · · do – apply iteration (13), stop at u and ς with proper criteria – if ς > θ, exit, end if » – H ∞u – get zµ+i = u, reset A with kAkA – update the QR decomposition A = QR end for

Upon exit, vectors zµ+1 , · · · , zµ+ν are remaining basis vectors of Nθ ( Sα ) in addition to z1 , · · · , zµ that are already – obtained. Furthermore, the QR decomposition of » ˆH Tˆ Z is a by-product from a proper accumulation of orSα ˆ ˜ thogonal transformations. Here Zˆ is z1 , · · · , zµ+ν with a column permutation and Tˆ is again a scaling matrix.

rank ( B ) .

The approxi-nullspace Nθ ( A ) of A is the (exact) nullspace of B that is nearest to A with rank ( B ) = rankθ ( A ). With this reformulation, approximate-rank/nullspace computing becomes well-posed. We refer to [10] for details. Approximate rank-revealing applies the iteration [10]. 8 – » –† » kAk∞ (uH < uk+1 = uk − 2kAk∞ uk k uk − 1) A Auk (13) : ςk+1 = kAuk+1 k2 , k = 0, 1, · · ·

6. COMPUTING A MULTIPLE ZERO Multiple zeros are infinitely sensitive to perturbation without regularization. Newton’s iteration as well as other standard iterative methods not only lose their superlinear convergence rate, they are also subject to a dismal barrier of

kuk+1 k2

119

J(x0 ) be its Jacobian matrix. Then

“attainable accuracy”. One of the remedies is the deflation approach [9, 17]. In particular, Leykin, Verschelde and Zhao [9] propose an effective deflation method, which we refer to as the LVZ method, with an objective to restore quadratic convergence for Newton’s iteration. From our perspective, deflation is in fact a regularization process that converts the ill-posed zero-finding into a well-posed least squares problem, and provides a mechanism to refine the multiple zero to high accuracy. The LVZ method can be described as follows. Let F (x) be a polynomial system [f1 (x), · · · , ft (x)]> in x as a ˆ is a multiple zero of F (x), then variable vector in Cs . If x ˆ is rank deficient. the Jacobian J(ˆ x) ∈ Ct×s of F (x) at x Let rank ( J(ˆ x) ) = r < s. Then, for almost all choices of an s × (r + 1) random matrix B, the matrix J(ˆ x)B is of r+1 nullity one. Therefore, for– almost , » –all choices of b ∈ C » J(ˆ x)B 0 has a unique solution y = the linear system H 1 b

J(x0 )x1 =

6 6 4

3 x1 · ∆x0 f1 (x0 ) 7 . 7 . 5 . x1 · ∆x0 ft (x0 )

= ∇x1 F (x0 )

1

3 F (x0 ) 4 J(x0 )x1 5 Rx1 − e1

≡

2

3 F (x0 ) 4∇x F (x0 )5 1 Rx1 − e1

(19)

ˆ 1 induces a functional ∇xˆ 1 [ˆ The value of x1 = x x0 ] ∈ Dxˆ 0 (I). If the zero ˆ z to F˜ remains multiple, then the ˜ z) of F˜ (z) at z ˆ has a nontrivial nullspace. The Jacobian J(ˆ deflation can be applied to F˜ the same way as (19) applied to F . Namely, we seek a solution in (x0 , x1 , x2 , x3 ) to „ « „ «» – » – x x x2 ˜ x2 = e 1 . F˜ x0 = 0, J˜ x0 = 0, R x x 1

1

3

„

Using (16) – (18), equation J˜ 2

(x2 · ∆x0 )F (x0 ) + 4 (x2 · ∆x )∇x F (x0 ) + 0 1 (x2 · ∆x0 )(Rx1 − e1 ) + 2 ∇x2 F (x0 ) = 4 (∇x2 ∇x1 + ∇x3 )F (x0 ) Rx3

3

«» – x2 x0 x3 x1

= 0 implies

3 (x3 · ∆x1 )F (x0 ) (x3 · ∆x1 )∇x1 F (x0 ) 5 3(x3 · ∆x1 )(Rx1 − e1 ) 5

= 0

Thus, the second deflation seeks a solution to the system F (x0 ) = 0, ∇x1 F (x0 ) = 0, (∇x2 ∇x1 + ∇x3 )F (x0 ) = 0.

∇x2 F (x0 ) = 0,

(20)

The third deflation adds variables x4 , · · · , x7 and equations ∇x4 F (x0 ) = 0, (∇x4 ∇x1 + ∇x5 )F (x0 ) = 0, (∇x4 ∇x2 ∇x1 + (∇x4 ∇x2 + ∇x6 )F (x0 ) = 0, ∇x4 ∇x3 + ∇x2 ∇x5 + ∇x6 ∇x1 + ∇x7 )F (x0 ) = 0.

(21)

ˆ 7 ) ∈ C8s to (20) and (21) induces Any solution (ˆ x0 , · · · , x eight differential functionals 1, ∇xˆ 1 , ∇xˆ 2 , ∇xˆ 4 and ∇xˆ 2 ∇xˆ 1 + ∇xˆ 3 , ∇xˆ 4 ∇xˆ 1 + ∇xˆ 5 , ∇xˆ 4 ∇xˆ 2 + ∇xˆ 6 , ∇xˆ 4 ∇xˆ 2 ∇xˆ 1 + ∇xˆ 4 ∇xˆ 3 + ∇xˆ 2 ∇xˆ 5 + ∇xˆ 6 ∇xˆ 1 + ∇xˆ 7

We shall use additional differential notation and operations. The original variables are in a vector form x = [x1 , · · · , xs ]> which will also be denoted by x0 in contrast to auxiliary (vector) variables x1 , x2 etc. For any fixed or variable vector y = [y1 , · · · , ys ]> , the directional differentiation operator along y is defined as

ˆ 0 . In general, the k-th deflation step that vanish on F at x seeks a collection of 2k differential functionals of order k or ˆ 0 . It remains to show less that vanish on the system F at x those functionals satisfy condition (4). For this purpose, we define differential operators Φα ’s as follows. P2ν −1 ν = 0, 1, · · · . (22) Φν+1 = ζ=0 x2ν +ζ · ∆xζ ,

(16)

When y is fixed in Cs , ∇y induces a functional ∇y [ˆ x] : p −→ (∇y p)(ˆ x). For any variable u = [u1 , · · · , us ]> , i> h ∂ ∂ , which is , · · · , ∂u the gradient operator ∆u ≡ ∂u s 1

Specifically, Φ1 Φ3

>

= =

x 1 · ∆ x 0 , Φ2 = x 2 · ∆ x 0 + x 3 · ∆ x 1 , x 4 · ∆x0 + x 5 · ∆x1 + x 6 · ∆x2 + x 7 · ∆x3

with operations such as

(17)

Φ1 F (x0 ) = ∇x1 F (x0 ), Φ2 F (x0 ) = ∇x2 F (x0 ), Φ2 ◦ Φ1 F (x0 ) = (∇x2 ∇x1 + ∇x3 )F (x0 ).

especially ∇y ≡ y · ∆x ≡ y · ∆x0 for any y of dimension s. For any f ∈ C[x0 ], let y and z be auxiliary variables, (y · ∆x0 )(∇z f (x0 )) = ∇y ∇z f (x0 ), z · ∆y f (x0 ) ≡ 0, (z · ∆y )(∇y f (x0 )) = (z · ∆y )(y · ∆x0 )f (x0 ) = ∇z f (x0 ).

=

2

2

F˜ (z) = 0, with F˜ (z) ≡

6.1 Duality analysis of the LVZ method

∂ ∂ , + · · · + vs ∂u v · ∆u ≡ v1 ∂u s 1

3 ∆x0 f1 (x0 )> 7 . 7 x1 . 5 . > ∆x0 ft (x0 )

1

ˆ whose x and y components has an isolated solution z = z ˆ and y ˆ respectively. If ˆ are x z is still a multiple zero to Fˆ , ˆ to one can repeat the same deflation technique on Fˆ and z ˆ ˆ ˆ obtain a further expanded system F and isolated zero ˆ z. Leykin, Verschelde and Zhao’s theory is that the multiˆ to system F . plicity of ˆ z to system Fˆ is less than that of x Since every deflation step deflates the multiplicity by at least one, recursive deflations will eventually exhaust the multiˆ . The plicity at a system G(u) = 0 with simple zero u number of deflation steps is strictly less than the multiplicˆ [9, Theorem 3.1]. The Gauss-Newton iteration ity m of x ˆ with a on the final system G(u) = 0 converges locally to u quadratic rate. More importantly as we see it, the multiple ˆ of system F can now be computed to high accuracy zero x ˆ of G. Furthermore, the LVZ as part of a simple zero u method also generates functionals in Dxˆ (I) as by-products.

mainly used in a “dot product” with v = [v1 , · · · , vs ]

6 6 4

We slightly modify the LVZ method. Let J(ˆ x0 ) be the ˆ 0 with rank r. For almost Jacobian of F (x0 ) at zero x (s−r)×s ˆ 1 ∈ Cs to all x – » solution » R –∈ C » – , there is a unique ˆ J(ˆ x0 ) 0 ˆ= x is a solution to x1 = e0 . As a result, z ˆ x R

ˆ . Consequently a new (2t + 1) × (s + r + 1) system y=y 2 3 » – F (x) x H ˆ ˆ 4 and F (z) ≡ b y − 15 (15) F (z) = 0, with z = y J(x)By

∂ ∇y ≡ y1 ∂x∂ 1 + · · · + ys ∂x . s

2

It is easy to verify that (20) and (21) can be written as F = 0, Φ1 F = 0, Φ2 F = 0, Φ2 ◦ Φ1 F = 0, Φ3 F = 0, Φ3 ◦ Φ1 F = 0, Φ3 ◦ Φ2 F = 0, Φ3 ◦ Φ2 ◦ Φ1 F = 0

(18)

Let F = [f1 , · · · , ft ]> be a polynomial system in C[x0 ] and

We have the following lemma.

120

Lemma 3. Let F = [f1 , · · · , ft ]> be a system of polynomials fi ’s in variable x = [x1 , · · · , xs ]> . Denote F0 = F and x0 = x. Then any isolated solution to the expanded system in the α-th deflation step described as in (19) solves Fα ≡

»

Fα−1 Φα Fα−1

–

= 0,

vector b ∈ Cs . Similar to the modified LVZ method, the first step deflation is to set up an expanded system: » – F0 (x0 ) G1 (x0 , x1 ) = (23) F1 (x0 , x1 )

α = 1, 2, · · ·

with F1 (x0 , x1 ) =

in (x0 , x1 , · · · , x2α −1 ), where Φα ’s are defined in (22).

Theorem 3. Let f1 , · · · , ft ∈ C[x1 , · · · , xs ] with an ˆ to the ideal I = hf1 , · · · , ft i. Then isolated multiple zero x the number k of deflation steps required for the modified LVZ method is bounded by the depth, namely k ≤ δxˆ (I). Furthermore, the method generates 2k differential functionals in the dual space Dxˆ (I) as by-products.

»

∇x1 F (x0 ) b H x1 − 1

–

.

3 F0 (x0 ) 4 5 G2 (x0 , x1 , x2 ) ≡ F1 (x0 , x1 ) (25) F2 (x0 , x1 , x2 ) » – (∇x1 ∇x1 + ∇x2 )F (x0 ) with F2 (x0 , x1 , x2 ) = b H x2

ˆ1, x ˆ 2 ). We define has an isolated zero (ˆ x0 , x P∞ Ψ = η=1 xη · ∆xη−1 .

(26)

Notice that the Ψf is in fact a finite sum for any particular polynomial f in (vector) variables, say x0 , · · · », xσ , since – ΨF (x ) ∆xµ f = 0 for µ > σ + 1. Thus F1 (x0 , x1 ) = bH x0 −0 1 ,

• Initialize F = [f1 , · · · , ft ]> , x0 = [˜ x1 , · · · , x ˜ s ]> • for i = 1, 2, · · · do − for k = 0, 1, · · · do ∗ calculate J(xk ) of F at xk ∗ if J(xk ) is approxi-rank deficient, » – reset x0 =

≡

2

We summarize the modified LVZ method below at an ap˜ = [˜ proximate zero x x1 , · · · , x ˜s ]> , to a system {f1 , · · · , ft }.

J(xk )y =0 Ry − e1 – xk , break y0

–

ˆ 0 is of nullity one, there is a Since the Jacobian of F at x ˆ 1 . Equation (24) along constant γ ∈ C such that v0 = γ x with βxˆ 0 (I) = 1 and (v0 , v1 ) 6= (0, 0) implies γ 6= 0. ˆ 1 . Setting Consequently we can choose γ = 1, namely v0 = x ˆ 2 = v1 , the system x

Proof. By Lemma 3 and the product rule Φα (f g) = (Φα f ) g + (Φα g) f in an induction, the α-th deflation step generates differential functionals of order α that satisfy condition (4). Therefore those functionals belong to Dxˆ (I). Since differential orders of all functionals in Dxˆ (I) are bounded by δxˆ (I), so is α.

»

J(x0 ) x1 b H x1 − 1

ˆ 1 ). If the The system G1 (x0 , x1 ) has an isolated zero (ˆ x0 , x ˆ 1 ), Jacobian J1 (x0 , x1 ) of G1 (x0 , x1 ) is of full rank at (ˆ x0 , x then the system is regularized. Otherwise, there is a nonzero vector (v0 , v1 ) ∈ C2s such that 3 2 – » ∇v0 F (ˆ x0 ) v0 5 = 0. (24) 4 (∇ ∇ + ∇ )F (ˆ x ) ˆ1) ≡ J1 (ˆ x0 , x v0 v1 0 ˆ1 x v1 b H v1

As a consequence, the following theorem improves the result in [9, Theorem 3.1] by bounding the number of deflation steps with depth. In other words, the LVZ method deflates depth, not just multiplicity.

then solve

»

1

F2 (x0 , x1 , x2 ) = ΨF1 (x0 , x1 ) and in general ν−1

}| { z Fν (x0 , · · · , xν ) = Ψ ◦ Ψ ◦ · · · ◦ ΨF1 (x0 , x1 ), ν ≥ 2.

(27)

For example, besides F1 and F2 in (23) and (25) respectively,

for y = y0 ,

F3 (x0 , x1 , x2 , x3 ) =

the k loop.

∗ if the accuracy of xk is satisfactory, ˆ = xk exit with x ∗ calculate xk+1 = xk − J(xk )† F (xk ) end for – construct F˜ in (19), reset F = F˜ end for

»

(∇x1 ∇x1 ∇x1 + 3∇x1 ∇x2 + ∇x3 )F0 (x0 ) b H x3

–

ˆ1, x ˆ2, x ˆ 3 ), we obtain a functional If, say, F3 = 0 at (ˆ x0 , x p −→ (∇xˆ 1 ∇xˆ 1 ∇xˆ 1 + 3∇xˆ 1 ∇xˆ 2 + ∇xˆ 3 ) p(x0 ) for p ∈ C[x0 ] that vanishes on the system F . The original system F = 0 provides a trivial functional ∂0···0 : p → p(ˆ x0 ). The lemma below ensures those functionals are in the dual space. Lemma 4. Let F = [f1 , · · · , ft ]> be polynomial system ˆ ∈ Cs . Denote F0 = generating an ideal I with a zero x ˆ0 = x ˆ and x0 = x. For any γ ∈ { 1, 2, · · · }, let F, x ˆ1, · · · , x ˆ γ ) be a zero of (ˆ x0 , x

The LVZ method appears to be robust in computing tests.

6.2 Special case: dual space of breadth one Consider the ideal I = hf1 , · · · , ft i in the case of breadth βxˆ (I) = 1. The Hilbert function is {1, 1, · · · , 1, 0, · · · }, making the depth δxˆ (I) = m − 1. Computing experiments show that LVZ method always require the maximal number of deflations in this case, with the final system being expanded to the size larger than (2m−1 t) × (2m−1 s) from t × s. A further modified version of the regularized system is of roughly (mt) × (ms). Upon solving the system, a complete basis is also obtained for Dxˆ (I) as a by-product. ˆ =x ˆ 0 . Notice that from Denote x = x0 and the zero x (–J(ˆ x0 )») –= 1 (11), the breadth βxˆ (I) = H(1) = nullity » J(ˆ x0 ) 0 x in implies the system (19), which becomes 1 = 1 bH

Gγ (x0 , x1 , · · · , xγ ) =

2 6 6 4

3 F0 (x0 ) 7 . .. 7. . 5 . . Fγ (x0 , · · · , xγ )

(28)

ˆ γ ) = 0 are Then the functionals derived from Gγ (ˆ x0 , · · · , x linearly independent members of the dual space Dxˆ 0 (I). ˆ be an isolated m-fold zero of the Theorem 4. Let x polynomial ideal I = hf1 , · · · , ft i with breadth βxˆ (I) = ˆ0 = x ˆ and x0 = x. 1. Denote F0 = [f1 , · · · , ft ]> , x Then there is an integer γ ≤ δxˆ (I) such that the system Gγ ˆ1, · · · , x ˆ γ ) that induces in (28) has a simple zero (ˆ x0 , x γ + 1 linearly independent functionals in Dxˆ 0 (I).

ˆ 1 ∈ Cs for almost all random x1 , has a unique solution x

121

Similar to Theorem 3, Theorem 4 can be proved for γ ≤ δxˆ (I). It is our conjecture that γ = δxˆ 0 (I) is always true in breadth one case. We have not seen an exception in our extensive computing experiment. An important impliˆ1, · · · , x ˆ γ ) to cation of γ = δxˆ 0 (I) is that the zero (ˆ x0 , x Gγ induces a complete basis for the dual space Dxˆ (I). ˜ = [˜ Given an approximate zero x x1 , · · · , x ˜s ]> to the > system F = [f1 , · · · , ft ] , the breadth-one algorithm can be summarized below.

4 | {z } LVZ 1 {z CAS

16

4

2

NR NR

}

3

2

1

1 1

1

1

1 2

1 |

1

1

1 NR

1

1

1

16 4

1

4

(0, 0, 0, 0)

(0, 0, −1)

Z3

DZ1

DZ2

DZ3

1, 4, 6, 4, 1

(1, 1, 1, 1, 1) √ √ (2, −i 3, 2, i 3) Z9 Caprasse Cyclic 9

KSS

1 1, 1 (−5/2, 5/2, 1)

2

1 1, 1

1, 2, 1

(1, 0, 0)

(0, 0, 1) Ojika3

(0, 0, 0) (0, 0, 0) (0, 1, 0) (0, 0) (0, 0, 1)

sy cmbs1 cmbs2 mth191 Decker2 Ojika2

em st

ro ze

1. cbms1 [23]: x3 − yz, y 3 − xz, z 3 − xy, (0, 0, 0) 2. cbms2 [23]: x3 − 3x2 y + 3xy 2 − y 3 − z 2 , z 3 − 3z 2 x + 3zx2 − x3 − y 2 , y 3 − 3y 2 z + 3yz 2 − z 3 − x2 , (0, 0, 0) 3. mth191 [9]: x3 + y 2 + z 2 − 1, x2 + y 3 + z 2 − 1, 2 2 3 x + y + z − 1, (0, 1, 0) 4. decker2 [3] x + y 3 , x2 y − y 4 , (0, 0) 5. Ojika2 [17]: x2 +y+z −1, x+y 2 +z −1, x+y+z 2 −1, (0, 0, 1), (1, 0, 0) 6. Ojika3 [17]: x + y + z − 1, 2x3 + 5y 2 − 10z + 5z 3 + 5, 2x + 2y + z 2 − 1, (0, 0, 1), (− 25 , 52 , 1) 7. Caprasse [16]: −x31 x3 + 4x1 x22 x3 + 4x21 x2 x4 + 2x32 x4 + 4x21 − 10x22 + 4x1 x3 − 10x2 x4 + 2, −x1 x33 + 4x2 x23 x4 + 4x1 x3 x24 + 2x2 x34 + 4x1 x3 + 4x23 − 10x2 x4 − 10x24 + 2, x22 x3 +√2x1 x2√ x4 − 2x1 − x3 , 2x2 x3 x4 + x1 x24 − x1 − 2x3 , (2, −i 3, 2, i 3)

t on er cti ilb un H f 1, 3, 3, 3, 1 1, 3, 3, 1 1, 2, 1 1, 1, 1, 1 1, 1

br 3 3 2 1 1

ea

Preliminary implementations of Algorithm MultStructure are conducted in both symbolic and numerical computation, as well as Algorithm BreadthOne with symbolic rank-revealing. Building a comprehensive test suite of polynomial systems is also underway. A large number of problems have been tested. We present results on the following benchmark problems with selected multiple zeros. Most of the systems are well documented in the literature for testing algorithms in polynomial system solving.

1, 2, 1 2 2 4 1, 2, 1 2 2 4 1, 4, 10, 16, 22, 25, 22, 4 10 131 16, 10, 4, 1 1, 2, 3, 3, 2, 2 7 16 2, 2, 1 1, 1, 1, 1, 1 1 4 5 | {z MultStructure and BreadthOne

2

4

2 1

h dt

de 4 3 2 3 1

COMPUTATIONAL EXPERIMENT

2

h pt

t ul m 11 8 4 4 2

i

ic pl

}

y it

Algorithm BreadthOne converges locally to the multiple zero beyond the “attainable accuracy” with a quadratic rate. Moreover, it also refines the basis for the dual space. An important application of Algorithm BreadthOne is that it regularizes the zero-finding problem for a univariate equation f (x) = 0 where the breadth is always one.

1

le ti ap ul M m 1 NR 1 4 1

ic pl

y it

a ic y at icit l m he tip at ul M m 3 NR 2 4 2

s on ti ded a fl e de ne 1 1 1 3 1

11. 12.

2

10.

1

9.

Algorithm BreadthOne ˜0 = x ˜ • Initialize F0 = [f1 , · · · , ft ]> , x • for γ = 1, 2, · · · do ˜ γ−1 , xγ ) = 0 for a least − solve Fγ (˜ x0 , · · · , x ˜γ squares solution xγ = x ˆ > ˜> ˜0 , · · · , x ˜> − set z0 = x γ − for k = 0, 1, · · · do ∗ calculate Jγ (zk ) of Gγ at zk ∗ if Jγ (zk ) is approxi-rank deficient, break the k loop ∗ if the accuracy of zk is satisfactory, exit γ loop with ˜ z = zk ∗ refine zk using the Gauss-Newton iteration zk+1 = zk − Jγ (zk )† Gγ (zk ) end for end for • multiplicity m = γ + 1 ˜ 0 of F0 from z ˜, get γ + 1 • extract the zero x ˜ functionals from components of z

7.

P fσ (x1 , · · · , x5 ) = x2σ + 5ν=1 xν − 2xσ − 4, for σ = 1, · · · , 5, at (1, 1, 1, 1, 1) P Q Cyclic nine [4]: fν = 8i=0 i+ν j=i xi , for ν = 0, · · · , 7 Q8 and f8 = 1 − j=0 xj with “cyclic” variables x8+µ = xµ for µ = 0, · · · , 7. The selected 4-fold zero Z9 = (z0 , z1 , z2 , z0 , −z2 , −z1 , z0 , −z2 , −z1 ) with z0 = −.9396926 − 0.3520201i, z1 = −2.4601472 − .8954204i and z2 = −.3589306 − .1306401i. DZ1: x41 − x2 x3 x4 , x42 − x1 x3 x4 , x43 − x1 x2 x4 , x44 − x1 x2 x3 , (0, 0, 0, 0). Modified from cbms1. DZ2: x4 , x2 y + y 4 , √z + z 2 − 7x3 − 8x2 , (0, 0, √ −1). DZ3: 14x + 33y − 3 5(x2 + 4xy + 4y 2 + 2) + 7+ √ x3 + 6x2 y + 12xy 2 + 8y√3 , 41x − 18y − 5 + 8x3 − 12x2 y + 6xy 2 − y 3 + 3 7(4xy − 4x2 − y 2 − 2), with coefficients rounded to 5 digits, at approximate zero Z3 = (1.5055, 0.36528).

8. KSS [8]:

Table 1: Test results on benchmark problems. Incorrect results from CAS’s are shaded. “NR” means no response in a reasonable amount of time. The CAS packages Maple and Mathematica are used to solve the systems with multiplicity extracted from the out-

122

¨ nemann, [7] G.-M. Greuel, G. Pfister, and H. Scho Singular 2.0. A Computer Algebra System for Polynomial Computations. Centre for Computer Algebra, University of Kaiserslautern (2001). http://www.singular.uni-kl.de. [8] H. Kobayashi, H. Suzuki, and Y. Sakai, Numerical calculation of the multiplicity of a solution to algebraic equations, Math. Comp., 67 (1998), pp. 257–270. [9] A. Leykin, J. Verschelde, and A. Zhao, Newton’s method with deflation for isolated singularities of polynomial systems. Preprint, (2004). [10] T. Y. Li and Z. Zeng, A rank-revealing method with updating, downdating and applications. to appear: SIAM J. Matrix Anal. Appl. [11] F. S. Macaulay, The Algebraic Theory of Modular Systems, Cambridge Univ. Press, 1916. [12] D. Manocha and J. Demmel, Algorithms for intersecting parametric and algebraic curves II: multiple intersections, Graphical Models and Image Processing, 57 (1995), pp. 81–100. ¨ ller, [13] M. G. Marinari, T. Mora, and H. M. Mo On multiplicities in polynomial system solving, Trans. AMS, 348 (1996), pp. 3283–3321. ¨ ller and H. J. Stetter, Multivariate [14] H. M. Mo polynomial equations with multiple zeros solved by matrix eigenproblems, Numer. Math., 70 (1995), pp. 311–329. [15] T. Mora, Solving Polynomial Equation Systems II. manuscript. [16] S. Moritsugu and K. Kuriyama, On multiple zeros of systems of algebraic equations. Proc. of ISSAC ’99, ACM Press, pp 23–30, 1999, 1999. [17] T. Ojika, Modified deflation algorithm for the solution of singular problems, J. Math. Anal. Appl., 123 (1987), pp. 199–221. [18] A. J. Sommese, J. Verschelde, and C. W. Wampler, Numerical decomposition of the solution sets of polynomial systems into irreducible components, SIAM J. Numer. Anal., 38 (2001), pp. 2022–2046. [19] R. P. Stanley, Hilbert function of graded algebras, Advances in Math., 28 (1973), pp. 57–83. [20] H. J. Stetter, Analysis of zero clusters in multivariate polynomial systems. Proc. of ISSAC ’96, ACM Press, pp 127–136, 1996. [21] , Matrix eigenproblems are at the heart of polynomial system solving, SIGSAM Bull., (1996), pp. 22–25. [22] , Numerical Polynomial Algebra, SIAM, Philadelphia, 2004. [23] B. Sturmfels, Solving Systems of polynomial Equations, Number 97 in CBMS Regional Conference Series in Mathematics, AMS, 2002. [24] G. H. Thallinger, Analysis of zero clusters in multivariate polynomial systems. Diploma Thesis, Tech. Univ. Vienna, 1996. [25] Z. Zeng and B. H. Dayton, The approximate GCD of inexact polynomials, II: a multivariate algorithm. Proc. of ISSAC ’04, ACM Press, pp 320–327, 2004.

put zeros. Table 1 lists the results in comparison with the multiplicity structures computed by Algorithm MultStructure. Also in the table are deflation steps required for LVZ method along with the modified version. The test shows that Maple either reports the correct multiplicity or skips on multiplicity identification, while Mathematica makes the attempt but often underestimates the multiplicities. Neither Maple nor Mathematica are implemented for calculating other structural invariances for multiple zeros. The solve function in Singular [7], if successfully terminated, accurately identifies the multiplicities for all above systems with exact coefficients. Like other CAS packages, it appears that Singular is not designed to handle approximate systems or zeros. For example, Singular outputs a cluster of “simple” zeros around Z3 for (approximate) system DZ3, while MultStructure correctly identifies the underlying multiplicity 5 with Hilbert Function {1, 1, 1, 1, 1, 0, · · · }. In summary, our codes for Algorithm MultStructure, as well as BreadthOne in breadth-one cases, accurately identify the multiplicity and output the complete multiplicity structure with bases for the dual spaces along with the Hilbert function. An example of analytic system. The primary objective of this paper is computing multiplicity structures for polynomial systems. The methods in this paper can nonetheless be applied to systems of analytic equations, since the construction of the multiplicity matrices only requires that partial derivatives be obtained. Consider a simple system f1 (x, y) = 1 − cos(x2 ) f2 (x, y) = sin(y) + x2 ex+y Algorithm MultStructure identifies multiplicity 4 at zero (0, 0) with the basis for the dual space ∂0 , ∂x , ∂x2 − ∂y , and ∂x3 − ∂xy + ∂y . Acknowledgment: We would like to thank Jan Verschelde and Anton Leykin for valuable discussions and for providing their results in [9]. We are grateful to Teo Mora and Michael M¨ oller for sharing their knowledge on the history of the duality approach to multiplicity. In particular, Teo Mora pointed out reference [11], kindly provided his manuscript [15] and made valuable suggestions on this paper. We also thank anonymous referees for their helpful comments.

8.

REFERENCES

[1] R. M. Corless, P. M. Gianni, and B. M. Trager, A reordered Schur factorization method for zero-dimensional systems with multiple roots. Proc. ISSAC ’97, AMS Press, pp 133–140. [2] D. Cox, J. Little, and D. O’Shea, Using Algebraic Geometry, Springer Verlag, 1998. [3] D. W. Decker, H. B. Keller, and C. T. Kelly, Convergence rate for Newton’s method at singular points, SIAM J. Numer. Anal., 20 (1983), pp. 296–314. [4] J. C. Faug` ere, A new efficient algorithm for computing G¨ obner bases, Journal of Pure and Applied Algebra, 139 (1998), pp. 61–88. [5] W. Fulton, Intersection Theory, Springer Verlag, Berlin, 1984. [6] G.-M. Greuel and G. Pfister, A Singular Introduction to Commutative Algebra, Springer Verlag, 2002.

123

Algorithms for the Non-monic Case of the Sparse Modular ∗ GCD Algorithm Jennifer de Kleine

Michael Monagan

Allan Wittkopf

Department of Mathematics Simon Fraser University Burnaby, B.C. Canada.

Department of Mathematics Simon Fraser University Burnaby, B.C. Canada.

Maplesoft 615 Kumpf Drive Waterloo, Ont. Canada.

[email protected].

[email protected].

[email protected].

ABSTRACT 2

2

on the running time of Brown’s algorithm (see[1]) when G is also sparse. Zippel’s algorithm is an output sensitive algorithm. Unlike Brown’s algorithm, the number of univariate images depends on the size of G, and not on A and B. Most computer algebra systems use either Zippel’s algorithm or Wang’s EEZ-GCD algorithm (see [10]) for multivariate gcd computation. Zippel’s algorithm is implemented in Macsyma, Magma, and Mathematica. A parallel implementation is described by Rayes, Wang and Weber in [8]. Previous work done to improve the asymptotic efficiency includes that of Zippel in [13], and Kaltofen and Lee in [5]. In this paper we present two new algorithms that extend Zippel’s algorithm to the case where G is not monic in the main variable x1 . In section 2 we give a description of Zippel’s algorithm and previous approaches made to extend it to the non-monic case. In section 3 we describe our first solution and in section 4 our second solution. We have implemented both algorithms in Maple. In section 5 we make some remarks about their efficiency and implementation. Although our algorithms do not require any polynomial factorizations, both require that the content of G in the main variable x1 is 1. The content of G can be computed efficiently by computing the gcd of one coefficient of A, the smallest, with a random linear combination of the other coefficients of A and all coefficients of B in x1 . This requires one recursive gcd computation in [x2, ..., xn ].

2

Let G = (4y + 2z)x + (10y + 6z) be the greatest common divisor (gcd) of two polynomials A, B ∈ [x,y, z]. Because G is not monic in the main variable x, the sparse modular gcd algorithm of Richard Zippel cannot be applied directly as one is unable to scale univariate images of G in x consistently. We call this the normalization problem. We present two new sparse modular gcd algorithms which solve this problem without requiring any factorizations. The first, a modification of Zippel’s algorithm, treats the scaling factors as unknowns to be solved for. This leads to a structured coupled linear system for which an efficient solution is still possible. The second algorithm reconstructs the monic gcd x2 + (5y 2 + 3z)/(2y 2 + z) from monic univariate images using a sparse, variable at a time, rational function interpolation algorithm. Categories and Subject Descriptors: I.1.2 [Symbolic and Algebraic Manipulation]: Algorithms – Algebraic algorithms; F.2.1 [Analysis of Algorithms and Problem Complexity]: Numerical algorithms and problems – Computations on Polynomials. General Terms: Algorithms. Keywords: Zippel’s Algorithm, Polynomial Greatest Common Divisors, Sparse Multivariate Polynomials, Modular Algorithms, Probabilistic Algorithms.

1.

2. ZIPPEL’S ALGORITHM

INTRODUCTION

There are two subroutines in the algorithm. Subroutine M, the main subroutine, computes G = gcd(A, B) where A, B ∈ [x1, ..., xn ]. It does this by computing gcd(A, B) modulo a sequence of primes p1 , p2 , ... then reconstructs G from these images by applying the Chinese Remainder Theorem. The first image G1 is computed by calling subroutine P with inputs A mod p1 and B mod p1 . Subroutine P, which is recursive, computes G = gcd(A, B) where A, B in p[x1 , ..., xn ] for a prime p as follows. If n = 1 it uses the Euclidean algorithm. If n > 1 it computes gcd(A, B) at a sequence of random points α1 , α2 , ... ∈ p for xn then reconstructs G ∈ p[x1 , ..., xn ] from the images using dense interpolation, e.g., Newton interpolation. The first image G1 is computed by calling subroutine P recursively with inputs A mod hxn − α1 i and B mod hxn − α1 i. In both subroutines, after the first image G1 is computed, subsequent images are computed using sparse interpolations. This involves solving a set of independent linear systems which are constructed based on the form of G1 . The algo-

Let A, B be polynomials in [x1, ..., xn ]. Let G be their ¯ = B/G greatest common divisor (gcd) and let A¯ = A/G, B ¯ and B. ¯ be their cofactors. Our problem is to compute G, A In [12] (see also [14] for a more accessible reference) Zippel presented a Las Vegas algorithm for computing G when G is monic in the main variable x1 . Zippel’s algorithm improves ∗Supported by NSERC of Canada and the MITACS NCE of Canada

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univariate images by the image of a known multiple of L. The solution used in the Macsyma implementation of Zippel’s algorithm is to use γ = gcd(lcx1 (A),lcx1 (B)). Now L divides γ hence γ = ∆ × L for some ∆ in [x2, ..., xn ]. If ∆ = 1, then this approach works very well. However, if ∆ is a non-trivial polynomial then Zippel’s algorithm would reconstruct ∆ × G which might have many more terms than G and it would need to remove ∆ from ∆ × G which would require another gcd computation. A non-trivial example where this will happen is when computing gcd(A, A0 ), the first gcd computation in a multivariate square-free factorization algorithm. An ingenious solution is presented by Wang in [9]. Wang determines L by factoring one of the leading coefficients of the input polynomials, A say, then heuristically determining ¯ If A and which factors belong to G and which belong to A. B are sparse, the factorization is usually not hard. Kaltofen in [4] shows how to reduce the factorization to a bivariate factorization and how to make Wang’s heuristic work for coefficient rings other than . We now present our solutions. Neither requires any factorizations.

rithm assumes that G1 is of the correct form, that is, all non-zero terms of the gcd G are present in G1 . This will be true with high probability if the primes are sufficiently large and the evaluation points are chosen at random from p. We identify three classes of primes and evaluation points which cause problems in the algorithm. Definition 1. (bad prime and evaluation) A prime p is bad if degx1 (G mod p) < degx1 (G). An evaluation point (α1 , ..., αn−1 ) ∈ n−1 is bad if degx1 (G mod I) < p degx1 (G) where I = hx2 − α1 , ..., xn − αn−1 i. For example, if A = (3yx + 1)(x − y) and B = (3yx + 1)(x + y + 1) then 3 is a bad prime and y = 0 is a bad evaluation point. These must be avoided so that the univariate images can be scaled consistently and so that unlucky primes and evaluations can be detected. We may avoid them by choosing p and (α1 , ..., αn−1 ) such that L(α1 , ..., αn−1 ) 6≡ 0 mod p where L = lcx1 A is the leading coefficient of A. Definition 2. (unlucky prime and evaluation) A prime p is unlucky if the cofactors are not relatively prime ¯ mod p, B ¯ mod p)) > 0. Simimodulo p, i.e., degx1 (gcd(A larly an evaluation point (α1 , ..., αn−1 ) ∈ n−1 is unlucky if p ¯ mod I, B ¯ mod I)) > 0 where I = hx2 − α1 , ..., degx1 (gcd(A xn − αn−1 i.

3. ALGORITHM LINZIP In Zippel’s algorithm, if any coefficient of G with respect to the main variable x1 is a monomial in x2 , ..., xn , then the normalization is straightforward. Consider the gcd problem from example 1. Notice that the O(x0 ) term in our first gcd image G1 = (y + 11)x3 + 9y has a single term coefficient, 9y. Since we know the exact form, we can scale our univariate gcd images based on this term. Our assumed form becomes Gf = (αy + β)x3 + (1)y for some α, β ∈ 17. Now we have two unknowns in our O(x3 ) term so we need two evaluation points, neither of which may be 0. We choose y = 5, 7, to get the univariate gcds x3 + 6 (mod 17) and x3 + 9 (mod 17), respectively. Now we scale the first univariate gcd by 65 (mod 17), and the second by 79 (mod 17) before equating, giving (5α + β)x3 + 5 = 15x3 + 5 and (7α + β)x3 + 7 = 14x3 + 7. Solving for α and β in 17, gives us the bivariate image, (8y + 9)x3 + y, which is a scalar multiple of the gcd modulo 17. Thus if (at any level in the recursion) an image has a coefficient in x1 which is a single term, the normalization problem is easily solved. The normalization problem essentially reduces to scaling of the univariate gcd images so that the solution of the linear system produces a correct scalar multiple of the gcd. The approach followed now for the general case is quite simple in concept. It is to treat the scaling factors of the computed univariate gcds as unknowns as well. This results in larger linear systems that may require additional univariate gcd images. We call this the multiple scaling case (as opposed to the single scaling case). Scaling of both the univariate gcds and the coefficients of the assumed form of the multivariate gcd results in a system that is under-determined by exactly 1 unknown (the computation is only determined up to a scaling factor). We fix the scaling factor of the first image to 1. The following example illustrates this approach.

¯ = 7x + 6y and B ¯ = 12x + y then p = 5 is For example, if A an unlucky prime and y = 0 is an unlucky evaluation point. These must be avoided if G is to be correctly reconstructed. Unlike bad primes and bad evaluation points, they cannot be ruled out in advance. Instead they identify themselves as we encounter a univariate image in x1 of higher degree than previous univariate images. Definition 3. (missing terms) A prime p is said to introduce missing terms if any integer coefficient of G vanishes modulo p. Similarly, an evaluation xn = α is said to introduce missing terms if any coefficient in p[xn ] of G vanishes at xn = α. For example, if G = x2 + 5y 3 x + 35 ∈ [x,y], the primes 5 and 7 and the evaluation y = 0 cause terms in G to vanish. Zippel’s algorithm cannot reconstruct G if it uses them. Example 1. (the normalization problem) Consider computing the non-monic bivariate gcd G = (y+ 50)x3 + 100y ∈ [x,y] from input polynomials A = (x − y + 1) G and B = (x + y + 1) G. Here G has leading coefficient y + 50 in the main variable x. Suppose we compute our first bivariate image modulo p1 = 13 and obtain G1 = (y+11)x3 + 9y (mod 13). We proceed to compute a second image using sparse interpolation working modulo 17. We assume G has the form Gf = (y + α)x3 + βy for some α, β ∈ 17. We have at most one unknown per coefficient in x so we evaluate at one random point, y = 5, and compute the univariate gcd x3 + 6 (mod 17). This image is unique up to a scaling factor m. We evaluate Gf at y = 5 and equate to obtain: (5 + α)x3 + 5β = m(x3 + 6). The normalization problem is to determine m. In our example, if we knew L(y) = y + 50, the leading coefficient of G, then m should be L(5) = 4 (mod 17) and we would have (5 + α)x3 + 5β = 4x3 + 7. Solving for α and β in 17, we would obtain G2 = (y + 16)x3 + 15y the second bivariate image.

Example 2. Consider the computation of the bivariate gcd (3y 2 − 90)x3 + 12y + 100. We obtain g ≡ x3 y 2 + 9x3 + 4y + 3 (mod 13), and assumed form of the gcd gf = αx3 y 2 + βx3 + γy + σ. Instead of computing two univariate gcd images for the new prime p2 = 17, we compute three,

Let L = lcx1 (G) be the leading coefficient of G. One solution to the normalization problem is to multiply the monic

125

choosing y = 1, 2, 3 and obtain the gcds x3 + 12, x3 + 8, and x3 respectively. We form the modified system as follows: αx3 + βx3 + γ + σ 4αx3 + βx3 + 2γ + σ 9αx3 + βx3 + 3γ + σ

Example 4. Consider the computation of the bivariate gcd (y + 2)x3 + 12y 2 + 24y. By a call to algorithm P we obtain our first image, g1 = x3 y+2x3 +12y 2 +11y (mod 13), and assumed form of the gcd , gf = αx3 y + βx3 + γy 2 + σy. We require at least three univariate gcd images for the new prime p2 = 17. Choosing y = 1, 2, 3 we obtain the gcds x3 + 12, x3 + 7, and x3 + 2 respectively, and form the modified system as follows:

m1 (x3 + 12) = x3 + 12, m2 (x3 + 8), m3 (x3 ),

= = =

where m2 , m3 are the new scaling factors, and we have set the first scaling factor m1 to 1. Solving this system yields α = 7, β = 11, γ = 11, σ = 1, with scaling factors m2 = 5, m3 = 6, so our new gcd image is given by g ≡ 7x3 y 2 + 11x3 + 11y + 1 (mod 17) which is consistent with our gcd.

αx3 + βx3 + γ + σ 2αx3 + βx3 + 4γ + 2σ 3αx3 + βx3 + 9γ + 3σ

4αx3 + βx3 + 16γ + 4σ









c c c c c c c c c



















1 c c c c c c c c c

c

1 c


































1






c

c c c c c c c c c








a2 a1 a0 b2 b1 b0 c1 c0 1 m2 m3






What is not necessarily obvious from example 4 is the cause of the failure, which is the presence of a content in the gcd with respect to the main variable x, namely y + 2. In example 4 the content in y is absorbed into the multipliers, so we are unable to obtain a solution for the coefficients in our candidate form as only the relative ratio between terms can be computed. Unfortunately, even if g, the gcd of a and b has no content, certain choices of primes and evaluation points can cause an unlucky content to appear in the gcd.




































= 


















0 0 0 0 0 0 0 0 0 0 0

Definition 4. (Unlucky Content). Given g ∈ [x1, ...,xn ] with contx1 (g) = 1, a prime p is said to introduce an unlucky content if cont x1 (g mod p) 6= 1. Similarly for g ∈ p[x1 , ..., xn ] with contx1 (g) = 1, an evaluation xi = αi is said to introduce an unlucky content if contx1 (g mod hxi − αi i) 6= 1. Consider, for example, g = x(y + 1) + y + 14. If we choose p = 13 then g mod p has a content of y + 1, while for any other prime no content is present. Since unlucky contents are rare, we design the algorithm so this problem is not detected in advance, but rather through its effect, so that its detection does not become a bottleneck of the algorithm. We now present the LINZIP M algorithm, which computes the gcd in [x1, ..., xn ] from a number of images in p[x1 , ..., xn ], and the LINZIP P algorithm, which computes the gcd in p[x1 , ..., xn ] from a number of images in p[x1 , ..., xn−1 ].

 

= m4 (x3 + 14).

The new system of equations is still under-determined. In fact, the system remains under-determined if we continue to choose new evaluation points for y. This is the case for any chosen prime and set of evaluations, so the algorithm fails to find the gcd for this problem.

Example 3. Consider the linear system that must be solved to compute the gcd for a problem with the assumed form gf = (a2 y 2 + a1 y + a0 )x2 + (b2 y 3 + b1 y + b0 )x + (c1 y 2 + c0 ). We require 3 images to have sufficiently many equations to solve for all unknowns. The resulting linear system has 9 equations in the 10 unknowns. It has the structure shown in figure 1 below. The equations are ordered by decreasing degree in x then by image number. The unknowns are in the same order as in the image followed by the scaling factors. The c’s denote (possibly) non-zero entries. All entries not shown are zero. 

x3 + 12, m2 (x3 + 7), m3 (x3 + 2),

In attempting to solve this system, we find that it is underdetermined, so we add a new evaluation point, y = 4, obtaining a gcd of x3 + 14, and the new equation

We explain why we fix the multiplier m1 to be 1 instead of fixing a coefficient in the assumed form of the gcd to be 1. In example 2, suppose we set α = 1. Suppose we then use the evaluation y = 0 which is not bad. Notice that α should be 0 if y = 0. Attempting to set β = 1 will have the same problem for the prime p = 5 which is not bad. Fixing a multiplier to be 1 cannot cause an inconsistency because the algorithm avoids bad primes and bad evaluation points. One might wonder about the efficiency of the multiple scaling case since we are constructing a system that ties together all unknowns through the multipliers. In the single scaling case, each degree in x1 has an independent subsystem. The trick is to realize that the resulting system is highly structured, and the structure can be exploited to put the solution expense of the multiple scaling case on the same order as the solution expense of the single scaling case.




= = =























Figure 1: Structure for the multiple scaling case.

Algorithm 1 (LINZIP M). Input: a, b ∈ [x1, ..., xn ] with gcd(contx1 a, contx1 b) = 1 and degree bounds dx on the gcd in x1 , ..., xn .

The solution can be easily computed by solution of a number of smaller subsystems corresponding to the rectangular blocks of non-zero entries augmented with the multiplier columns. Once the subsystems are upper triangular, remaining rows, only involving the multipliers, can be used to compute the multiplier values, which can then be back-substituted into the subsystems to obtain the image coefficients.

Output: g = gcd(a, b) ∈

[x1, ..., xn ].

1 Compute the scaling factor: γ = gcd(lcx1 ,...,xn (a), lcx1 ,...,xn (b)) ∈

.

2 Choose a random prime p such that γp = γ mod p 6= 0, and set ap = a mod p, bp = b mod p, then compute from these a modular gcd image gp ∈ p[x1 , ..., xn ] with a call to LINZIP P. If the algorithm returns Fail, repeat, otherwise set dx1 = degx1 (gp ) and continue.

This approach solves the normalization problem but it also introduces another difficulty which is illustrated by the following example.

126

gc . Test if gc | a and gc | b. If yes, return gc . Otherwise we need more primes, so goto 5.1.

3 Assume that gp has no missing terms, and that the prime is not unlucky. We call the assumed form gf . There are two cases here:

Algorithm 2 (LINZIP P). Input: a, b ∈ p[x1 , ..., xn ], a prime p, and degree bounds dx on the gcd in x1 , ..., xn .

3.1 If there exists a coefficient of x1 in gf that is a monomial, then we can use single scaling and normalize by setting the integer coefficient of that monomial to 1. Count the largest number of terms in any coefficient of x1 in gf , calling this nx .

Output: g = gcd(a, b) ∈

γp lcx1 ,...,xn (gp )

or Fail.

0 If the gcd of the inputs has content in xn return Fail.

3.2 If there is no such coefficient, then multiple scaling must be used. Compute the minimum number of images needed to determine gf with multiple scaling, calling this nx . 4 Set gm =

p[x1 , ..., xn ]

1 Compute the scaling factor: γ = gcd(lcx1 ,...,xn−1 (a), lcx1 ,...,xn−1 (b)) ∈

p[xn ].

2 Choose v ∈ p \ {0} at random such that γ mod hxn − vi 6= 0. Set av = a mod hxn − vi, bv = b mod hxn − vi, then compute gv = gcd(av , bv ) ∈ p[x1 , ..., xn−1 ] with a recursive call to LINZIP P (n > 2) or via the Euclidean algorithm (n = 2). If for n > 2 the algorithm returns Fail or for n = 2 we have deg x1 (gv ) > dx1 then return Fail, otherwise set dx1 = degx1 (gv ) and continue.

× gp mod p and m = p.

5 Repeat 5.1 Choose a new random prime p such that γp = γ mod p 6= 0, and set ap = a mod p, bp = b mod p. 5.2 Set S = ∅, ni = 0.

3 Assume that gv has no missing terms, and that the evaluation is not unlucky. We call the assumed form gf . There are two cases here:

5.3 Repeat 5.3.1 Choose α2 , ..., αn ∈ p \ {0} at random such that degx1 (ap mod I) = deg x1 (a), degx1 (bp mod I) = degx1 (b) where I = hx2 − α2 , ..., xn − αn i. Set a1 = ap mod I, b1 = bp mod I.

3.1 If there exists a coefficient of x1 in gf that is a monomial, then we can use single scaling and normalize by setting the integer coefficient of that monomial to 1. Count the largest number of terms in any coefficient of x1 in gf , calling this nx .

5.3.2 Compute g1 = gcd(a1 , b1 ). 5.3.3 If deg x1 (g1 ) < dx1 our original image and form gf and degree bounds were unlucky, so set dx1 = degx1 (g1 ) and goto 2.

3.2 If there is no such coefficient, then multiple scaling must be used. Compute the minimum number of images needed to determine gf with multiple scaling, calling this nx .

5.3.4 If deg x1 (g1 ) > dx1 our current image g1 is unlucky, so goto 5.3.1, unless the number of failures > min(2, ni ), in which case assume p is unlucky and goto 5.1.

4 Set gseq =

5.3.5 For single scaling, check that the scaling term in the image g1 is present. If not, the assumed form must be wrong, so goto 2.

γ(v) lcx1 ,...,xn−1 (gv )

× gv mod p and vseq = v.

5 Repeat 5.1 Choose a new random v ∈ p \ {0} such that γ mod hxn − vi 6= 0 and set av = a mod hxn − vi, bv = b mod hxn − vi.

5.3.6 Add the equations obtained from equating coefficients of g1 and the evaluation of gf mod I to S, and set ni = ni + 1.

5.2 Set S = ∅, ni = 0. 5.3 Repeat 5.3.1 Choose α2 , ..., αn−1 ∈ p \ {0} at random such that deg x1 (av mod I) = degx1 (a) and degx1 (bv mod I) = deg x1 (b) where I = hx2 − α2 , ..., xn−1 −αn−1 i. Set a1 = av mod I, b1 = bv mod I.

Until ni ≥ nx . 5.4 We may now have a sufficient number of equations in S to solve for all unknowns in gf mod p so attempt this now, calling the result gp . 5.5 If the system is inconsistent our original image is incorrect (missing terms or unlucky), so goto 2.

5.3.2 Compute g1 = gcd(a1 , b1 ).

5.6 If the system is under-determined, then record the degrees of freedom, and if this has occurred twice before with the same degrees of freedom then assume that an unlucky content problem was introduced by the current prime p so goto 5.1. Otherwise we need more images so goto 5.3.1.

5.3.3 If degx1 (g1 ) < dx1 then our original image and form gf and degree bounds were unlucky, so set dx1 = degx1 (g1 ) and goto 2. 5.3.4 If deg x1 (g1 ) > dx1 then our current image g1 is unlucky, so goto 5.3.1, unless the number of failures > min(1, ni ), in which case assume xn = v is unlucky and goto 5.1.

5.7 The system is consistent and determined. Scale γp the new image. Set gp = lcx ,...,x ×gp mod p. n (gp ) 1 Apply the Chinese remainder theorem to update gm by combining the coefficients of gp ∈ p[x1 , ..., xn ] with gm ∈ m[x1 , ..., xn ], updating m = m×p.

5.3.5 For single scaling, check that the scaling term in the image g1 is present. If not, the assumed form must be wrong, so goto 2. 5.3.6 Add the equations obtained from equating coefficients of g1 and the evaluation of gf mod I to S, and set ni = ni + 1.

Until gm has stopped changing for one iteration. 7 Remove integer content from gm placing the result in

127

Until ni ≥ nx .

this approach only requires computation of univariate contents to detect the problem as any content in the gcd will eventually show up as a univariate content as we evaluate xn , xn−1 , ....

5.4 We should now have a sufficient number of equations in S to solve for all unknowns in gf mod p so attempt this now, calling the result gv .

4. The check in step 5.6 of either algorithm is intended to check for an unlucky content introduced by the evaluation (LINZIP P ) or prime (LINZIP M ) chosen in step 5.1 of both algorithms. Since it is possible that a new random image from step 5.3.1 does not constrain the form of the gcd (even without the content problem) we check for multiple failures before rejecting the current iteration of loop 5.

5.5 If the system is inconsistent our original image is incorrect (missing terms or unlucky), so goto 2. 5.6 If the system is under-determined, then record the degrees of freedom, and if this has occurred twice before with the same degrees of freedom then assume the content problem was introduced by the evaluation of xn so goto 5.1. Otherwise we need more images so goto 5.3.1.

5. The LINZIP P algorithm performs one probabilistic univariate division test in step 7 instead of testing if gc | a and gc | b. This check is substantially less expensive than a multivariate trial division, though there is still a chance that the test fails to detect an incorrect answer, so the termination division test in LINZIP M must be retained.

5.7 The system is consistent and determined. Scale γ(v) the new image gv . Set gseq =gseq , lcx ,...,x (g ) 1 n−1 v × gv , vseq = vseq , v. Until we have dxn + degxn (γ) + 1 images. 6 Reconstruct our candidate gcd gc using Newton interpolation (dense) on gseq ,vseq , then remove any content in xn .

6. Random evaluation points are chosen from p\{0} rather than p because zero evaluations are likely to cause missing terms in the assumed form, and possibly scaling problems when normalizing images.

7 Probabilistic division test: Choose α2 , ..., αn ∈ p at random such that for I = hx2 − α2 , ..., xn − αn i and g1 = gc mod I we have degx1 (g1 ) = degx1 (gc ). Then compute a1 = a mod I, b1 = b mod I and test if g1 | a1 and g1 | b1 . If yes return gc , otherwise goto 2.

To verify the correctness of this algorithm, in addition to the standard issues with modular algorithms we need also verify that the images are scaled consistently to allow the image reconstruction to proceed. We need to consider 4 main problems, namely bad primes or evaluations, unlucky contents, unlucky primes or evaluations, and missing terms in an initial image.

We make some remarks before discussing the correctness and termination of the algorithm. 1. The degree bound of the gcd in the main variable x1 is used to detect unlucky primes and evaluations, but only those that involve x1 . We update this degree bound whenever we compute a gcd of lower degree in x1 . The degree bounds of the gcd in the non-main variables x2 , ..., xn are used to compute the number of images needed for the interpolation in step 6 of LINZIP P. They are not updated by the algorithm. The degree bound for a variable can be obtained by evaluating the inputs mod a random prime and set of evaluations for all but that variable, then as long as the prime and evaluations are not bad, the degree of the univariate gcd is a bound on the degree of the multivariate gcd for that variable.

Bad primes and bad evaluations: The treatment of bad primes and bad evaluations is straightforward. It is handled for the first prime or evaluation by the check that γ does not evaluate to 0 in step 2 of the algorithms, handled for subsequent primes or evaluations by the check that γ does not evaluate to 0 in step 5.1 of the algorithms, and handled for the univariate images in step 5.3.1 of the algorithms. Unlucky content: The unlucky content problem for the first prime or first evaluation is treated in step 0 of LINZIP P by the single variable content check. As in point 3 above we emphasize that this check will always detect the problem at some level of the recursion, specifically the level containing the last variable contained in the unlucky content (as all the other variables in the content have been evaluated, so the content becomes univariate). There is no efficient way to detect where such an unlucky content was introduced. It may have been introduced by the prime chosen in LINZIP M or any evaluation in prior calls (for xj with j > n) to LINZIP P in the recursion. Thus LINZIP P fails all the way back up to LINZIP M which restarts with a new prime. This strategy is efficient, as only evaluations (modular and variable) and other single variable content checks have been performed before such a failure is detected. The introduction of an unlucky content by the prime or evaluation chosen in step 5.1 of either algorithm will be handled in the combination of steps 5.4 and 5.6. The result is a system with additional degrees of freedom, so this always results in an under-determined system. The check in step 5.6 handles this, as eventually we will obtain a solution for all variables but the free ones resulting from the unlucky

2. The number of required images for the multiple scaling case computed in step 3.2 can be the same as the number of required images for the single scaling case computed in step 3.1, and no more than 50% higher. The worst case is quite infrequent. It will only occur when there are only two coefficients with respect to the main variable, each having exactly the same number of terms. The extra expense of this step can usually be reduced by an intelligent choice of the main variable x1 . The exact formula for the number of images needed for a problem with coefficients having term counts of n1 , ..., ns and a maximum term count of nmax is given by max(nmax , d( si=1 ni − 1)/(s − 1)e). The complexity of LINZIP is otherwise the same as that of Zippel’s original algorithm. For a detailed asymptotic analysis of the LINZIP algorithm, the interested reader may consult [11]. 3. The check in step 0 of LINZIP P is used to detect an unlucky content in the initial gcd introduced higher up in the recursion by either a prime or evaluation. We note that

128

detect this problem, we arrive at LINZIP M. Now we will compute new images in LINZIP M until gc divides both a and b, so the problem must eventually be detected. Note that the missing term case is the most likely failure case of both algorithms, that is, more likely than unlucky primes, unlucky evaluations, and unlucky contents. The probability of choosing a prime or evaluation that causes a term to vanish is O(t/p), where t is the number of terms in the polynomial, and p is the prime. Thus the primes need to be much larger than the number of terms.

content, so the degrees of freedom will stabilize, and we will go back to step 5.1 choosing a new prime or evaluation. Unlucky primes and unlucky evaluations: The treatment of unlucky primes and evaluations is less straightforward. First we consider an unlucky evaluation in step 2 of LINZIP P for xn for which the factor added to the gcd depends upon x1 . If the degree bound dx1 is tight, then this will be detected at a lower level of the recursion by step 2 of LINZIP P when n = 2. If the degree bound dx1 is not tight, then the gcd computed in that step may be unlucky, but we proceed with the computation. Once we reach loop 5, we begin to choose new evaluation points for xn . With high probability we will choose a new point that is not unlucky in step 5.1, the problem will be detected in step 5.3.3. In the worst case, all evaluations in step 5.1 may also be unlucky, introducing the same factor to the gcd, and we will proceed to step 6, and reconstruct an incorrect result. Note that if the factor is in fact different, then the equations accumulated in step 5.3.5 will most likely be inconsistent, and this problem will most likely be detected in steps 5.4 and 5.5. Step 7 will again perform checks much like those in step 5.3.3, and will detect this problem with high probability, but if it does not, an invalid result may be returned from LINZIP P. If we continue to choose unlucky evaluations we will eventually return an incorrect image to LINZIP M. This problem (as well as the unlucky prime case for step 2 of LINZIP M ) is handled by the structure of LINZIP M. Since the steps are essentially the same, the same reasoning follows, and we need the computation to be unlucky through all iterations of loop 5. Since the form of the gcd is incorrect, it is unlikely that gm will stabilize, and we will continue to loop. If gm does stabilize, the invalid image will not divide a and b, so step 7 will put us back into the loop. Now within that loop, which cannot terminate until we have found the gcd, step 5.3.4 will eventually detect this problem, as we must eventually find a prime that is not unlucky. Now consider the case where the unlucky evaluation or prime is chosen in step 2 of either algorithm, and the factor added to the gcd is independent of x1 , that is, it is a content with respect to x1 . This is handled by the same process as the unlucky content problem, specifically it is handled on the way down by step 0 of LINZIP P. Now if an unlucky prime or evaluation occurs in step 5.1 of either algorithm, it must either raise the degree in x1 , in which case it will be detected in step 5.3.4, or it results in an unlucky content. If this content is purely a contribution of the cofactors, then this will not cause a problem for the algorithm, as it reconstructs the new gcd image without that content present (as a result of the assumed form). Hence, the only type of unlucky evaluation that can occur in step 5.3.1 must raise the degree of the gcd in x1 , and thus is handled by step 5.3.4.

4. ALGORITHM RATZIP An alternative way of handling the non-monic case is to use sparse rational function interpolation. The idea is as follows. Suppose we are computing the gcd of two polynomials in [x,w, y, z] with x as the main variable. We will compute the monic gcd in (w,y, z)[x] in the form: n−1

xn + i=0

ai (w, y, z) i x, bi (w, y, z)

where ai , bi ∈ [w, y, z], by interpolating the rational function coefficients using a sparse interpolation. For example, if our gcd is (y + 14)yx3 + 12y 2 x + y + 14, we compute the monic gcd 12y 1 x3 + x+ . y + 14 y We then recover the non-monic gcd by multiplying through by the least common multiple of the denominators. In our example, we multiply through by lcm(y + 14, y) = (y + 14)y to get our non-monic gcd (y + 14)yx3 + 12y 2 x + y + 14. To illustrate how sparse rational function reconstruction works in general, suppose one of the rational function co3 +∗zy 2 efficients is C = ∗z∗w 2 +∗y 2 +wy 3 , here ∗ indicates an integer. Suppose we have reconstructed C at w = 5 to get 2 C1 = ∗z2∗+∗zy . Notice we have normalized the lead+∗y 2 +y 3 ing coefficient of the denominator to be 1, essentially dividing through by w. We then assume the form to be α(w)+β(w)zy 2 Cf = δ(w)z 2 +γ(w)y 2 +y 3 , where α(w), β(w), δ(w), γ(w) are rational functions in w. We have 4 unknowns so we need 4 equations to solve for the next image, C2 . We do this for as many w values as we need, then perform rational function interpolation in w to obtain

∗ zy 2 ∗w2 + w ∗ z 2 + ∗ y 2 +y 3 w w

fractions in w gets us what we want, namely

. Clearing the ∗w3 +∗zy 2 . ∗z 2 +∗y 2 +wy 3

Example 5. Consider the computation of the above gcd G = (y + 14)yx3 + 12y 2 x + y + 14 from input polynomials A = (yx + 1) G and B = (yx + 2) G. Using p1 = 11 we compute our first monic gcd image in 11(y)[x] using dense rational function interpolation. Given a degree bound in y, dy = 2, we need N = 2dy + 1 = 5 evaluation points to ay 2 +by+c interpolate a rational function of the form dy 2 +ey+f in y. If we do this by constructing a linear system, the rational function interpolation will cost O(N 3 ). Instead we use the Euclidean Algorithm. We first apply the Chinese Remainder Theorem to reconstruct polynomial coefficients in y followed by rational function reconstruction (see [2]). This reduces the cost to O(N 2 ). We choose y = 1, 4, 9, 3, 6, to get the gcd images in 11[x], x3 + 3x + 1, x3 + 10x + 3, x3 + 9x + 5, x3 + 6x + 4 and x3 + 8x + 2, respectively. We

Missing terms: If the initial image of g (in either algorithm) has missing terms, the resulting system will likely be inconsistent which will be detected by step 5.5 with high probability. If it is not detected in any iteration of loop 5, then an incorrect image will be reconstructed in step 6 of LINZIP P. The additional check in step 7 of LINZIP P will detect this problem with the new images with high probability, but if this also fails, then we return an incorrect image from LINZIP P. Again assuming a sequence of failures to

129

interpolate in y to get x3 + (6y 4 + 9y 3 + 9y 2 + 10y + 2)x + 10y 4 + y 3 + 5y 2 + 3y + 4 and then apply rational function reconstruction to the coefficients of x to get our first monic y gcd image G1 = x3 + y+3 x+ y1 ∈ 11(y)[x], and our assumed αy x + yδ . form Gf = x3 + y+β Working modulo p2 = 13 we compute a second monic gcd image in 13(y)[x] using sparse rational function interpolation. We have at most two unknowns per coefficient in our main variable x so we need two evaluation points. We evaluate at y = 1, 6, and compute the univariate gcd images in 13[x], x3 + 6x + 1 and x3 + x + 11, respectively. We evaluate Gf at our chosen y values and equate by coefficient to get the following system. 6 1

= =

α , 1+β 6α , 6+β

1 11

= =

δ 1 δ 6

For each coefficient of x1 in gf , count the number of terms in the numerator nt and the number of terms in the denominator dt. Take the maximum sum nt + dt over all coefficients and set nx = nt + dt − 1. The −1 is because we normalize the leading coefficients of the denominators to be 1.

4

5 Repeat 5.1 Choose a new random v ∈ p \ {0} such that γ mod hxn − vi 6= 0 and set av = a mod hxn − vi, bv = b mod hxn − vi.

⇒ α = 12, β = 1, δ = 1

5.2 Set S = ∅, ni = 0.

Substituting back into Gf we get our second monic image in 12 y 3 1 13(y)[x], G2 = x + y + 1 x + y . We then apply the Chinese Remainder Theorem to the integer coefficients of the rational functions of G1 and G2 y to reconstruct our monic gcd in (y)[x], x3 + y12 x + y1 . + 14 Clearing fractions gives us our non-monic gcd in [x,y].

5.3 Repeat 5.3.1 Choose α2 , ..., αn−1 ∈ p \ {0} at random such that deg x1 (av mod I) = degx1 (a) and degx1 (bv mod I) = deg x1 (b) where I = hx2 − α2 , ..., xn−1 −αn−1 i. Set a1 = av mod I, b1 = bv mod I.

Algorithm RATZIP M computes the gcd in [x1, ..., xn ] from a number of images in p(x2 , ..., xn )[x1 ]. After applying the Chinese remainder theorem it must clear the fractions in (x2, ..., xn )[x1 ] which requires further multivariate gcds which is the disadvantage of this algorithm in comparison with LINZIP. Algorithm RATZIP P computes the gcd in p(x2 , ..., xn )[x1 ] from a number of images in p(x2 , ..., xn−1 )[x1 ]. As for the LINZIP M algorithm any content of the gcd with respect to x1 must be removed before the initial call to the RATZIP M algorithm. Unlike in the LINZIP algorithms, we do not use single scaling. It is plausible that it may be applied here but it is not straightforward and we have yet to work out the details. For lack of space, only subroutine RATZIP P is presented. Since it is similar to LINZIP P, the differences are highlighted. Algorithm 3

Set gm = gv , m = xn − v, and Ni = 1.

5.3.2 Compute g1 = gcd(a1 , b1 ). 5.3.3 If degx1 (g1 ) < dx1 then our original image and form gf and degree bounds were unlucky, so set dx1 = degx1 (g1 ) and goto 2. 5.3.4 If deg x1 (g1 ) > dx1 then our current image g1 is unlucky, so goto 5.3.1, unless the number of failures > min(1, ni ), in which case assume xn = v is unlucky and goto 5.1. 5.3.5 Add the equations obtained from equating coefficients of g1 and the evaluation of gf mod I to S, and set ni = ni + 1. Until ni ≥ nx . 5.4 We should now have a sufficient number of equations in S to solve for all unknowns in gf mod p so attempt this now, calling the result gv .

(RATZIP P).

Input: a, b ∈ p[x1 , ..., xn ], a prime p, and degree bounds dx on the gcd in x1 , ..., xn .

5.5 If the system is inconsistent our original image is incorrect (missing terms or unlucky), so goto 2.

Output: g = gcd(a, b) ∈

5.6 If the system is under-determined, then record the degrees of freedom, and if this has occurred twice before with the same degrees of freedom then assume the content problem was introduced by the evaluation of xn so goto 5.1. Otherwise we need more images so goto 5.3.1.

p(x2 , ..., xn )[x1 ]

or Fail.

0 If the gcd of the inputs has content in xn return Fail. 1 Compute the scaling factor: γ = gcd(lcx1 ,...,xn−1 (a), lcx1 ,...,xn−1 (b)) ∈

p[xn ].

If γ = 1 then set RR = F alse else set RR = T rue.

5.7 The system is consistent and determined, so we have a new image gv .

2 Choose v ∈ p \ {0} at random such that γ mod hxn − vi 6= 0. Set av = a mod hxn − vi, bv = b mod hxn − vi, then compute gv = gcd(av , bv ) ∈

Solve f ≡ gm mod m(xn ) and f ≡ gv mod (xn − v) using the Chinese remainder algorithm for f ∈ p[xn ](x2 , ..., xn−1 )[x1 ] mod m(xn )×(xn −v). Set gm = f, m = m(xn )× (xn − v), and Ni = Ni + 1.

p(x2 , ..., xn−1 )[x1 ]

with a recursive call to RATZIP P (n > 2) or via the Euclidean algorithm (n = 2). If for n > 2 the algorithm returns Fail or for n = 2 we have deg x1 (gv ) > dx1 then return Fail, otherwise set dx1 = degx1 (gv ) and continue.

Until

3 Assume gv has no missing terms, and that the evaluation is not unlucky. We call the assumed form gf .

Ni ≥ dxn + 1 and (RR = F alse or Ni ≥ 3).

6 Reconstruct

130

think of G as a polynomial in x1 and x2 (main variables) with coefficients in [x3, ..., xn ]. If the cost of computing a bivariate image is less than the cost of evaluation mod I = hx3 − α3 , ..., xn − αn i, overall efficiency is not compromised. If G mod I is dense in x1 and x2 then we expect a significant reduction in the maximum of the number of terms of the coefficients in x1 and x2 , hence, a reduction in the maximum size of the linear systems and a reduction in the number of images needed for the sparse interpolations. We also increase the likelihood of not needing to apply the multiple scaling or rational reconstruction methods. Furthermore, we simplify the multivariate gcd computation for the content of G and, in RATZIP M, the final lcm computation.

6.1 If RR = T rue then apply rational function reconstruction in xn and assign the result to gc . If it fails then we need more points, goto 5.1. For n > 2, clear the rational function denominators of gc ∈ p(xn )(x2 , ..., xn−1 )[x1 ] to obtain gc ∈ p(x2 , ..., xn )[x1 ]. 6.2 If RR = F alse then set gc = gm . 7 Probabilistic division test: Choose α2 , ..., αn ∈ p at random such that for I = hx2 − α2 , ..., xn − αn i and g1 = gc mod I we have degx1 (g1 ) = degx1 (gc ). Then compute a1 = a mod I, b1 = b mod I and test if g1 | a1 and g1 | b1 . If yes return gc , otherwise goto 2.

6. REFERENCES [1] W. S. Brown. On Euclid’s Algorithm and the Computation of Polynomial Greatest Common Divisors. J. ACM 18, 478–504, 1971. [2] J. von zur Gathen and J. Gerhard. Modern Computer Algebra. Cambridge University Press, UK, 1999. [3] M. van Hoeij, M. B. Monagan. A Modular GCD Algorithm over Number Fields Presented with Multiple Field Extensions. P. ISSAC ’2002, ACM Press, 109–116, 2002. [4] E. Kaltofen. Sparse Hensel lifting. P. EUROCAL 85, Springer-Verlag LNCS 2, 4–17, 1985. [5] E. Kaltofen, W. Lee. Early Termination in Sparse Interpolation Algorithms. J. Symbolic Comp. 36 (3-4), 365–400, 2003. [6] E. Kaltofen and B. Trager. Computing with polynomials given by black boxes for their evaluations: Greatest common divisors, factorization, separation of numerators and denominators. J. Symbolic Comp. 9, 301–320, 1990. [7] M. B. Monagan. Maximal Quotient Rational Reconstruction: An Almost Optimal Algorithm for Rational Reconstruction. P. ISSAC ’2004, ACM Press, 243–249, 2004. [8] M. O. Rayes, P. S. Wang, K. Weber. Parallelization of The Sparse Modular GCD Algorithm for Multivariate Polynomials on Shared Memory Multiprocessors, P. ISSAC ’94, ACM Press, 66–73, 1994. [9] P. S. Wang. An Improved Multivariate Polynomial Factorization Algorithm. Math. Comp. 32 (144), 1215–1231, 1978. [10] P. S. Wang. The EEZ-GCD algorithm. ACM SIGSAM Bull. 14, 50–60, 1980. [11] A. D. Wittkopf, Algorithms and Implementations for Differential Elimination, Ph.D. Thesis, Simon Fraser University. (http://www.cecm.sfu.ca/~wittkopf/WittThesis.pdf), 2004. [12] R. Zippel, Probabilistic Algorithms for Sparse Polynomials, P. EUROSAM ’79, Springer-Verlag LNCS, 2, 216–226, 1979. [13] R. Zippel. Interpolating Polynomials from their Values. J. Symbolic Comp. 9 (3), 375–403, 1990. [14] R. Zippel, Effective Polynomial Computation, Kluwer Academic, 1993.

Our implementation of the RATZIP algorithm includes the following enhancement. To reconstruct the rational functions in some variable y with degree bound dy , we need 2dy + 1 evaluation points. In fact, we may need fewer points than this, depending on the form of the rational functions being reconstructed. In our implementation we use the Maximal Quotient Rational Reconstruction algorithm [7], which uses at most one more evaluation point than the minimum number of points required for the reconstruction to succeed. For example, to reconstruct the rational functions in y of y G = x3 + y+3 x + y1 , we would need 4 points, not 5.

5.

IMPLEMENTATION

We have implemented algorithm LINZIP in Maple and have compared it with Maple’s default algorithm, an implementation of the EEZ-GCD algorithm of Wang [10]. The linear algebra over p and univariate polynomial computations over p and the integer arithmetic are all coded in C. The rest is coded in Maple. Algorithm LINZIP is generally faster when the evaluation points used by the EEZ-GCD algorithm cannot be 0. It is also much less sensitive to unlucky primes and evaluations than the EEZ-GCD algorithm. Otherwise it is generally slower, sometimes more than a factor of 3 slower. We have also implemented algorithm LINZIP and RATZIP in Maple using the “recden” [3] data structure. This data structure supports multiple field extensions over and p, and hence, will allow us to extend our implementations to work over finite fields and algebraic number fields. The data structure is currently being implemented in the kernel of Maple for improved efficiency. On our data, the two algorithms perform within a factor of 2 of each other. Algorithm LINZIP is generally faster than RATZIP. A disadvantage of Zippel’s algorithm is the large number of univariate images that must be computed for the sparse interpolations which means a large number of evaluations of the inputs. On our test data we find that the percentage of time spent on evaluations was on average 68% and 75% for LINZIP and RATZIP, respectively. The multivariate trial division in LINZIP M (step 7) and RATZIP M took 19% and 11% of the time, respectively. To improve the efficiency of LINZIP and RATZIP, we are implementing the following idea. Instead of evaluating out all but one variable x1 in LINZIP P and RATZIP P, consider evaluating out all but 2 variables x1 , x2 and computing the bivariate images using a dense gcd algorithm. Thus

131

Computing µ-Bases of Rational Curves and Surfaces Using Polynomial Matrix Factorization Jiansong Deng

Falai Chen

Liyong Shen

Department of Mathematics University of Science and Technology of China Hefei, Anhui 230026, P. R. of China

[email protected],[email protected] ABSTRACT

equation of the rational curve/surface. Thus it provides a connection between the parametric form and the implicit form of a curve/surface. Furthermore, the µ-basis was successfully applied in reparametrizing a rational ruled surface [4], in computing the singular points of a rational curve [7] and in finding more compact representation for the implicit equation of a rational curve with high order of singularities [2]. There are several methods to compute the µ-basis of a rational curve. The first method is based on undetermined coefficients by solving linear system of equations [14]. This method needs O(n3 ) arithmetic operations, where n is the degree of the curve, and it is a trial-and-error approach. The second method was developed by Zheng and Sederberg [18], and it is similar to the Buchberger’s algorithm for computing the Gr¨ obner basis of a module. The computational cost of the method is about 81 n2 + O(n) multiplications in generic 4 case. In [3], Chen and Wang applied vector elimination technique to improve the efficiency of the second algorithm by a factor of two. For a rational ruled surface, an efficient algorithm similar to curve case was developed to compute the µ-basis [1]. However, we do not have a rigorous algorithm to compute the µ-basis of a general rational surface so far. Currently, we use the Gr¨ obner basis technique to compute a generator for the syzygy module of the rational surface, and then try to find the µ-basis by forming linear combinations of the elements in the generator. This is totally a non-automatic approach and fails in most circumstances. In this paper we apply the theory of polynomial matrices developed by researchers in linear systems [11, 12, 13] to the computation of a µ-basis. Using some polynomial matrix operations, such as primitive factorization and GCD extraction, we are able to compute a µ-basis of a rational curve/surface rigorously. The computed µ-basis is further simplified by lowering its degree using vector elimination technique [3]. For curve case, a µ-basis can be computed in 33 2 n + O(n) operations, which is superior than any existing 4 algorithms. The organization of the paper is as follows. In Section 2, some preliminary knowledge about the µ-basis of a rational curve or surface is introduced. In Section 3, some basic concepts and results in the theory of polynomial matrices are reviewed, including the primitive factorization algorithm, Hermite form, and GCD extraction algorithm. Sections 4 and 5 apply the results of Section 3 to the computation of the µ-basis of a rational curve and surface respectively. Some examples are illustrated to demonstrate the detailed process

The µ-bases of rational curves/surfaces are newly developed tools which play an important role in connecting parametric forms and implicit forms of the rational curves/surfaces. They provide efficient algorithms to implicitize rational curves/surfaces as well as algorithms to compute singular points of rational curves and to reparametrize rational ruled surfaces. In this paper, we present an efficient algorithm to compute the µ-basis of a rational curve/surface by using polynomial matrix factorization followed by a technique similar to Gaussian elimination. The algorithm is shown superior than previous algorithms to compute the µ-basis of a rational curve, and it is the only known algorithm that can rigorously compute the µ-basis of a general rational surface. We present some examples to illustrate the algorithm.

Categories and Subject Descriptors I.1.2 [Symbolic and Algebraic Manipulation]: Algorithms—Algebraic algorithms; I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling—Curve, surface, solid, and object representations

General Terms Algorithms

Keywords µ-basis, syzygy module, implicitization, primitive factorization algorithm, Hermite form, GCD extraction algorithm

1.

INTRODUCTION

The µ-basis was first introduced in [9] to provide a compact representation for the implicit equation of a rational parametric curve. Then it was generalized by one of the present authors to general rational surfaces [1, 5, 6]. The µ-basis can be used not only to recover the parametric equation of a rational curve/surface but also to derive the implicit

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. ISSAC’05, July 24–27, 2005, Beijing, China. Copyright 2005 ACM 1-59593-095-705/0007 ...$5.00. Copyright2005ACM1-59593-095-7/05/0007...5.00.

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of the algorithm. We end the paper with some conclusions and future works in Section 6.

2.

Definition 2. Given a rational parametric surface in homogeneous form: P(s, t) := (a(s, t), b(s, t), c(s, t), d(s, t)) ∈ K 4 [s, t],

µ-BASES OF RATIONAL CURVES AND

where a, b, c, d are relatively prime. A basis {p(s, t), q(s, t), r(s, t)} ⊂ K 4 [t] of the syzygy module Syz(a, b, c, d) is called a µ-basis of the surface P(s, t). If in addition, p, q, r satisfy

SURFACES Throughout the paper, we work over the field K of real numbers or rational numbers. K[x1 , . . . , xk ] = K[x] and K(x) are the polynomial ring and the rational function field, respectively, in variables x = (x1 , . . . , xk ) with coefficients in K. K m×l [x] denotes the set of m×l matrices with entries in K[x]. If m = 1, we write it in K l [x] for short. For any F := (f1 , . . . , fl ) ∈ K m×l [x], the set n

l X

Syz(F ) := (h1 , . . . , hl ) ∈ K l [x]

1. among all the bases of Syz(a, b, c, d), degt p + degt q + degt r is smallest, and 2. among all the bases of Syz(a, b, c, d) which satisfy item 1, degs p + degs q + degs r is smallest, then {p, q, r} is called a minimal µ-basis of P(s, t).

o

hi fi ≡ 0

Remark 2 The existence of µ-basis of a rational surface was proved in [6]. It can be also seen from Corollary 3.2 in the next section. However, except for parametrizations with no base points, standard computational methods only give generating sets for the syzygy module. The main task of the current paper is to describe how to compute a basis of this module, i.e., a µ-basis. Remark 3 From [6], the parametric equation of a rational surface can be recovered by the outer-product of a µ-basis, i.e.,

i=1

is a module over K[x], called a syzygy module [8]. If we can find a generating set {b1 , . . . , bm }, bi ∈ K l [x], of a syzygy module, then the matrix M = (b1 , . . . , bm ) is called the generating matrix of the syzygy module. It follows that F M = 0. A generating set of a module over K[x] is called a basis if the elements in the generating set are K[x]-linearly independent. If a module has a basis, then it is called a free module. Conditions for a syzygy module being free will be given in the next section. We just mention that, if k = 1 or 2, i.e., we are working with univariate or bivariate polynomials, then the syzygy module is free.

[p, q, r] = κ(a, b, c, d) for some nonzero constant κ in K. Here 0 [p, q, r] = 

Now we review the definitions of µ-bases of a planar rational curve [3] and a rational surface [6]. Consistent with the notation in [6, 3, 5, 1, 9], we use t and s, t as the variable names for univariate and bivariate cases, respectively. Definition 1 ([9]). Given a planar rational curve of degree n in homogeneous form:

p2 p3 p4

q2 q3 q4

r2 r3 r4

p1 p2 p4

q1 q2 q4

r1 r2 r4

,− ,−

p1 p3 p4

q1 q3 q4

r1 r3 r4

p1 p2 p3

q1 q2 q3

r1 r2 r3

, 1 A .

On the other hand, the implicit equation of a rational surface can be obtained by computing the Gr¨ obner basis for the ideal hp·v, q·v, r·vi : g N , where v = (x, y, z, 1), g ∈ K[s] is defined by ha, b, c, di ∩ K[s] = hgi and N is a sufficiently large integer. Though it is relatively more efficient than the method by direct computation of Gr¨ obner basis of the ideal hdx − a, dy − b, dz − c, dw − 1i ∩ K[x, y, z, s, t], finding more efficient method to derive the implicit representation from a µ-basis is a problem worthy of further investigation.

P(t) := (a(t), b(t), c(t)) ∈ K 3 [t], where max(degt a, degt b, degt c) = n and gcd(a, b, c) = 1. The syzygy module Syz(a, b, c) has a basis {p(t), q(t)} ⊂ K 3 [t] with degree µ and n − µ respectively, where µ 6 n2 . {p(t), q(t)} is called a µ-basis of the rational curve P(t). Remark 1 A µ-basis has the following properties [3]: (1) The µ-basis has the lowest possible degree among all the bases of the syzygy module Syz(a, b, c).

3. PRELIMINARY RESULTS IN THE THEORY OF POLYNOMIAL MATRICES

(2) The parametric equation of P(t) can be recovered from a µ-basis. In fact, for any basis p, q of Syz(a, b, c), we have

Given a matrix M in K m×m [x], its determinant det M is a polynomial in K[x]. If the polynomial is nonzero, the matrix M is nonsingular, otherwise it is singular. If the determinant det M is a nonzero constant in K, then we called M a unimodular matrix. A matrix M ∈ K m×l [x] is of rank r if there exists at least one minor of order r being nonzero polynomial, and all the minors of order r+1 being zero polynomials. We use rank M to denote the rank of M . If rank M = min(l, m), we call the matrix full-rank. For a nonsingular matrix M ∈ K m×m [x], we can calculate its inverse matrix, whose entries are in K(x). The inverse matrix is also in K m×m [x] if and only if M is unimodular.

[p, q] = κ(a, b, c) for some nonzero constant κ in K. (3) A basis {p(t), q(t)} of Syz(a, b, c) is a µ-basis if and only if degt (p(t)) + degt (q(t)) = n, and if and only if LCV(p(t)) and LCV(q(t)) are linearly independent. Here LCV(p(t)) is the leading coefficient vector of vector polynomial p(t) which is defined by LCV(p(t)) := (p1µ , p2µ , p3µ ) if we write p(t) = (p1µ , p2µ , p3µ )tµ + . . . + (p10 , p20 , p30 ). LCV(q(t)) is defined similarly. (4) The implicit equation of P(t) can be obtained by taking the resultant of p · v and q · v with respect to t, where v = (x, y, 1).

Definition 3. Let F ∈ K m×l [x] with m 6 l. Then F is said to be :

133

1. minor left prime (MLP) if all m × m minors of F are coprime.

on how to extract GCRDs from A and B. The approach consists of some important algorithms in the theory of bivariate polynomial matrices, including primitive factorization, Hermite form, and GCD extraction. We will review them in the following subsections.

2. factor left prime (FLP) if any polynomial decomposition F = F1 F2 in which F1 is square, then F1 is a unimodular matrix, i.e., det F1 = k0 ∈ K\{0}.

3.1 Primitive factorization algorithm

Minor right prime (MRP) and factor right prime (FRP) can be similarly defined.

For a bivariate polynomial a(s, t) ∈ K[s][t], we write it into the following form:

In [16], the authors proved that for k = 1, 2, MLP ≡ FLP, MRP ≡ FRP; for k > 3, MLP 6≡ FLP, MRP 6≡ FRP; and for all k > 1, MLP ⇒ FLP, MRP ⇒ FRP. Here k is the number of variables.

a(s, t) =

where ai (s) ∈ K[s]. The content of a(s, t) with respect to K[s][t] is the gcd of the ai ’s. Suppose p(s) is irreducible in K[s]. Then a(s, t)

(mod p(s)) = 0

(i.e., p(s)|ai (s), i = 0, . . . , n), or a(s, t)

˜ ˜ (A = AC, B = BC),

(mod p(s)) =

n1 X

αi (s)ti ,

i=0

then C is a unimodular matrix.

where αi (s) ∈ K[s] with degs αi (s) < degs p(s), n1 6 n, and αn1 (s) 6≡ 0. The αi ’s can be obtained by means of the Euclidean division algorithm. In [13] a primitive factorization (PF) algorithm of bivariate polynomial matrices is proposed, which extracts the content of a full-rank matrix with entries in the ring K[s, t] of bivariate polynomials over some algebraically closed field K. [11] eliminates the restriction on K, such that we can do the factorization over the real field or even the field of rational numbers, provided the coefficients start out in the same field. We describe the PF algorithm as follows. Further details can be found in [11].

Definition 5. Given two polynomial matrices A, B with the same number of rows (columns), where entries are in K[x], a square polynomial matrix D is their greatest common left (right) divisor (GCL(R)D) if there exist two left ˜ B ˜ such that (right) coprime polynomial matrices A, ˜ B = DB ˜ A = DA,

ai (s)ti ,

i=0

Definition 4. Given two polynomial matrices A, B with the same number of rows (columns), where entries are in K[x], we call them to be left (right) coprime if whenever ˜ B ˜ and a square polynomial matrix there are two matrices A, C, where entries of all the three matrices are in K[x] such that ˜ B = CB ˜ A = C A,

n X

˜ ˜ (A = AD, B = BD).

The following theorem due to Lin [12] describes when a syzygy module is free, i.e., it has a basis. ˜ , D] ˜ ∈ K m×l [x] be of Theorem 3.1. [12] Let F = [−N m×m ˜ rank m, with l > m, D ∈ K [x] being nonsingular, and r = l − m. Then Syz(F ) has a generating matrix of dimension l × r (i.e., Syz(F ) is free) if and only if there exists an MRP matrix H ∈ K l×r [x] such that F H = 0m×r . Furthermore, H is the generating matrix.

Algorithm 1

(PF Algorithm).

Input F : an m×l full-rank matrix with entries in K[s, t], m 6 l. Output L, R: m × m and m × l matrices, respectively, with entries in K[s, t], such that F = LR and det L = g(s), where g(s) ∈ K[s] is the content of the greatest common divisor (GCD) of the set of m × m minors of F . Step

Based on Theorem 3.1, Lin derived the following corollary, the proof of which is constructive. We copy its proof to help the reader to more easily understand the algorithm in Sections 4 and 5.

1. Calculate the GCD of the set of m × m minors of F . g(s) is its content as a polynomial in t.

˜ , D] ˜ ∈ K m×l [s, t] be of Corollary 3.2. Let F = [−N m×m ˜ ∈ K rank m, with l > m, D [x] being nonsingular, and r = l − m. Then, there exists a generating matrix H ∈ K l×r [s, t] of Syz(F ).

2. Let L be an identity matrix of order m, and R = F . Factorize g(s) into a list of irreducible factors in K[s]. For every factor p(s) do the following steps:

˜. ˜ −1 N Proof. Associate F with a rational matrix P = D By a well-known result in bivariate polynomial matrix theory [11, 13], P has a right matrix fraction description (MFD) P = N D−1 , where N and D, whose entries are in K[s, t], are right coprime. Let H = (DT , N T )T ∈ K l×r [s, t], which ˜ −1 N ˜ = N D−1 gives rise is FRP, hence MRP. Clearly, P = D to F H = 0. By Theorem 3.1, H is a generating matrix of Syz(F ). According to the proof of Corollary 3.2, to find a basis of Syz(F ), we need to get the MFD of a rational matrix. For any rational matrix M , it is easy to write it into M = AB −1 , where A, B are polynomial matrices. Hence it is important

(a) Set the current row and column indices i, j to be ¯ = R (mod p(s)). 1. R ¯ if there exists a (b) Among rows from i to m in R, row with all entries zeros, say row i0 , then i0

D0 = diag(1, . . . , 1, p(s), 1, . . . , 1) is a left divisor of F , and we let L ← LD0 , and R ← D0−1 R. Continue Step 2 for the next factor. ¯ from rows i to m is zero, then go If no row of R to the next sub-step.

134

(c) From columns j to l, find the first column (say column j0 ) with at least one nonzero entry from rows i to m. Set j ← j0 .

13] is a matrix (aij (t)) or (aij (s, t)) with aij ≡ 0, j < i, and degt ajj > degt aij , j > i. [10] and [13] presented algorithms to compute the Hermite form of a full-rank matrix in univariate case and bivariate case respectively. For univariate case, based on Gaussian elimination technique and Euclid division algorithm, one can find a unimodular matrix U such that H = U F is the Hermite form of F . For bivariate case, the algorithm consists of two steps. First, we work over K(s)[t] to find U with entries ˜ =U ˜ F is a Hermite form with respect in K(s)[t] such that H ˜ ∈ K(s). Second, let pi (s, t) be the to K(s)[t], where det U ˜, least common multiple of the denominators in row i of U ˜ ˜. and D = diag(p1 (s, t), . . . , pm (s, t)), H = DH and U = DU Then it follows that H = U F ∈ K[s, t] is the Hermite form of F with respect to K[s][t], and det U ∈ K[s]. It is obvious that the two steps can be merged into one by using the pseudo division algorithm for two polynomials.

(d) In the column j, from rows i to m, find the entry with the smallest degree of t, say i1 . Interchange ¯ This is equivalent to premultirow i and i1 of R. ¯ with matrix D1 , where D1 comes from plying R Im by interchanging rows i and i0 . Let L ← LD1 , R ← D1−1 R = D1 R. (e) Suppose the entries in column j to be ∗, . . . , ∗, ai (s, t), . . . , am (s, t)


T

.

The leading coefficients of ai (s, t), . . . , am (s, t) in t are bi (s), . . . , bm (s). Since bi (s) and p(s) are relatively prime, by Euclidean algorithm we can find x(s) and y(s) in K[s] such that x(s)bi (s) = 1 − y(s)p(s).

3.3 GCD extraction algorithm

Let = x(z)ai (s, t) (mod p(s)).1 There exist qk (s, t) and rk (s, t) in K[s, t] such that

In [13], a GCD extraction algorithm of bivariate polynomial matrices is presented. We describe it as follows.

ak (s, t) = qk (s, t)a∗i (s, t) + rk (s, t),

Algorithm 2 (GCRD extraction algorithm). Input A, B: two bivariate polynomial matrices A(s, t) and B(s, t) with the same number of columns, such that (AT , B T ) is of full rank. Output D: GCRD of A and B. Step

a∗i (s, t)

k = i + 1, . . . , m, where degt rk (s, t) < degt a∗i (s, t), or rk (s, t) ≡ 0. Then for k = i + 1, . . . , m, we add to row k with row i multiplied by −x(s)qk (s, t). This is ¯ with the matrix equivalents to premultiplying R D3 = diag(Ii−1 , E), where 0 B B B E=B B 

1 −xqi+1 −xqi+2 .. . −xqm

1. Use the RPF algorithm with respect to K[s][t] on the ¯ B ¯ and R0 such right side of (AT , B T )T , i.e., find A, that     ¯ A A = ¯ R0 , B B

1 C C C C. C A

1 1 ..

.

where det R0 ∈ K[s].

1

2. Find U with entries in K[s, t] and det U ∈ K[s] to get ¯T , B ¯ T )T , i.e., the Hermite form of (A

LD3−1 ,

Let L ← R ← D3 R. Then the j column ¯ is with the form of R ∗, . . . , ∗, ai (s, t), ri+1 (s, t), . . . , rm (s, t)


T



U

.





=

R 0



.

3. Use the LPF algorithm to R,

¯←R ¯ mod p(z). Let R If ri+1 (s, t) ≡ · · · ≡ rm (s, t) ≡ 0, then j ← j + 1, i ← i + 1, and go to sub-step (b). Otherwise, repeat the current sub-step (e).

¯ ∗. R = RR Then D = R∗ R0 is the GCRD of A and B. Remark 6 To save some unnecessary primitive factorization in Step 3, we make a little modification to the above algorithm. In the computation of the Hermite form of a ˜ and U ˜ , let bivariate polynomial matrix, after we get H qi (s, t) be the least common multiple of the denominators ˜ then qi (s, t) is a factor of pi (s, t). Let D ¯ = in row i of H, ¯ =D ¯ H, ˜ and take H ¯ in place diag(q1 (s, t), . . . , qm (s, t)) and H of H in Step 3 of the GCD extraction algorithm.

Remark 4 There is of course a similar primitive factorization algorithm for m > l, where an l × l matrix is extracted on the right. We denote these two algorithms as the LPF and RPF algorithms with respect to K[s][t], respectively. Remark 5 The LPF and RPF algorithms terminate after finitely many steps. In fact, the complexity is predictable after given the degree of polynomials in F . For a given F , the factorization is unique up to a unimodular matrix.

4. COMPUTING µ-BASES OF A RATIONAL CURVE

3.2 Hermite form

Suppose we are given a planar rational curve

Given a univariate or bivariate m × l full-rank polynomial matrix F , m > l, we are interested in finding its Hermite form with respect to K[t] or K[s][t]. The Hermite form [10, 1

¯ A ¯ B

P(t) := (a(t), b(t), c(t)), where a, b, c ∈ K[t] and gcd(a, b, c) = 1. Computing a µ-basis of P(t) is equivalent to computing a basis of the

Note that a∗i (s, t) is monic and degt a∗i (s, t) = degt ai (s, t).

135

syzygy module Syz(a, b, c) with lowest possible degree. We first compute a basis of Syz(a, b, c) based on the proof of Corollary 3.2. ˜ = (c), N ˜ = (−a, −b). Construct a matrix P and Set D compute its MFD:

and

The column-reduced form of M is 0

5(t + 3) t M′ =  −13t − 25

where A = (−a, −b) and B = diag(c, c). To compute a generating matrix of Syz(a, b, c), we need to find the GCRD of A and B. The GCRD extraction algorithm in Section 3.3 is described for bivariate polynomial matrices, but it works also for univariate case with minor modifications. In the univariate case, we do not need to do the primitive factorization in Steps 1 and 3 of Algorithm 2. The key step is to find the Hermite form of matrix A B



0

1 −k  0 1 0 0

1

−a −b 0 A. = c 0 c

10

10

0 1 0 0 A  0 µ1 1 0 c/e

Then

0 λ1 A  c/d µ2 bc/(de) 0

0

1

0

λ2 a/d 0

5. COMPUTING µ-BASES OF A RATIONAL SURFACE To compute a µ-basis of a rational surface, we just follow what we did to compute a µ-basis of a rational curve. Here the main computational complexity comes from the GCRD extraction algorithm. However, since Steps 1 and 3 of Algorithm 2 can’t be omitted, it is difficult to write down the generating matrix of the syzygy module explicitly. Given a rational parametric surface in homogeneous form

1

0 0 A 1

P(s, t) := (a(s, t), b(s, t), c(s, t), d(s, t)),

1

where a, b, c, d ∈ K[s, t] and gcd(a, b, c, d) = 1. A µ-basis of rational surface P(s, t) is a basis of the syzygy module Syz(a, b, c, d). ˜ = (d), N ˜ = (−a, −b, −c). Construct a matrix P Set D and compute its MFD:

−a −b d r 0 A= 0 e A D c 0 c 0 0 is the Hermite form. Hence the GCRD of A and B is 

R=

d 0

r e



˜ −1 N ˜ = (−a/d, −b/d, −c/d) = AB −1 , P =D

,

and the generating matrix of Syz(a, b, c) is 

M=

B A

0



R

−1

e = 0 −a/d

where A = (−a, −b, −c) and B = diag(d, d, d). Suppose the ¯ B ¯ ∈ K[s, t] such GCRD of A and B is G, i.e., there exist A, that ¯ ¯ A = AG, B = BG,

1

−r A, d (ar − bd)/c

(1)

¯ and B ¯ are right coprime. Then (B ¯T , A ¯T )T is the where A generating matrix of Syz(a, b, c, d), and the three columns p, q, r of the generating matrix are a µ-basis of the rational surface P(s, t). Similar to curve case, the µ-basis obtained may not be a minimal µ-basis. To lower the degree, we proceed as follows. Rewrite the µ-basis as

˜ (t), q ˜ (t) of matrix M are a since c = de. The two columns p basis of the syzygy module Syz(a, b, c). Note that the basis obtained so far is possibly of higher degree than a µ-basis. To get the µ-basis, we need re˜ (t), q ˜ (t). Suppose n2 := deg(˜ duce the degree of p q(t)) ≥ n1 := deg(˜ p(t)) and n1 + n2 > n. Then LCV(˜ p(t)) and ˜ (t) × LCV(˜ q(t)) must be linearly dependent (otherwise p ˜ (t) 6= k(a, b, c)), that is, there exists some constant α such q ˜ (t) by q ˜ (t) := that LCV(˜ q(t)) = α LCV(˜ p(t)). Update q ˜ (t) − αtn2 −n1 p ˜ (t). This process can be continued until q ˜ q(t) ˜ are a µ-basis. Let deg(˜ p(t)) + deg(˜ q(t)) = n, i.e., p(t), us use an example to illustrate the process.

p=

i=0

P(t) = (2t2 + 4t + 5, 3t2 + t + 4, t2 + 2t + 3). Then 2

e(t) = t + 2t + 3,

λ2 (t) = 2,

dp X

pi (s)ti , q =

dq X i=0

qi (s)ti , r =

dr X

ri (s)ti ,

i=0

where pi (s), qi (s) and ri (s) in K 4 [s]. Without loss of generality, we assume dp > dq > dr . Let mp , mq , mr ∈ K 4 [s] be the leading coefficient vectors of p, q and r with respect to t, respectively, i.e., mp (s) = pdp (s), etc. From the recovery equation of Remark 3 in Section 2, it is easy to see that dp + dq + dr = degt (P(s, t)) if and only if mp , mq and mr are K[s]-linearly independent. Now if mp , mq and mr are K[s]-linearly independent, then dp + dq + dr reaches minimum and the process is terminated. Otherwise, consider the syzygy module Syz(mp , mq ,

Example 1 Suppose a rational curve is parametrized by

λ1 (t) = 1,

1

−5(t + 1) A. 1 10t + 7

The two columns of M ′ are a µ-basis of the rational curve (a(t), b(t), c(t)). The main computational costs of the µ-basis algorithm lie in computing GCDs of univariate polynomials using Euclidean algorithm and column-reduction of matrix M . One can easily prove that the computational complexity is less than 33 2 n + O(n) multiplications, which is faster than fastest 4 known algorithm [3].

Suppose gcd(−a, c) = d, then there exist λ1 , λ2 ∈ K[t] such that λ1 (−a) + λ2 c = d. Assume gcd(−bc/d, c) = e, then there exist µ1 , µ2 ∈ K[t] such that µ1 (−bc/d) + µ2 c = e. Finally suppose the quotient and remainder of λ1 b divided by e are k and r. Denote the following matrix as D, 0

1

t2 + 2t + 3 −5(t + 1) A. 0 1 M = −2t2 − 4t − 5 10t + 7

˜ −1 N ˜ = AB −1 . P =D



0

d = 1,

r(t) = 5(t + 1),

136

mr ), the basis of which can be found based on the results in [12]. Find a vector α := (αp , αq , αr ) in the basis of the syzygy module Syz(mp , mq , mr ) such that one of αp , αq , αr is a non-zero constant, if possible. If not, terminate the process. Set βp = αp , βq = αq sdp −dq , βr = αr sdp −dr and u = βp p + βq q + βr r. If αp is a non-zero constant and degs (u) < degs (p), update p by u. Otherwise if αq is a non-zero constant, and dp = dq and degs (u) < degs (q), update q by u. Otherwise if αr is a non-zero constant, and dp = dq = dr and degs (u) < degs (r), update r by u. This process can be continued until dp + dq + dr = degt (P(s, t)) or one of the above conditions fails to hold. The next step is to reduce the degree of p, q, and r with respect to s while keeping the degree of p, q, and r with respect to t unchanged. This can be done by applying the vector elimination technique in [3] to p, q, and r. We should note that, while the above process generally reduces the degree of a µ-basis, it doesn’t necessarily produce a minimal µ-basis. Now we present some examples to demonstrate the detailed process of the algorithm.

Example 3 Given a bi-quadratic surface defined by a(s, t) = t2 + st + 2s2 − 2s2 t, b(s, t) = t2 + 2st + st2 + 2s2 − s2 t + 2s2 t2 , c(s, t) = −t2 + st + 2st2 + 2s2 − s2 t − 2s2 t2 , d(s, t) = 2st − 2st2 − 2s2 t − s2 t2 . The content of the GCD of all the major minors of 

A B



0

0

−2t −2s C 0 0 C. 2 2 A s +t +1 0 2 2 0 s +t +1

0

0

t/2 0 0

B U =B 

0

s2 +t2 +1 2s

After Step 2, it follows that 0

2s(3s4 + 5s3 + s2 − 2s + 2) R= 0 0

1

1 1 0

0 0 C C 1 A. 1

t s

¯ −1 R

2

− t s+1 = 1 − s2st+1 0

1 s

0 0

α(s, t) = −(s + 2)(3s2 − 5s − 4)st, β(s, t) = 12s4 + 20s3 + 4s2 − 8s + 8 + 5s4 t + 4s3 t − 4s2 t + 12st − 8t, γ(s, t) = t(2s + st + 2t − 2).

1

0 0

A,

1 s2 +1

−(t2 + 1)(s2 + 1) ∗  R = (s2 + 1)s −s2 t

−st2 s2 + 1 −st

1

The results of the rest steps are omitted since they are a little clumsy to write down. Finally, we obtain a µ-basis for the biquadratic surface as follows: 1 p = 35412 ·

t(t2 + 1) A. −st t2 + 1

Then the GCRD of A and B is R∗ R0 = R∗ . Therefore the generating matrix of Syz(a, b, c, d) is 

M=

B A

0

 ∗ −1

(R )

1

α(s, t) β(s, t) A, γ(s, t) 0 0 γ(s, t)

where

Here U ∈ K(s, t), but ∈ / K[s, t]. Since det(R) = (s2 + 1)s(s2 + t2 + 1)2 , its irreducible factor list of the content with respect to t is s2 + 1, s. Applying the LPF algorithm in Algorithm 1, we get 0

1

−s 1 −1 ¯ = t(st + 2s + 2t − 2)  0 −1 0 A . B 0 0 −1

s2 + 1 −st 2 A, s + t2 + 1 0 0 s2 + t 2 + 1 s 0 0 t

1T

2s2 t − 2s2 − st − t2 ¯  A , −t(2st + s + t + 1) A= 2st2 + 3st − 4s − 2t2 − 2t

1

(s2 + 1)s  R= 0 0

1

0

The content of the GCD of all the minors of order 3 with respect to K[s][t] is 1, so we skip Step 1 in the GCRD extraction algorithm, i.e., R0 = I3 . In Step 2 of the GCRD extraction algorithm, we make use of the discussion in Remark 6 of Section 3.3, and obtain 0

1

−a −b −c B d 0 0 C B C = 0 d 0 A 0 0 d

1 1 −1 R0 =  0 s 0 A , 0 0 s

1

−2st

B s2 + t 2 + 1 =B  0

0

with respect to K[s][t] is s2 . Then the irreducible factor list is {s, s}. In Step 1 of GCRD extraction algorithm, one can compute

Example 2 The Steiner surface is defined by (a, b, c, d) = (2st, 2t, 2s, s2 + t2 + 1). The matrices A and B are then 

A B



−1

B s =B  0

0

0

1

B B −35412, B B
B −2 s 18264 s3 + 49451 s2 + 46965 s + 4705 , B B  12176 s4 + 18762 s3 + 30440 s2 t − 11887 s2 − 4843 st

C C C C C C C A

30440 ts4 + 36528 s4 + 56037 s3 t + 98902 s3 + 16316 s2 t B C + 93930 s2 + 52004 st + 9410 s + 35412 B C

1

0 t C t2 + 1 0 C. 2 st s +1 A −2t −2s

The three columns of M gives a µ-basis of the Steiner surface. One can show that it is a minimal µ-basis.

+ 35412 s + 26002 t + 17706

137

q= 0

1 · 256176

Example 4 Consider the surface parameterized by

−46308 st − 74337 t2 s4 − 96066 ts4 − 216726 s2 t − 209666 s3 t + 59836 s2 t2 − 40826 s3 t2 − 311720 st2 + 92216 s + 100316 t − 36272 + 21680 s2 − 54752 s3 − 22872 s4 ,

B B B B B B B
B 4 −4534 + 6993 s + 9703 s2 + 2859 s3 B B (2 s + st + 2 t − 2) , B B
B 32022 t 3 s4 + 5 s3 + s2 − 2 s + 2 , B B B −20586 ts4 + 22872 s4 + 55203 s3 t + 73812 s3 B  − 74337 s2 t2 + 226126 s2 t + 11240 s2 + 107848 st2

1

a(s, t) = −3s2 t2 + 5s2 t − 5t2 − 4st + 5,

C C C C C C C C C C C C C C C C C A

b(s, t) = −3s2 t2 + 3s2 t + s2 + st2 − s − 2t2 − 5st + 1, c(s, t) = −5s2 t2 + 6s2 t + 2st − t2 − t − 5, d(s, t) = −4s2 t2 + 3s2 t − st + 6t2 − t + 1. If we use the computer algebra system Singular or the package CASA in Maple to compute a generator of syzygy module Syz(a, b, c, d), then we get four or five vector polynomials (depending on different orderings), and it is very difficult to find proper combinations of them to form a µ-basis. By our algorithm, we can easily compute a µ-basis. The result is omitted.

− 68094 st − 118252 s − 155860 t2 − 82180 t − 18136 0

1

B

C C C C C C C A

6. CONCLUSION AND FUTURE WORKS

6 s4 + 10 s3 + 2 s2 − 4
s + 4 + 5 ts4 + 4 ts3 B C −4 ts2 + 12 ts − 8 t s, B C

B 1 B 0, r= B
6 B −2 s 3 s4 + 5 s3 + s2 − 2 s + 2 , B B  2 s5 + s4 + 5 s3 t − 4 s3 − 6 s2 t

In this paper, we apply the theory of polynomial matrices to compute µ-bases of rational curves and surfaces. The algorithm is based on several important techniques in the theory of polynomial matrices, such as primitive factorization, Hermite form and GCD extraction. This is the only known algorithm to compute the µ-bases of general rational surfaces, and it is superior than any existing algorithms for computing the µ-bases of rational curves. In the future, µ-bases of a spatial rational parametric curve will be considered. It is expected that the implicit equation of a space curve can be computed from a µ-bases. On the other hand, finding an efficient method to compute a minimal µ-basis of a rational surface and the complexity analyzing of the algorithm are problems worthy of further research.

+ 10 s2 + 8 st − 4 t Now we apply degree reduction algorithm to reduce the degree of the µ-basis. The new µ-basis is shown below. 1 p′ = 236661622380753 · 0

−300729067167523 st + 56467802265703 s2 t + 203543640533634 s + 228386226979701 t − 196295646522670 s2 − 140869867499516,

B B B B B B −279870932485122 s + 90277583448339 st B B − 125135373778758 t + 140869867499516 B B + 48465134115412 s2 − 97142080550445 s2 t, B B B −58285248330267 s2 t + 81366332623128 st B B + 103250853200943 t + 147830512407258 s2 B B + 76327291951488 s, B B B −40462746679845 st − 77713664440356 s2 t B  − 207436107658841 s − 105225741859592 t

1 C C C C C C C C C C C C C C C C C C C C C A

7. ACKNOWLEDGMENTS The authors are support by the Outstanding Youth Grant of NSF of China (No. 60225002), NSF of China (10201030 and 60473132), a National Key Basic Research Project of China (2004CB318000), the TRAPOYT in Higher Education Institute of MOE of China, and SRF for ROCS, SEM. Special thanks go to Dr. Zhiping Lin from Nanyang Technological University, Singapore, for his novel results in the paper [12] and for the helpful discussions with him.

− 811703353674 s2 − 70434933749758 ′

q =

8. REFERENCES

[1] Falai Chen, Jianmin Zheng, and T. W. Sederberg, The µ-basis of a rational ruled surface, Computer Aided Geometric Design, Vol.18, 2001, 61–72. [2] Falai Chen and Thomas W. Sederberg, A new implicit representation of a planar rational curve with high order of singularity, Comput. Aided Geom. Design, vol.19, 2002, 151–167. [3] Falai Chen and Wenping Wang, The µ-basis of a planar rational curve — properties and computation, Graphical Models, Vol.64, 2003, 368–381. [4] Falai Chen, Reparameterization of a rational ruled surface by µ-basis, Computer Aided Geometric Design, Vol.20, 2003, 11–17. [5] Falai Chen and Wenping Wang, Revisiting the µ-basis of a rational ruled surface, Journal of Symbolic Computation, Vol.36, 2003, 699–716. [6] Falai Chen, David Cox, and Yang Liu, The µ-basis of a rational parametric surface, Journal of Symbolic Computation, Vol.39, 2005, 689–706.

1 · 56167843795

0

1

B B −326160947200 − 360111336704 s2 + 487462361088 s, B B B −410518028544 s2 − 164118258432 s, B B  163080473600 − 223271993856 s2

C C C C C C C A

326160947200 + 162042688768 s2 t + 770629365248 s2 B C + 324085377536 st − 323344102656 s, B C

+ 82429766656 s + 162042688768 t r′ = 0

1 · 2039538

23600164 − 47200328 s3 + 64900451 s2 − 23600164 s,

B B −23600164 + 47200328 s + 29500205 s3 − 35400246 s2 , B B B 17700123 s3 − 29500205 s2 − 23600164 s, 

1 C C C C C A

11800082 + 23600164 s3 + 5900041 s2

138

[7] Falai Chen and Wenping Wang, Computing the singular points of a planar rational curve using the µ-basis, preprint, 2004. [8] David Cox, John Little, and Donal O’Shea, Using Algebraic Geometry, New York, Springer-Verlag, 1998. [9] David Cox, T. W. Sederberg, and Falai Chen, The moving line ideal basis of planar rational curves, Computer Aided Geometric Design, Vol.15, 1998, 803–827. [10] F. R. Gantmacher, Theory of Matrices, New York: Chelsea Publishing Co., 1959. [11] John P. Guiver and N. K. Bose, Polynomial matrix primitive factorization over arbitrary coefficient filed and related results, IEEE Transactions on Circuits and Systems, Vol.CAS-29, No.10, 1982, 649–657. [12] Zhiping Lin, On syzygy modules for polynomial matrices, Linear Algebra and Its Application, Vol.298, 1999, 73–86. [13] Martin Morf, Bernard C. L´evy, and Sun-Yuan Kung, New results in 2-D system theory, Part I: 2-D polynomial matrices, factorization, and coprimeness, Proceedings of the IEEE, Vol.65, No.6, 1977, 861–872.

[14] T. W. Sederberg and Falai Chen, Implicitization using moving curves and surfaces, Computer Graphics Proceedings, Annual Conference Series, Vol.2, 1995, 301–308. [15] T. W. Sederberg, T. Saito, D. Qi, and K. Klimaszewski, Curve implicitization using moving lines, Computer Aided Geometric Design, Vol.11, 1994, 687–706. [16] D. C. Youla and G. Gnavi, Notes on n dimensional systems, IEEE Transaction on Circuits and System, Vol.26, 1979, 105–111. [17] Fangling Zeng and Falai Chen, Degree reduction of rational curves by µ-basis, Computer Mathematics, Proceedings of the Sixth Asian Symposium (ASCM’2003), Lecture Notes Series on Computing, Vol.10, ed. Ziming Li and William Sit, World Scientific, 2003, 265–275. [18] Jianmin Zheng and T. W. Sederberg, A direct approach to computing the µ-basis of planar rational curves, Journal of Symbolic Computation, Vol.31, 2001, 619–629.

139

Efficient Computation of the Characteristic Polynomial Jean-Guillaume Dumas

Clement ´ Pernet

Zhendong Wan

Universite´ Joseph Fourier, LMC-IMAG B.P. 53, 38041 Grenoble Cedex 9, France.

Universite´ Joseph Fourier, LMC-IMAG B.P. 53, 38041 Grenoble Cedex 9, France.

Department of Computer and Information Science University of Delaware, Newark, DE 19716, USA.

[email protected]

[email protected]

[email protected]

ABSTRACT

algorithms to compute it are based on computations of characteristic polynomial (see for example [21, §9.7]). Using classical matrix multiplication, the algebraic time complexity for the computation of the characteristic polynomial is optimal : several algorithms require only O(n3 ) algebraic operations (to our knowledge the oldest one is due to Danilevski [11, §24]). Now considering that the determinant can be deduced from the characteristic polynomial, and that its computation is proven to be as hard as matrix multiplication [2] the optimality is then straightforward. But with fast matrix arithmetic (O(nω ) with 2 ≤ ω < 3), the best asymptotic time complexity is O(nω logn), given by Keller-Gehrig’s branching algorithm [15]. Now the third algorithm of Keller-Gehrig has a O(nω ) algebraic time complexity but only works for generic matrices. In this paper we focus on the practicability of such algorithms applied on matrices over a finite field. Therefore we used the techniques developped in [4, 5], for efficient basic linear algebra operations over a finite field. We propose a new O(n3 ) algorithm designed to take benefit of the block matrix operations; improve KellerGehrig’s branching algorithm and compare these two algorithms. Then we focus on Keller-Gehrig’s third algorithm and prove that its generalization is not only of theoretical interest but is also promising in practice. As an application, we show that these results directly lead to an efficient computation of the characteristic polynomial of integer matrices using chinese remaindering and an early termination criterion adapted from [6]. This basic application outperforms the best existing softwares on many cases. Now better algorithms exist for the integer case, and can be more efficients with sparse or structured matrices. Therefore, we also propose a probabilistic algorithm using a blackbox computation of the minimal polynomial and our finite field algorithm. This can be viewed as a simplified version of the algorithm described in [22] and [14, §7.2]. Its efficiency in practice is also very promising. In the following we will denote by Ai1 ...i2 ,j1 ...j2 the submatrix of A located between rows i1 and i2 and columns j1 and j2 and by Ak,1...n the kth row vector of A.

We deal with the computation of the characteristic polynomial of dense matrices over word size finite fields and over the integers. We first present two algorithms for finite fields: one is based on Krylov iterates and Gaussian elimination. We compare it to an improvement of the second algorithm of Keller-Gehrig. Then we show that a generalization of KellerGehrig’s third algorithm could improve both complexity and computational time. We use these results as a basis for the computation of the characteristic polynomial of integer matrices. We first use early termination and Chinese remaindering for dense matrices. Then a probabilistic approach, based on integer minimal polynomial and Hensel factorization, is particularly well suited to sparse and/or structured matrices.

Categories and Subject Descriptors G.4 [Mathematics and Computing]: Mathematical Software—Algorithm Design and Analysis; I.1.2 [Computing Methodologies]: Symbolic and Algebraic Manipulation

General Terms Algorithms, Experimentation

Keywords Characteristic polynomial, minimal polynomial, Keller-Gehrig, probabilistic algorithm, finite field, integer, Magma

1.

INTRODUCTION

Computing the characteristic polynomial of an integer matrix is a classical mathematical problem. It is closely related to the computation of the Frobenius normal form which can be used to test two matrices for similarity. Although the Frobenius normal form contains more information on the matrix than the characteristic polynomial, most

2. KRYLOV’S APPROACH Among the different techniques to compute the characteristic polynomial over a field, many of them rely on the Krylov approach. A description of them can be found in [11]. They are based on the following fact: the minimal linear dependance relation between the Krylov iterates of a vector v (i.e. the sequence (Ai v)i ) gives the minimal polynomial min PA,v of this sequence, and a divisor of the minimal polyno-

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mial of A. Moreover, if X is formed by the the first independent column vectors of this sequence and CP min is the com-

to avoid the computation of the last n − k Krylov iterates with an early termination. This reduces the time complexity to O(nω logk) for fast matrix arithmetic, and O(n2 k) for classical matrix arithmetic. Note that if the finite field is reasonably large, then choosing v randomly makes the algorithm Monte-Carlo for the computation of the minimal polynomial of A.

A,v

min panion matrix associated to PA,v , we have AX = XCP min . A,v

2.1 Minimal polynomial We show here an algorithm to compute the minimal polynomial of the sequence of the Krylov iterates of a vector min v and a matrix A. This is the monic polynomial PA,v of least degree such that P (A).v = 0. We firstly presented it in [19, 18], however, we recall it here for clarity and accessibility purposes. A similar algorithm was also simultaneously published in [16, Algorithm 2.14], but it does not take advantage of fast matrix multiplication as we do here. The idea is to compute the n × n matrix KA,v (called the Krylov matrix), whose ith column is the vector Ai u, and to perform an elimination on it. More precisely, one computes the LSP min t . So [12] factorization of KA,v . Let k be the degree of PA,v the first k columns of KA,v are linearly independent, and the n − k following ones are a linear combination of the first k ones. Therefore S is triangular with its last n − k rows t equals to 0. Thus, the LSP factorization of KA,v can be viewed as in figure 1.

t

K(A,v) =

vt (Av) t (A 2v) t ..... (A k+1 v)t

=

2.2 LU-Krylov algorithm From algorithm 2.1, we then derive the computation of the characteristic polynomial since it produces the k first independent Krylov iterates of v. They form a basis of an invariant subspace under the action of A (the first invariant min subspace of A if PA,v = PAmin ). The idea is to use the elimination performed on this basis to compute a basis of its supplementary subspace. Then a recursive call on a n − k × n − k matrix provides the remaining factors. The algorithm follows, where k, P , and S are as in algorithm 2.1. Algorithm 2.2 LUK : LU-Krylov algorithm Require: A a n × n matrix over a field A Ensure: Pchar the characteristic polynomial of A 1: Pick a random vector v A,v 2: Pmin =» MinPoly(A, v) of degree k – L1 {X = [S1 |S2 ]P is computed} L2 3: if (k = n) then A,v A 4: return Pchar = Pmin 5: else » 0 – A11 A012 6: A0 = P A T P T = where A011 is k × k. A021 A022

L 1..k .

S

. P

L k+1 .....

Figure 1: LSP factorization of the Krylov matrix

−1

A0 −A021 S1

22 Pchar

7:

8: return 9: end if

Now the trick is to notice that the vector m = Lk+1 L−1 1...k min contains the opposites of the coefficients of PA,v . Indeed, T let us define X = KA,v therefore Xk+1,1...n = (Ak v)T = Pk−1 i T min = m · X1...k,1...n where PA,v (X) = X k − i=0 mi (A v) k−1 mk X − · · · − m1 X − m0 . Thus Lk+1 SP = m · L1...k SP and finally m = Lk+1 .L−1 1...k . Algorithm 2.1 is then straightforward. The dominant operation in this algorithm is the

A Pchar

S2

(X) = LUK(A022 − A021 S1−1 S2 ) −1 A0 −A021 S1 S2

A,v 22 = Pmin × Pchar

Theorem 2.1. Algorithm LU-Krylov computes the characteristic polynomial of a n × n matrix A in O(n3 ) field operations. Proof. The first k rows of X (X1...k,1...n ) form a basis of the invariant subspace generated by v. Moreover we have X1..k AT = CPT A,v X1..k Indeed ∀i < k we have min ` ´T ` ´T Xi AT = Ai−1 v AT = Ai v = Xi+1 and Xk AT = ` k−1 ´T T ` k ´T ` i ´T Pk−1 A v A = A v = The idea is i=0 mi A v now to complete this basis into a basis of the whole space. Viewed as a matrix, this basis form the n × n invertible matrix X: – » –» » – X L1 0 S1 S2 X= P = ˆ 1...k,1...n˜ 0 In−k 0 In−k 0 In−k P | {z }| {z }

Algorithm 2.1 MinPoly : Minimal Polynomial of A and v Require: A a n × n matrix and v a vector over a field A,v Ensure: Pmin (X) minimal polynomial of (Ai v)i 1: K1...n,1 = v 2: for i = 1 to log2 (n) do i−1 3: K1...n,2i ...2i+1 −1 = A2 K1...n,1...2i −1 4: end for 5: (L, S, P ) = LSP(K T ), k = rank(K) 6: m = Lk+1 .L−1 1...k P A,v i 7: return Pmin (X) = X k − k−1 i=0 mi X

L

S

Let us compute

computation of K, in log2 n matrix multiplications, i.e. in O(nω logn) algebraic operations. The LSP factorization requires O(nω ) operations and the triangular system resolution, O(n2 ). So the algebraic time complexity of this algorithm is O(nω logn). Now with classical matrix multiplications (ω = 3), one should prefer to compute the Krylov matrix K by k successive matrix vector products. The complexity is then O(n3 ). One can also merge the construction of the Krylov matrix and its LSP factorization so as

T

XA X

−1

= =

"

# CT 0 ˜ ˆ −1 −1 0 In−k P AT P T S L " # » CT 0 CT ˆ 0 ˜ −1 −1 = Y A21 A022 S L

0 X2

–

with X2 = A022 − A021 S1−1 S2 . By a similarity transformation, we thus have reduced A to a block triangular matrix.

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Algorithm 2.3 ColReducedForm Require: A a m × n matrix of rank r Ensure: r linearly independent columns of A 1: (L, Q, U, P, r) = LQUP(AT ) (r = rank(A)) 2: return ([Ir 0](QT AT ))T

Then the characteristic polynomial of A is the product of the characteristic polynomial of these two diagonal blocks: A0 −A0 S −1 S

A,v 2 A 22 21 1 Pchar = Pmin × Pchar . Now for the time complexity, we will denote by TLUK (n) the number of field operations for this algorithm applied on a n×n matrix, by Tminpoly (n, k) the cost of the algorithm 2.1 applied on a n × n matrix having a degree k minimal polynomial, by TLSP (m, n) the cost of the LSP factorization of a m × n matrix, by Ttrsm (m, n) the cost of the simultaneous resolution of m triangular systems of dimension n, and by TMM (m, k, n) the cost of the multiplication of a m × k matrix by a k × n matrix. The values of TLSP and Ttrsm can be found in [5]. Then, using classical matrix arithmetic, we have: TLUK (n) = Tminpoly (n, k) + TLSP (k, n)+Ttrsm (n−k, k)+Tmm (n−k, k, n−k)+TLUK (n−l) = 2 O(n k2 n + k2 (n − k) + k(n − k)2 ) + TLUK (n − k) = Pk + 2 2 O( n) = O(n3 ), The latter being true since P i n ki + kiP 2 2 k = n and i i i ki ≤ n .

LQUP factorization indicates the positions of the linearly independant blocks of iterates in W . To each of these blocks, one can associate a block column in L. Now applying the triangular system resolution of algorithm 2.1 to this block column will compute the coefficients of the first linear dependency between these iterates. Since the Krylov iterates are already computed, and the last call to ColReducedForm performed the elimination on them, there only remains to solve triangular systems. We thus get the coefficients of each polynomial, for a total cost of O(n2 ). Algorithm 2.4 shows

We have thus derived a deterministic algorithm from a probabilistic one. When algorithm 2.1 fails, it still returns a factor of the true minimal polynomial and the next recursive calls then compute the forgotten factors. Note also that when using fast matrix arithmetic, it is no longer possible to sum the log(ki ) into log(n) nor the kiω−2 n2 into nω . Therefore the best known algebraic time complexity, O(nω logn), can not be reached by such an algorithm. We thus focus on the second algorithm of Keller-Gehrig achieving this best known time complexity.

Algorithm 2.4 KGB: Keller-Gehrig Branching algorithm Require: A a n × n matrix over a field A Ensure: Pchar (X) the characteristic polynomial of A 1: i = 0 2: V0 = In = (V0,1 , V0,2 , . . . , V0,n ) 3: B = A 4: while (∃k, Vk has 2i columns) do 5: for all j do 6: if ( Vi,j has strictly less than 2i columns ) then 7: Wj = Vi,j 8: else 9: Wj = [Vi,j |BVi,j ] 10: end if 11: end for 12: W = (Wj )j 13: Vi+1 = ColReducedForm(W ) (remember L and Q from LQUP) {Vi+1,j are the remaining vectors of Wj in Vi+1 } 14: B = B ×B 15: i=i+1 16: end while 17: for all j do 18: Let s, t be the indexes of the first and last column of linearly independent iterates of the vector ej in W (given by Q) 19: m = Lt+1,s...t .L−1 s...t,s...t P 20: Pj (X) = X t−s − t−s−1 mi X i i=0 21: end for 22: return Πj Pj

2.3 Improving Keller-Gehrig branching algorithm In [15], Keller-Gehrig presents a so called branching algorithm, computing the characteristic polynomial of a n × n matrix over a field K in the best known time complexity : O(nω logn) field operations. The idea is to compute the Krylov iterates of a several vectors at the same time, so as to replace several matrix vector products by a fast matrix multiplication. More precisely, the algorithm computes a sequence of n×n matrices (Vi )i whose columns are the Krylov iterates of vectors of the canonical basis. V0 is the identity matrix (every vector of the canonical basis is present). At the i-th iteration, the algorithm computes the next 2i Krylov iterates of the remaining vectors. Then a Gaussian elimination determines the linear dependencies between them so as to form Vi+1 by picking the n linearly independent vectors. The algorithm ends when no more iterate can be added (Vi+1 = Vi ). Then the matrix Vi−1 AVi is block upper triangular with companion blocks on the diagonal. The polynomials of these blocks are the minimal polynomials of each of the sequence of Krylov iterates, and their product is the characteristic polynomial of the input matrix. The removal of the linear dependencies is performed by a step-form elimination algorithm defined by Keller-Gehrig. Its formulation is rather sophisticated, and we propose to replace it by the column reduced form algorithm (algorithm 2.3) using the more standard LQUP factorization [12]. More precisely, the step form elimination of Keller-Gehrig, the LQUP factorization of Ibarra & Al. and the echelon form elimination (see e.g. [21]) are equivalent and can be used to determine the linear dependencies in a set of vectors. Our second improvement is to apply the idea of algorithm 2.1 to compute the polynomials associated to each companion block, instead of computing V −1 AV . The permutation Q of the

these modifications. The operations in the while loop have a O(nω ) algebraic time complexity. This loop is executed at most logn times and the overall algebraic time complexity is therefore O(nω logn). More precisely it is O(nω logkmax ) where kmax is the degree of the largest invariant factor.

2.4 Experimental comparisons We implemented these two algorithms, using a finite field representation on double size machine floating point numbers : modular<double> (see [5]), and the efficient routines for finite field linear algebra FFLAS-FFPACK presented in [5, 4]. We also only considered classic matrix arithmetic. We ran them on a series of matrices of order 300 whose Frobenius normal forms had different number of diagonal com-

142

Kω nω + o(nω ) algebraic operations, where » 2ω−2 1 Kω = Cω − − ω ω−2 2(2 − 1)(2ω−1 − 1)(2ω − 1) 2 −1 1 3 2 + ω−2 − ω−1 + ω−2 (2 − 1)(2ω−1 − 1) 2 −1 2 −1

panion blocks. Figure 2 shows the computational time on a Pentium IV 2.4Ghz with 512Mb of RAM. It appears that

KGB vs LU−Krylov for a 300x300 matrix over GF(101) 1.6

LU−Krylov Keller−Gehrig

1.4

1 2ω−2 + ω−2 + ω ω−2 (2 − 1)(2 − 1) 2(2 − 1)(2ω−1 − 1)2

Time (s)

1.2 1

and Cω is the constant in the algebraic time complexity of the matrix multiplication.

0.8 0.6

The proof and a description of the algorithm are given in appendix A. In particular, with classical matrix arithmetic (ω = 3, Cω = 2), we have on the one hand Kω = 176/63 ≈ 2.794. On the other hand, the algorithm 2.2 called on a generic matrix simply computes the n Krylov vectors Ai v (2n3 operations), computes the LUP factorization of these vectors (2/3n3 operations) and the coefficients of the polynomial by the resolution of a triangular system (O(n2 )). Therefore, the constant for this algorithm is 2+2/3 ≈ 2.667. These two algorithms have thus a similar algebraic complexity, LU-Krylov being slightly faster than Keller-Gehrig’s third algorithm. We now compare them in practice.

0.4 0.2 0 0

5

10

15

20 25 30 35 Number of blocks

40

45

50

Figure 2: LU-Krylov vs. KGB LUK is faster than KGB on every matrix. This is due to the extra logn factor in the time complexity of the latter. One can note that the computational time of KGB is decreasing with the number of blocks. This is due to the fact that the log(n) is in fact log(kmax where kmax is the size of the largest block. This factor is decreasing when the number of blocks increases. Conversely, LUK computational time is almost constant. It slightly increases, due to the increasing number of rectangular matrix operations: their computation are less efficient than square matrix operations, due to BLAS optimizations of memory accesses.

3.2 Experimental comparison We claim that the study of precise algebraic time complexity of these algorithms is worthwhile in practice. Indeed these estimates directly correspond to the computational time of these algorithms applied over finite fields. Therefore KG3 vs LU−Krylov over Z/65521Z 1800

LU−Krylov Keller−Gehrig Fast algorithm

1600

3.

–

1400

TOWARD AN OPTIMAL ALGORITHM

1200 Mfops

As mentioned in the introduction, the best known algebraic time complexity for the computation of the characteristic polynomial is not optimal in the sense that it is not O(nω ) but O(nω logn). However, Keller-Gehrig gives a third algorithm (let us name it KG3), having this time complexity but only working on generic matrices. In the following, we will use Keller-Gehrig’s definition of a generic matrix : each of its coefficients can be considered as an independent indeterminate. To get rid of the extra log(n) factor, it is no longer based on a Krylov approach. The algorithm is inspired by a O(n3 ) algorithm by Danilevski (described in [11]), improved into a block algorithm. The genericity assumption ensures the existence of a series of similarity transformations changing the input matrix into a companion matrix.

1000 800 600 400 200 0

500

1000

1500 2000 Matrix order

2500

3000

3500

Figure 3: LUK vs. KG3: speed comparison we ran these algorithms on a word size prime finite field with modular arithmetic. Again we used modular<double> and FFLAS-FFPACK. These routines can use fast matrix arithmetic, we, however, only used classical matrix multiplication so as to compare two O(n3 ) algorithms having similar constants (2.67 for LUK and 2.794 for KG3). We used random dense matrices over the finite field Z65521 , as generic matrices. We report the computational speed in Mfops (Millions of field operations per second) for the two algorithms on figure 3. It appears that LU-Krylov is faster than KG3 for small matrices (better algebraic time complexity), but for matrices of order larger than 1500, KG3 is faster. Indeed, the O(n3 ) operations are differently performed: LUKrylov computes the Krylov basis by n matrix-vector products, whereas KG3 only uses matrix multiplications. Now, as the order of the matrices increases, the BLAS routines

3.1 Comparing the constants The optimal “big-O” complexity often hides a large constant in the exact expression of the time complexity. This makes these algorithms impracticable since the improvement induced is only significant for huge matrices. However, we show in the following lemma that the constant of KG3 is very close to the one of LUK. Lemma 3.1. The computation of the characteristic polynomial of a n×n generic matrix using KG3 algorithm requires

143

4.

“

”

1 √ ≤ . Therefore for n > 4, log 2 1+(n) 12n+1 2π n −1.296. Then i(n−i) is decreasing in i for i < bn/2c so that n . its maximum is 2(n−2) ` ´i “ n ”n−i p (n−i) Consider now K(n, i) = ni (n − i)B 2 . n−i 2 n−i n n We have log 2 (K(n, i)) = 2 log2 (B )+ 2 log2 (n)+ 2 T (n, i), n where T (n, i) = log 2 ( n−i ). Well T (n, i) is ) + ni log2 ( n−i i2 √ −1+ 1+4en maximal for i = as announced in the lemma. 2e n We end with the fact that T (n, i) − n2 1.296 + n1 log2 ( 2(n−2) ) 1 12n

provide better efficiency for matrix multiplications than for matrix vector products. Once again, algorithms exclusively based on matrix multiplications are preferable: from the complexity point of view, they make it possible to achieve O(nω ) time complexity. In practice, they promise the best efficiency thanks to the BLAS better memory management.

OVER THE INTEGERS

There exist several algorithms to compute the characteristic polynomial of an integer matrix. A first idea is to perform the algebraic operations over the ring of integers, using exact divisions [1] or by avoiding divisions [3, 13, 14]. We focus here on field approaches. Concerning the bit complexity of this computation, a first approach, using Chinese remaindering gives O˜ (nω+1 logkAk) bit operations (O˜ is the “soft-O” notation, hiding logarithmic and polylogarithmic factors in n and kAk). Baby-step Giant-step techniques applied by Kaltofen [13] improves this complexity to O˜ (n3.5 logkAk) (using classical matrix arithmetic). Lastly, the recent improvement of [14, §7.2], combining Coppersmith’s block-Wiedemann techniques set the best known exponent for this computation to 2.697263 using fast matrix arithmetic. Our goal here is not to give an exhaustive comparison of these methods, but to show that a straightforward application of our finite field algorithm LU-Krylov is already very efficient and can outperform the best existing softwares. A first dense deterministic algorithm, using Chinese remaindering is given in section 4.1. Then we propose in section 4.2 a probabilistic algorithm that can be adapted for dense or for sparse and structured matrices. It combines the early termination technique of [6, §3.3], and a recent alternative to chinese remaindering in [22], also developed in [14, §7.2]. Lastly we compare implementations of these algorithms in practice.

< (n) <

is maximal over Z for n = 15 where it is lower than 0.208935. The latter is lower than 0.21163175. "1 1 1 1 1 # Well,

1 1 1 1

1 −1 −1 −1

−1 1 −1 −1

−1 −1 1 −1

−1 −1 −1 1

has X 5 − 5X 4 + 40X 2 − 80X + 48 for

` ´√ 4 characteristic polynomial and 80 = 51 4 is greater than Hadamard’s bound 55.9, and less than our bound 80.66661. Note √ that the complexity finding the maximal value is ` ´ ` n of ´ n−i+1 only O( n), since ni = i−1 . Note also that the nui merical bound improves the one used in [9, lemma 2.1] since 0.21163175 < 2 + log2 (e) ≈ 3.4427. Now, using fast integer arithmetic and the fast Chinese remaindering algorithm [8, Theorem 10.25], one gets the overall complexity for the dense integer characteristic polynomial via Chinese remaindering of O(n4 (log(n) + log(B))). In the following, we focus on probabilistic methods that can improve drastically the computation time for both dense and structured matrices.

4.2 Probabilistic improvements The idea here, is to limit the chinese remaindering to the computation of the minimal polynomial. It is then factored by Hensel lifting algorithm. One modular characteristic polynomial is then computed (resuming algorithm LUK from one modular minimal polynomial). Lastly the multiplicities of each factor are recovered from this last polynomial.

4.2.1 Early termination

4.1 Dense deterministic : Chinese remainder

To reduce the number of homomorphic computations of the minimal polynomial, we use the early termination of [6, §3.3], to stop the remaindering :

The first naive way of computing the characteristic polynomial is to use Hadamard’s bound [8, Theorem 16.6] to show that any integer coefficient of the characteristic polynomial has the order of n bits:

Lemma 4.2. [6] Let v ∈ Z be a coefficient of the characteristic polynomial, and U be a given upper bound on |v|. Let P be a set of primes and let {p1 . . . pk , p∗ } be a random subset ofQ P . Let l be a lower bound such that p∗ > l and let M = ki=1 pi . Let vk = v mod M , v ∗ = v mod p∗ and vk∗ = vk mod p∗ as above. Suppose now that vk∗ = v ∗ . Then

n×n

Lemma 4.1. Let A ∈ Z , with n > 4, whose coefficients are bounded in absolute value by B > 1. The coefficients of the characteristic polynomial of A are denoted by ` ´p (n−i) cj . Then ∀j |cj | ≤ maxi=0.. −1+√1+4en ni (n − i)B 2 2e ` ´ and log 2 (|cj |) ≤ n2 log2 (n) + log 2 (B 2 ) + 0.21163175

v = vk with probability at least 1 −

Proof. ci , the i-th coefficient of the characteristic polynomial, is an alternate sum of all the (n − i) × (n − i) diagonal minors of A. It is therefore bounded by H(n, i) = `n´p (n−i) (n − i)B 2 . First note, that from the symmetry of i the binomial coefficients we only need to explore the bn/2c p √ (n−i) i first ones, since (n − i)B 2 > iB 2 for i < bn/2c. Now, the lemma claims √ that actually the maximal value must occur within the O( n) first ones. The lemma is true for j = 0 by Hadamard’s bound. And for j = 1, we have log 2 (H(n, i)) < n2 (log2 (n) + log2 (B 2 ) + 0.21163175) as soon as n > 4, since the difference is decreasing in n. Then √ n from Stirling’s formula (n! = (1 + (n)) 2πn nen ), we have q ` ´ ` n ´i “ n ”n−i n √ ∀i ≥ 2 ni < 1+(n) . Now first i(n−i) i n−i 2π

U −v logl ( M k ) . |P |

The proof is that of [6, lemma 3.1]. The probabilistic algorithm is then straightforward: after each modular computation of a minimal polynomial, the algorithm stops if every coefficient is unchanged. It is of the Monte-Carlo type: always fast with a controlled probability of success. This probability is bounded by the probability of lemma 4.2. In practice it is much higher, since the n coefficients are checked. But since they are not independent, we are not able to produce a tighter bound. Note that one could have just applied this early termination to the remaindering of the algorithm of section 4.1. But the improvement here, is that the number of homomorphic computations is reduced if the minimal polynomial has a small degree. The recovery of the rest of the integer characteristic polynomial is then

144

cheap (an integer factorization and one modular computation of the whole characteristic polynomial as shown next).

any certification of success, it can not be of the type “LasVegas”. The first case is when P min = P˜ min and a factor of P min divides another factor modulo p. In such a case, the exponent of this factor will appear twice in the reconstructed characteristic polynomial. The overall degree being greater than n, FAIL will be returned. Now, if P min 6= P˜ min , the tests αi > 0 will detect it unless P˜ min is a divisor of P min , say P min = P˜ min Q. In that case, on the one hand, if Q does not divide P˜ min modulo p, the total degree will be lower than n and FAIL will be returned. On the other hand, a wrong characteristic polynomial will be reconstructed, but the trace test will detect most cases. The overall complexity is not better than e.g. [22, 14] but its practical timings shown in section 4.3 prove its efficiency.

4.2.2 Multiplicities search and sparse/structured For dense matrices, the modular computation can be done by algorithm 2.1. The integer coefficients are recovered by chinese remaindering. The idea is to compute first the integer minimal polynomial and to factor it. Then only the multiplicities of the factors remain to be found. To recover them, we perform a modular computation of the characteristic polynomial and compute the multiplicities that match with this modular polynomial. By structured or sparse matrices we mean matrices for which the matrix-vector product can be performed with less than n2 arithmetic operations. In this case one can use the specialized methods of [6, §3] (denoted by IMP). Note that one modular computation of the characteristic polynomial will still be dense. Thus it is not a pure sparse or structured algorithm (memory storage can become the limiting factor), but since the dominant operation is usually the computation of the minimal polynomial, this approach still make sense.

4.3 Experimental results We now compare implementations of the previously described algorithms. ILUK-det from section 4.1 is dense deterministic. We declined the probabilistic algorithm 4.1 into a dense version, CIA-dense, (using algorithm 2.1) and a sparse version, CIA-prob, (using algorithm [6, §3]). We used the same programming environment as in section 2.4. The polynomial factorization is computed with NTL1 via Hensel’s factorization. The choice of moduli is there linked to the constraints of the matrix multiplication of FFLAS. Indeed, the wrapping of numerical BLAS is only valid if n(p − 1)2 < 253 (the result can be stored in the 53 bits of the double mantissa). Therefore, we chose to sample the primes between 2m and 2m+1 (where m = b25.5− 12 log2 (n)c). This set was always sufficient in practice. Even with 5000 × 5000 matrices, m = 19 and there are 38658 primes between 219 and 220 . Now if the coefficients of the matrix are between −1000 and 1000, the upper bound on the coefficients of the minimal polynomial is log2m (U ) ≈ 4267.3. Therefore, the probability of finding a bad prime is lower than 4267.3/38658 ≈ 0.1104. Then performing a couple a additional modular computations to check the result will improve this probability. In this example, only 18 more computations (compared to the 4268 required for the deterministic computation) are enough to ensure a probability of error lower than 2−55 . At this rate, there is e.g. more chances that cosmic radiations perturbed the output! 2 In the following, the probabibilistic algorithms will always ensure a probability of failure less than 2−55 . Table 1 focuses on dense matrices with coefficients chosen uniformly between 0 and 10. Their minimal polynomial equals their characteristic polynomial. ILUK-det and CIA-dense are compared Maple-v9 and Magma-2.11. We ran these tests on an Athlon 2200 (1.8 Ghz) with 2Gb of RAM, running Linux-2.4.3 The implementation of Berkowitz algorithm used by Maple has prohibitive computational timings. Magma uses a p adic algorithm that combines a probabilistic computation and a proof of correctness to make it deterministic. For these matrices, this proof is free, since the minimal polynomial equals the characteristic polynomial [20]. Comparing deterministic algorithms, ILUK-det is slightly faster than magma on most cases. For matrices of order over 800, magma tries to allocate

Algorithm 4.1 CIA : Characteristic polynomial over Integers Algorithm Require: A ∈ Zn×n , even as a blackbox, . Ensure: The characteristic polynomial of A with a probability of √ 1 − . 1: η = 1 − 1 −  A 2: Pmin = IMP(A, η) via [6, §3] or CRA(MinPoly)(A, η) A 3: Factor Pmin over the integers, e.g. by Hensel’s lifting. n log (n)+log (||A||2 )+0.21163175 ( ) 2 2 4: B = 2 2 5: Choose√ a random prime p in a set of 1 log2 ( n + 1 2n+1 B + 1) primes. η 6: Compute Pp the characteristic polynomial of A mod p via LUK. A 7: for all fi irreducible factor of Pmin do 8: Compute f i ≡ fi mod p. 9: Find αi the multiplicity of f i within Pp . 10: if αi == 0 then Return “FAIL”. end if 11: end for Q P i A 12: Compute Pchar = fiαi = X n − n−1 i=0 ai X . P 13: if ( αi degree(fi ) 6= n) or (T race(A) 6= an−1 ) then 14: Return “FAIL”. 15: end if A 16: Return Pchar . Theorem 4.3. Algorithm 4.1 is correct. It is probabilistic of the Monte-Carlo type. Proof. Let P min be the integer minimal polynomial of ˜ min the result of the call to IMP. With a probability A and √ P of 1 − , P min = P˜ min. Then the only problem that can occur is that an irreducible factor of P min divides another factor when taken modulo p, or equivalently, that p divides the resultant of these polynomials. Now from [8, Algorithm 6.38] and lemma √ 4.1 an upper bound on the size of this resultant is log2 ( n + 1 2n+1 B + 1). Therefore, the probability of choosing a bad prime is less than η. Thus the result will be correct with a probability greater than 1 − 

1 2

www.shoup.net/ntl

Indeed, cosmic rays only can be responsible for 105 software errors in 109 chip-hours at sea level[17] . At 1GHz, this makes 1 error every 255 computations. 3 We are grateful to the Medicis computing center hosted by the CNRS STIX lab : medicis.polytechnique.fr/medicis.

This algorithm is also able to detect many erroneous results and return “FAIL” instead. In that case, one could perform the computation again. However, since we don’t have

145

n 100 200

Maple 163s 3355s

1500

Magma 0.34s 4.45s 11.1Mb 69.8s 56Mb 1546s 403Mb 8851s 1368Mb MT

400

74970s

3000

MT

800 1200

ILUK-det 0.23s 3.95s 3.5Mb 91.4s 10.1Mb 1409s 36.3Mb 7565s 81Mb 21010s 136Mb 349494s 521Mb

CIA 0.20s 3.25s 3.5Mb 71.74s 10.1Mb 1110s 36.3Mb 5999s 81Mb 16705s 136Mb

is still rather efficient ( the best on B T B ), but magma appears to be extremely fast on AT A. We report in table 3 on Matrix TF12 Tref500 mk9b3

n 552 500 1260

ω 7.6 16.9 3

magma-sparse 10.12s 112s 48.4s

CIA-sparse 6.8s 65.14s 31.25s

CIA-dense 61.77s 372.6s 433s

Table 3: CIA on other sparse matrices some comparisons using other sparse matrices available at There, ILUK-det is rather expensive, although it is slightly faster than magma. The probabilistic approach of algorithm CIA specialized for dense or sparse matrices appears to reach the best computation times in almost every cases. www-lmc.imag.fr/lmc-mosaic/Jean-Guillaume.Dumas.

Table 1: Characteristic polynomial of dense integer matrices (comp. time and mem. allocation)

5. CONCLUSION We presented a new algorithm for the computation of the characteristic polynomial over a finite field, and proved its efficiency in practice. We also considered Keller-Gehrig’s third algorithm and showed that its generalization would be not only interesting in theory but produce a practicable algorithm. There, improvements of the dense computation over a finite field could be achieved and, at least, some heuristics could be built when a row-reduced form elimination can ensure generic rank profile. We applied our algorithm for the computation of the integer characteristic polynomial in two ways: a simple deterministic use of Chinese remaindering for dense matrix computations, and a probabilistic one, using integer minimal polynomial computation. The latter can be specialized for dense or sparse and structured matrices. This last algorithm outperforms existing softwares for this task. Moreover we showed that the recent improvements of [22, 14] should be highly practicable since the successful CIA algorithm is inspired by their ideas. It remains to show how much they improve the simple approach of CIA. Lastly, concerning the sparse computations, the search for multiplicities could be done by a pure blackbox algorithm, and would make it possible to handle problems of much higher size. These techniques have to be studied and compared to the blackbox algorithms of [23] and of [7], where preconditionners are introduced which use can be prohibitive in practice.

more than 2Gb of RAM to store the corresponding integer Krylov space, and the computation crashes (MT stands for Memory Thrashing). The memory usage of our implementations is much smaller than that of magma, and makes it possible to handle larger matrices. Concerning the probabilistic algorithms, CIA improves the computational time of the deterministic one of roughly 25% and is faster than magma-prob. In table 2, we denote by d the degree of the integer minimal polynomial and by ω the average number of nonzero elements per row within the sparse matrix. We Matrix n d ω Magma-prob Magma-det ILUK-det CIA-sparse CIA-dense IMP-sparse IMP-dense Fact Mul

A 300 75 1.9 1.14 1.31 1.1 0.32 1.22 0.03 0.93 0.04 0.25

U −1 AU 300 75 300 7.11 10.55 93.3 4.32 1.3 4.03 1.0 0.04 0.25

AT A 300 21 2.95 0.23 0.24 64.87 0.81 0.87 0.02 0.08 0.01 0.78

B 600 424 4 6.4 6.4 68.4 4.4 38.9 1.62 36.2 0.6 2.17

U −1 BU 600 424 600 184.7 185 2305 352.6 42.6 349 39.9 0.58 2.08

BT B 600 8 13 6.04 6.07 155.3 2.15 2.57 0.08 0.5 0.01 2.06

Table 2: CIA on sparse or structured matrices show the computational times of algorithm 4.1 (CIA), decomposed into the time for the integer minimal polynomial computation (IMP-sparse for [6, §3] and IMP-dense for 2.1 and chinese remaindering), the factorization of this polynomial (Fact) and the computation of the multiplicities (Mul). They are compared to the timings of the algorithm of section 4.1 and that of magma with and without the proof of correctness. We used two sparse matrices A and B of order 300 and 600, having a minimal polynomial of degree respectively 75 and 424. A is the almost empty matrix Frob08blocks and is in Frobenius normal form with 8 companion blocks and B is the matrix ch5-5.b3.600x600.sms presented in [6]. Again ILUK-det is faster than magma but still expensive. On the two matrices A and B, CIA-sparse is the fastest thanks to its usage of sparsity. Then, we made these matrices dense with an integral similarity transformation. The lack of sparsity slows down both magma and CIA-sparse, whereas CIA-prob maintains similar timings and outperforms every other algorithms. Lastly, we used symmetric matrices with small minimal polynomial (AT A and B T B). CIA-sparse

Acknowledgement We are indebted to the two anonymous referee for their worthy suggestions and especially to Allan Steel, for his cooperation and his helpful explanations about Magma.

References [1] J. Abdeljaoued and G. I. Malaschonok. Efficient algorithms for computing the characteristic polynomial in a domain. J. of Pure and Applied Algebra, 156:127–145, 2001. [2] W. Baur and V. Strassen. The complexity of partial derivatives. Theor. Computer Science, 22(3):317–330, 1983. [3] S. J. Berkowitz. On computing the determinant in small parallel time using a small number of processors. Inf. Process. Lett., 18(3):147–150, 1984. [4] J.-G. Dumas, T. Gautier, and C. Pernet. Finite field linear algebra subroutines. In T. Mora, editor, ISSAC’2002. ACM Press, New York, July 2002.

146

[5] J.-G. Dumas, P. Giorgi, and C. Pernet. FFPACK: Finite field linear algebra package. In Gutierrez [10]. [6] J.-G. Dumas, B. D. Saunders, and G. Villard. On efficient sparse integer matrix Smith normal form computations. J. Symb. Comp., 32(1/2):71–99, July–Aug. 2001. [7] W. Eberly. Reliable krylov-based algorithms for matrix null space and rank. In Gutierrez [10]. [8] J. v. Gathen and J. Gerhard. Modern Computer Algebra. Cambridge University Press. 1999.

to compute the 1-Frobenius form A0 of A by computing the sequence of matrices Ar = A, . . . , A0 , where Ai has the 2i Frobenius form and r = dlog ne. The idea is to compute Ai from Ai+1 by slicing the block M of Ai+1 into two n × 2i columns blocks B and h C. Then,i similarity transformations 0 C with the matrix U = Idn−2i C12 will “shift” the block B to the left and generate an identity block of size 2i between B and C. More precisely, one computes the sequence ˚ of matriˇ ces Ai,0 = Ai+1 , Ai,1 , . . . , Ai,si = Ai , where si = n/2i − 1, −1 by the relation Ai,j+1 = Ui,j Ai,j Ui,j , with the following notations :

[9] M. Giesbrecht and A. Storjohann. Rational forms of integer matrices. J. Symb. Comp., 34(3):157–172, 2002. [10] J. Gutierrez, editor. ISSAC’2004. Santander, Spain. ACM Press, New York, July 2004. [11] A. Householder. The Theory of Matrices in Numerical Analysis. Blaisdell, Waltham, Mass., 1964. [12] O. H. Ibarra, S. Moran, and R. Hui. A generalization of the fast LUP matrix decomposition algorithm and applications. J. of Algorithms, 3(1):45–56, Mar. 1982. [13] E. Kaltofen. On computing determinants of matrices without divisions. In P. S. Wang, editor, ISSAC’92. ACM Press, New York, July 1992.

C i,j1

0

0

0 Ai,j = Id B

i,j

0

i,j C2 Ai,j+1=

i

i,j+1

C2

i,j+1

2i

j2

0

Id

0

Id 2i

i,j+1

C1

B

Id

2i

(j+1)2 i

2i

[14] E. Kaltofen and G. Villard. On the complexity of computing determinants. Comp. Complexity, 13:91–130, 2004. As long as C1 is invertible, the process will carry on, and make at last the block B disappear from the matrix. This last condition is restricting. This is why this algorithm is only valid for generic matrices.

[15] W. Keller-Gehrig. Fast algorithms for the characteristic polynomial. Theoretical computer science, 36:309–317, 1985. [16] H. Lombardi and J. Abdeljaoued. M´ethodes matricielles - Introduction a ` la complexit´e alg´ebrique. Springer, 2004. [17] T. J. O’Gorman, J. M. Ross, A. H. Taber, J. F. Ziegler, H. P. Muhlfeld, C. J. Montrose, H. W. Curtis, and J. L. Walsh. Field testing for cosmic ray soft errors in semiconductor memories. IBM J. of R&D, 40(1):41–50, Jan. 1996.

A.2 Proof of lemma 3.1 We will denote by Xa...b the submatrix composed by the rows from a to b of the block X. For a given i, KG3 performs n/2i similarity transformations. Each one of them can be described by the following operations: −1 0 1: Bn−2 i +1...n = C1...2i B1...2i 0 0 2: B1...n−2i = −C2i +1...n Bn−2 i +1...n + B2i +1...n 3: C 0 = B 0 Cλ+1...λ+2i 4: C20 i +1...2i +λ + = C1...λ 5: C20 i +λ+1...n + = C2i +λ+1...n The first operation is a system resolution, and consists in a LUP factorization and two triangular system solve with matrix right hand side. The two following ones are matrix multiplications, and we do not consider the two last ones, since their cost is dominated by the previous ones. The cost of a similarity transformation is then: Ti,j = TLUP(2i , 2i ) + 2TTRSM (2i , 2i ) + TMM (n − 2i , 2i , 2i ) +“ TMM (n, 2i , 2i ).” Thus,

[18] C. Pernet. Calcul du polynˆ ome caract´eristique sur des corps finis. Master’s thesis, U. J. Fourier, June 2003. www-lmc.imag.fr/lmc-mosaic/Clement.Pernet. [19] C. Pernet and Z. Wan. L U based algorithms for characteristic polynomial over a finite field. SIGSAM Bull., 37(3):83–84, 2003. Poster available at www-lmc.imag.fr/ lmc-mosaic/Clement.Pernet. [20] A. Steel. Personnal communication. 2005 [21] A. Storjohann. Algorithms for Matrix Canonical Forms. PhD thesis, Institut f¨ ur Wissenschaftliches Rechnen, ETHZentrum, Z¨ urich, Switzerland, Nov. 2000. [22] A. Storjohann. Computing the frobenius form of a sparse integer matrix. Manuscript, Apr. 2000. [23] G. Villard. Computing the Frobenius normal form of a sparse matrix. In V. G. Ganzha, E. W. Mayr, and E. V. Vorozhtsov, editors, CASC’00, Oct. 2000.

we have TLUP(m, n) =

ω−2

Cω mω−1 2ω−1 −2

−1 n − m 22ω−1 −1 i

i

[5, Lemma 4.1] and [18]) and TTRSM (2 , 2 ) =

APPENDIX A. A.1

(from

Cω mnω−1 . 2(2ω−1 −1)

Cω 2ω−2 Cω (2i )ω + 2ω−2 (2i )ω + 2(2ω−2 −1)(2ω−1 −1) −1) ( Cω (n − „ 2i )(2i )ω−1 + Cω n(2i )ω−1 . « Rewrite this as Ti,j = 2ω−3 + 2ω−1 − 1 i ω − 1 +2nCω (2i )ω−1 and, Cω (2 ) (2ω−2 − 1) (2ω−1 − 1)

Therefore Ti,j =

ON KELLER-GEHRIG’S THIRD ALGORITHM Principle of the algorithm

|

We first recall the principle of this algorithm, so as to determine the exact constant in its algebraic time complexity. This advocates for its practicability. First, let us h define a mi 0 M Frobenius form as a n × n matrix of the shape: Idn−m M21 . Note that a 1-Frobenius form is a companion matrix, whose characteristic polynomial is given by the opposites of the coefficients of its last column. The aim of the algorithm is

{z

Dω

}

Plog(n/2) Pn/2i −1 as the total cost is T = i=1 Ti,j , it becomes j=1 ´ Plog(n/2) ` n i ω T = − 1 C D (2 ) + 2nC (2i )ω−1 . Simω ω ω i i=1 2 Plog(n/2) plifying, we obtain T = Cω i=1 (Dω − 3)n(2i )ω−1 + Plog(n/2) i x x 2 i ω−2 i ω −1 = 2n (2 ) − Dω (2 ) and since (2 ) = n2x −1 i=1 x x n + o(n ) this ends the proof. 2x −1

147

´ Selfintersections of a Bezier Bicubic Surface Andre´ Galligo, Jean Pascal Pavone∗ Laboratoire J-A. Dieudonne, ´ UMR 6621, Universite´ de Nice Sophia-Antipolis, Parc Valrose, 06108 Nice Cedex 2, France {

galligo, jppavone}@math.unice.fr

ABSTRACT

1

We present the computation of selfintersections as a major problem in Computer Aided Geometric Design (CAD) and Geometric Modeling, and particularly for patches of parametrized bicubic surfaces. Then we expose two complementary contributions on that subject with Computer Algebra tools: First, a specific sparse bivariate resultant adapted to the corresponding elimination problem, second a semi-numeric polynomial solver able to deal with large system of equations with floating point coefficients. Examples and timings are provided.

0

2

3

(a)

111111111111111 000000000000000 1 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 2 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 0 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 3 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 4 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111

(b)

Figure 1: A border with a selfintersection

Categories and Subject Descriptors I.1.2 [Computing Methodologies]: Symbolic and Algebraic Manipulation—Algorithms; J.6 [Computer Applications]: Computer-Aided Engineering—computer-aided design.

General Terms Algorithms.

Keywords surfaces of bidegree (3, 3), implicit equations, sparse resultant, singular locus, polynomial solver, floating point coefficients, implementations.

1.

INTRODUCTION

A common representation of surfaces in Solid Modeling and Computer Aided Geometric Design uses parametrized bicubic patches, i.e. images of maps Φ : [0, 1] × [0, 1] → R3 (t, u) → Φ(u, v) =

„

Φ1 (u, v) Φ2 (u, v) Φ3 (u, v) , , Φ0 (u, v) Φ0 (u, v) Φ0 (u, v)

«

∗ The authors were supported by the european project GAIA II (IST-2001-35512).

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. ISSAC’05, July 24–27, 2005, Beijing, China. Copyright 2005 ACM 1-59593-095-7/05/0007 ...$5.00.

where Φ0 , Φ1 , Φ2 , Φ3 are polynomials with real coefficients and bidegree (3, 3). They are called B´ezier when Φ0 (u, v) = 1. These patches are encountered in many applications [13]. Spline surfaces are made by gluing together such patches. So the result we present is a main step towards the computation of the selfintersections of a bicubic spline surface. There are many articles presenting methods and algorithms to intersect two patches (see e.g. [28], [25],[22],[26]), but very few papers address the computation of selfintersections. Moreover selfintersection points of these patches are often difficult to detect [1, 15]. This difficulty became a major problem in CAD as we will explain in section 2. Here we propose two distinct contributions towards the solving of this problem using tools from Computer Algebra. First in section 3 and 4, we recall useful bounds from projective algebraic geometry and set the system of equations via a suited change of variables. Then in section 5, we present a specific bivariate resultant adapted to the corresponding elimination procedure. Our approach seems new as it mixes, in an elementary way, sparse resultant and Groebner bases techniques. Alternatively, one can use Khetan’s bivariate resultant [24]. This allows to get in section 6 a bivariate equation of bidegree (44, 44) describing the selfintersection locus. In section 7, we describe a polynomial solver for systems with floating point coefficients that we implemented, then we apply it to determine points needed to draw accurately the selfintersection locus. In the conclusion we propose different directions of research.

2. MOTIVATIONS A domain in the plane is represented in C.A.D. by the list of the curve segments delineating its border. This list represents a well-defined domain if, as in the right picture

148

Figure 2: A surface with selfintersections

Figure 3: A thin selfintersection locus

of figure 1, each segment is free of loop. Now if, as in the left picture of figure 1, there is a loop and a selfintersection, the domain is ill-defined. To correct this defect, one should compute the selfintersection points and reparametrize the border without the loop. This procedure has been applied to the left picture in order to get the right picture, which becomes correct and well suited for further processing. C.A.D. systems use a similar representation for volumes. A volume is delimited by the list of its faces with adjacency conditions. Here also each face should not contain any loop or protruding part. To correct this defect, one should be able to compute the selfintersections of the surface patch. In that case, selfintersections do not appear as isolated points but along a curve. So we should find all such curves (that we call double points curves) lying on the surface. In general, a designer does not produce selfintersections on purpose when she (or he) uses a C.A.D. system. They are created when she calls some built-in function (such as offset or draft or sweep) or when she proceeds to the junction between two already constructed surfaces. Often the loops formed by these selfintersections are very small, hence they are hard to detect on the screen. In figure 2, we show a bicubic patch with a selfintersection loop. This corresponds to an example which will be defined and studied in section 8, we also show in figure 3 another example with a thin selfintersection locus. When it detects a possibility of selfintersection that it cannot correct, a C.A.D. system blocks the call of the corresponding function and sends a warning to the user. If this happens several times in a session, the user gets frustrated. So, the question became an issue for C.A.D. systems developers. This problem is one of the main topics of the European project GAIA II (cf. [14]). Several strategies have been developed to address it: either via multivariate resultant [3], or via special cases study [16] or via approximate implicitization [8] and more recently [32], or via numerical methods ([15]). The numerical method algorithm is very efficient. We used it to compute the complicated selfintersection locus

shown in figure 3. However it relies on samplings. Hence it has the weakness common to all sampling techniques: if the size of a loop in the selfintersection locus is smaller than the step-size of the sampling, it might become invisible and the program is unable to compute it. There are heuristics to choose the step-size, but they do not really overcome this difficulty. Therefore, symbolic (or semi-numeric) methods are essential in order to completely solve the question. For that purpose, the representation of the parametrization should be also algebraic. The most commonly used algebraic representation in C.A.D. is the so called bicubic NURBS and particularly the B´ezier bicubic patches. The core of the geometric problem we address, is to determine a sharp lower bound of the size of the loops in order to fix a step-size. Thus, approaches relying on Computer Algebra might be conclusive. It would be even better to determine a point on each such loop as well. Here, we will present two different tools developed for this purpose: • First, a specific sparse resultant allowing to compute a bivariate polynomial, which is the equation of a plane projection of the 4D-double points locus. This polynomial is represented via a (21, 21) determinant and has bidegree (44, 44). So it is a challenging question in Computer algebra to study the corresponding curve and bound the size of its real loops. • Second, a semi-numeric polynomial solver. This will be useful to determine the critical points of a projection of the selfintersection locus, hence ease the solution of our C.A.D. problem.

3. BOUNDS ON THE DEGREES We first present a general analysis in the complex projective case, then it will be restricted to the real and affine setting. The complex projective space of dimension r ∈ N \ {0} is denoted by Pr = Pr (C). In this section, we derive the formula for the degree of the double curve of a general bicubic surface. (See also [31],

149

m

[18], [16]). Let the image X of 1

1

Φ : Q = P × P −→ P

3

be a projective bicubic surface. Proposition 3.1. For generic choices of Φ, X is a surface of degree 18. The double points curve Y in P3 has degree 132. The curve in P1 × P1 which maps 2-1 to it via Φ has bidegree (44, 44). Proof. Let X denote the image of the map Φ. It suffices to count the number of intersection points between X and a general line. Such a line is defined by two linear equations. Their pull-back in P1 × P1 provide two general equations of bidegree (3, 3). So the corresponding curves intersect in 18 points. A generic hyperplane section of X is the image of a curve of type (3, 3) on the quadric Q, and so has geometric genus 4. However, this hyperplane section is a plane curve of degree 18, since X has degree 18. If the hyperplane section is generic, then the only singularities will be double points, and these double points will lie on the double point curve Y of X. The genus g, degree d, and number of double points δ on a plane curve (whose only singularities are double points) are related by the formula g = (d−1)(d−2) − δ. Thus, the number 2 of double points on our double curve is 136 − 4 = 132. The inverse image of the double point curve Y will be the double of a curve of bidegree (a, b) on Q, for some a, b. If Φ maps this curve 2-1 to P3 , then 3a + 3b = 2(132), and so a + b = 88. For generic choice of φ, a and b must be equal, and so the curve in Q which maps 2-1 to the double curve has bidegree (44, 44). ✸ We will also consider the curve in (P1 × P1 ) × (P1 × P1 ) which maps 1-1 to the double points curve in P3 and its projections D on the second and third P1 component. This curve D also maps 2-1 to Y , so the same reasoning applies to it, then D has bidegree (44, 44). In the real and affine case, the previous loci can break into several components and some components can lie at infinity. Therefore the previous numbers 18, 44, 44 are a priori only bounds for the respective degrees. However in the generic case, we expect that these bounds are effectively reached. This will be confirmed in section 6 below.

4.

k Figure 4: A very sparse support

We let: u = u1 + l; v = v1 ; u = u1 ; v  = v1 + lk. Then we replace the previous condition Φ(u, v) = Φ(u , v  ) by the 3 equations: T :=

Each component Ti of T is a polynomial of multidegree (3, 3, 2, 3) in (u1 , v1 , l, k). Our geometric task is equivalent to describe the curve corresponding to C in this new setting. More precisely we consider its projection on the (u1 , v1 )-plane, which is the same that the projection of C on the (u , v)-plane. It is an easy procedure to pass from one of these curve to another one, once we have determined it. So the algebraic question amounts to eliminate l and k in T . For that purpose, we look more closely on the monomial support in (l, k), common to each Ti . We observe that only 6 monomials do appear: k3 l2 , k2 l, l2 , k, l, 1. To decrease the total degree of these monomials, we set: 1 T := m2 ∗ subs(l = m ,T) and get a polynomial in the 6 monomials k3 , k2 m, 1,km2 , m, m2 , hence of total degree ≤ 3. Thus the 3 equations Ti share the same monomial support which is very sparse. See figure 4. In the next section we will present a bivariate resultant adapted to this support, then we will be able to eliminate the two variables l and m between the 3 equations Ti .

SETTING THE EQUATIONS We start with a parametrization

Φ : [0, 1] × [0, 1] → R3 (u, v) → Φ(u, v) =

Φ(u1 + l, v1 ) − Φ(u1 , v1 + lk) = 0. l

„ « Φ1 (u, v), Φ2 (u, v), Φ3 (u, v)

5. A SPECIFIC SPARSE RESULTANT A resultant of 3 general polynomials in 2 variables g1 , g2 , g3 having given monomial supports (N is the number of terms of these 3 supports) is, by definition, a polynomial R with integer coefficients in N variables which satisfies the following property: R is irreducible, and

where Φ1 , Φ2 , Φ3 are polynomials with real coefficients and bidegree (3, 3) in the variables (u, v). We denote by S the image of Φ. we will assume that the bidegree is not smaller (e.g. (2, 3) or (2, 2)), otherwise simpler specific equation should be used. The set of double points of Φ can be characterized by a curve C in R4 or its algebraic closure or simply by its projection (generically 2 to 1) on R2 .

R(coef f (g1 ), coef f (g2 ), coef f (g3 )) = 0 if and only if these 3 polynomials have a common root in a specified domain. In our situation it is variety (a kind of toric variety) embedded in P5 (see [20] for a formal presentation). Several families of multivariate resultants have been studied and some implementations are available. One of them called MULTIRES [4] is a Maple package, with a function

C = {(u, v, u , v  ) s.t. (u, v) = (u , v  ), Φ(u, v) = Φ(u , v  )}. In order to discard the unwanted trivial solution, we assume u > u and we perform the following change of variables, (l is in ]0, 1] and k is in R).

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C3 := coef f (f3 , x ∗ y 2 ) = determinant([a1 a2 a3 ], [b1 b2 b3 ], [c1 c2 c3 ]). Then we choose the 21 polynomials: L := [g1 , g2 , g3 , y ∗ g1 , y ∗ g2 , y ∗ g3 , x ∗ g1 , x ∗ g2 , x ∗ g3 , x ∗ y ∗ g1 , x ∗ y ∗ g2 , x ∗ y ∗ g3 , y 2 ∗ g1 , y 2 ∗ g2 , y 2 ∗ g3 , x2 ∗ g3 , x2 ∗ g1 , x2 ∗ g2 , x ∗ y ∗ g4 , y 2 ∗ g4 , y 3 ∗ g5 ]. We observe that they all have degree at most 5. We denote by M the matrix of their coefficients in the basis C formed by the set of all 21 monomials of degree at most 5. C := [1, x, y, x ∗ y, x2 , y 2 , y 3 , x2 ∗ y, x ∗ y 2 , x3 , y 4 , x ∗ y 3 , x2 ∗ y 2 , x3 ∗ y, x4 , x ∗ y 4 , x2 ∗ y 3 , x3 ∗ y 2 , x4 ∗ y, x5 , y 5 ]. The matrix M is square and it is generically of full rank 21. This last assertion is easily checked on a randomly chosen set of coefficients. We denote by R1 the determinant of this matrix M . It is a polynomial in the 18 indeterminate a1 , ..., c6 with integer coefficients. We claim that this polynomial R1 is divisible by the following product which is also a polynomial in the same ring: R2 := B22 ∗ C32 ∗ B5 . 1 and we claim that R is We call R the exact quotient R R2 an irreducible polynomial and is the resultant of (g1 , g2 , g3 ). In this article we will not give a traditional mathematical proof of these claims. Indeed, this approach and these observations need to be generalized and compared with other more sophisticated and efficient approaches such that of [24]. This will be conduced in a future work. However we present a convincing experimental argument relying on the comparison with the output of the program MULTIRES [4] which can be launched with Maple. It was developed by I. Emiris, widely tested, and founded on published papers. We also more briefly compared our output with those of Khetan’s software, and they agree. We performed 100 tests where the coefficients of the gi were affine polynomial in a variable z with random integer coefficients (with no common divisors). For each test the polynomial R was irreducible and was a factor of the polynomial output by MULTIRES. If we conduced this comparison on N such tests, for a very big N which can be precised, this will lead to a certification. Now we detail the characteristics of the polynomial R. R is antisymmetric in (g1 , g2 , g3 ), e.g. R(g1 , g2 , g3 ) = −R(g2 , g1 , g3 ). R is a polynomial of total degree 24 in Z[a1 , . . . , c6 ], it has degree 8 in each variable, and also partial total degree 8 in each 3 sets of variables such as (a1 , . . . , a6 ). It has also partial total degree 8 in each 6 sets of variables such as (a1 , b1 , c1 ). All these computations of resultants are fast and, even not optimized, take less than a second.

spres which is easy to handle and computes a sparse resultant. For a multivariate polynomial, sparseness is generally expressed by its Newton polytope. In our case, we would like to take into account that the support of the polynomials Ti , defined in the previous section, has only 6 monomials while its Newton polytope has 9 terms. So we decided to build a resultant adapted to that situation. MULTIRES only output a multiple of the aimed resultant, moreover the extra factor is not given. This is not satisfactory with floating point coefficients because ill-conditioned data can be produced. This weakness has been successfully treated in an article by C. d’Andrea [2]. Also, let us remark that the case of resultants of bivariate polynomials have been extensively studied by A. Khetan [24] who used the so-called Toric-Tate resolution and some adapted cohomology techniques [12]. These papers are not at all elementary. We found very recently that Khetan has on his web page a Maple implementation with a function bires which computes a sparse resultant in 2 variables via a determinant and without extra factor. We could apply successfully this function on our data, it gives the same final result that our homemade resultant, that we now briefly present. Our approach is elementary as it relies on simple observations. Let us first illustrate our first observation on a basic univariate case. Let n > p be two integers, and F1 , F2 be two univariate polynomials with the following form: F1 := a ∗ xn + ap ∗ xp + . . . + a0 ; and similarly for F2 but with the letters bi . Then the usual (often called Sylvester) resultant is given by a 2n × 2n determinant. If we proceed to the inter-reduction between F1 and F2 (this corresponds to the spoly function in a Groebner package) we get a polynomial F3 = b ∗ F1 − a ∗ F2 of degree p. The resultant of F1 and F3 is given by a (n + p) × (n + p) determinant. Up to the multiplication by ap , the two resultants are of course equal. In order to diminish the exponent of that power of a we can replace, in the last determinant, the first p (out of n) lines containing the coefficients of F3 by the coefficients of F2 . Indeed F3 = b ∗ F1 − a ∗ F2 and the corresponding monomial products of b ∗ F1 are multiple of lines already in that determinant. So we form a new determinant that we denote by R1 , it has p lines of F1 , p lines of F2 and n − p lines of F3 . It turns out that the Sylvester resultant of F1 , F2 equals this specific more compact resultant R1 (F1 , F2 , F3 ). Now, we present our recipe for computing our sparse resultant adapted to our C.A.D. problem. We consider 3 polynomials in 2 variables denoted g1 (x, y),g2 (x, y),g3 (x, y) with the same monomial support x3 , x2 y, xy 2 , 1, y, y 2 . g1 = [a1 , a2 , a3 , a4 , a5 , a6 ] ∗ transpose([x3, x2 y, xy 2 , 1, y, y 2 ]), and similarly for g2 and g3 with letters b and c instead of a. We assume that the coefficients are generic. We first construct two following polynomials g4 and g5 of a Groebner basis completion of the ideal spanned by g1 , g2 and g3 , with respect to the total degree ordering tdeg(x, y). With an abbreviated Maple syntax, f2 := reduce(g2 , [g1 ]), f3 := reduce(g3 , [g1 , g2 ]), g4 := reduce(xf3 , [f1 , f2 ]), g5 := spoly(g4 , reduce(xg4 , [f2 , f3 ]). We will also need the following notations: B2 := coef f (f2 , x2 ∗ y) = a1 ∗ b2 − a2 ∗ b1 ; B5 := coef f (f2 , y) = a1 ∗ b5 − a5 ∗ b1 ;

6. IMPLICIT EQUATIONS We let R denote the resultant defined in the previous section, and Ti the 3 polynomials defined in section 4 that we first view as polynomials in the main two variables (k, m) and coefficients polynomials in (u, v) of bidegree (3, 3). We form the polynomial P (u1 , v1 ) := R(T1 (u1 , v1 ), T2 (u1 , v1 ), T3 (u1 , v1 )). Following the description of R given at the end of the previous section, we can expect that the degree of P (u1 , v1 ) in u1 is bounded by 3×24 = 72. But due to the special form

151

of the coefficient of the Ti , there are some simplification, and generically the degree of P (u1 , v1 ) in u1 is 44, and similarly for v1 . This is exactly the maximal number predicted by our geometric analysis in section 3. Indeed P (u1 , v1 ) = 0 is the equation of a plane projection of the curve C. Therefore to certify that this bidegree (44, 44) is generically obtained, it is enough to check that it is reached only on one example. Moreover we can see that the total degree of P (u1 , v1 ) is 64. This P is still a large polynomial with more than 500 monomials, thus it is not easy to handle. We were not able to draw the implicit curve D defined by P , using the available drawing devices. In order to draw the portion of the curve D contained in [0, 1]2 , we should find a point on each of its connected components. The ones crossing the border of the square are easily found, so the difficulty is essentially concentrated on the internal loops. A common way to find a point on such a loop is to compute all the critical points of the projection of D on an axis, e.g. the u1 -axis. This amounts to solve the bivariate system: (u1 , v1 ) = 0. P (u1 , v1 ) = 0 ; ∂P u1 As this system is rather large, we provide an alternative approach in the next section. Indeed these critical points are projections on the plane (u1 , v1 ) of the critical points of the projection of C on the u1 -axis. In order to compute these points we express that the Jacobian determinant of the following (3 × 3)-matrix A vanishes. A := matrix([

Figure 5: Principle of the IPP projection

Figure 6: IPP reduction (a), our solver (b)

the graphs of the equations to reduce the domain for each variable involved in the process. The projection used by IPP is illustrated in figure 5.

∂T ∂T ∂T , ]). , ∂v1 ∂l ∂m

Our solver presents significant improvements compared to the IPP ; firstly in the use of these projections. The IPP solver computes the convex hull (figure 6 (a)) when we introduce the curve image of the polynomials defined by the projection (figure 6 (b)). This new procedure induces an acceleration of the convergence. It allows earlier exclusion of unwanted sub-domains as we see on figure 7 (a), and proceeds to much less subdivisions as illustrated on figure 7 (b). The second main point, which differentiates our solver from the IPP, is the introduction of a preconditioner inspired by the one developed for interval analysis methods and called mid-point Jacobian preconditioner[23]. The preconditioner allows our solver to reach a quadratic local convergence, which is not the case for IPP. This is proved in [27].

The determinant T4 := det(A) is a polynomial of multidegree (9, 8, 5, 8) in (u1 , v1 , l, m). Therefore we have to solve a system of 4 polynomials in 4 variables: (T1 , T2 , T3 , T4 ) = 0. An alternative possibility is to proceed to a partial elimination, say of the variable l in T , then solve a system of 3 equations in 3 unknowns. We developed a solver based on multivariate Bernstein representation of polynomials which is adapted to our setting.

7.

A POLYNOMIAL SOLVER

Several kinds of multivariate polynomials system solvers have been developed by different authors. However here we are essentially interested by real solutions in a bounded domain, basically an hyper-box in the parameters space. A natural approach is to choose solvers based on the Bernstein/B´ezier representation of polynomials commonly used in C.A.D. (see e.g. [13]) which better reflects what might happen in the box. That representation of polynomials is detailed on an example in the next section: a polynomial surface patch is given by its so called control polyhedron. The basic principle of our solver, developed by the second author in his PhD thesis ([27]), is similar to that of the IPP solver [29], considered as one of the best solver for C.A.D. problems. A description of our solver can be found in [27] and an article will appear soon. For other references about Bernstein Basis subdivision solvers see [30], [17] and [9]. The solver performs a sequence of carefully chosen subdivisions and reductions to determine a finite number of very small boxes which may contain roots of the system. These two solvers rely on projections of the control polyhedrons of

Another significant point to be considered is the cost of an iteration. For our solver, it is very close to the cost of the application of de Casteljau’s algorithm to express the equation on the current domain; it is about O(ndn+1 ) when each of the n variables is of degree d. Let us also note that,

(a)

(b)

Figure 7: exclusion (a) and multiple reductions (b)

152

because of the use of projections, the number of iterations increases strongly with the dimension of the problem. Finally, let us remark that it is difficult to say a priori which from the two following strategies is the best: • First eliminate symbolically a variable in a system of equations, and thus increase the degrees, and then apply the solver, or • Keep the degrees small but accept a larger number of variables. In the next section we provide some timings for systems in 4 variables.

8.

EXAMPLES AND TIMINGS

We apply the preceding solver and the procedure described in section 6 to the following example. As usual in C.A.D., the basis chosen to express the polynomials is not the monomial basis but the Bernstein basis: B30 (u) = (1 − t)3 , B31 (t) = 3(1 − t)2 t, B32 (t) = 3(1 − t)t2 , B3i (t) = t3 . Hence each coordinate function (x, y, z) of the parametrization is expressed by an array (4, 4) of coefficients. At the intersection of the line i + 1 and column j + 1, one finds the coefficient of B3i (u)B3j (v). Here are these coefficients for the example we consider. The corresponding surface patch appeared in figure 2 and was referenced in section 2. If we form the 3-vectors obtained by collecting the coefficients for x, y, z corresponding to the same term B3i (u)B3j (v), we get 16 vertices in R3 , they define the so-called control polyhedron of the surface patch. 3 2 1.718019 0.045451 1.220627 −8.437654 6 2.245235 0.572671 −0.528670 −1.472482 7 7 x=6 4 −2.706853 −3.948356 −0.572671 −2.245235 5 2 −2.565772 −4.284708 −3.824920 −1.718019 3 −5.030251 −3.145330 −2.801745 −4.636559 6 −3.561677 −1.676751 0.410812 1.075778 7 7 y=6 4 −2.077333 −0.843524 1.676751 3.561677 5 3.530333 5.030251 3 2 −1.994069 0.916993 1.935424 2.807696 0.101320 −2.755474 6 −0.227130 0.645141 −0.162758 0.015489 7 7 z=6 4 0.790476 −1.114797 −0.645141 0.227130 5 −1.779410 −0.718161 −2.442728 −1.935424 We tested the procedure relying on our solver on this example, and computed the coordinates of the critical points. It worked efficiently (about 2 to 5 seconds on a PC1 ), and we performed several such computations, considering critical point of different projections. In the first picture (figure 8) we represented the results for 4 directions of projections in the (u1 , v1 )-plane. The 4 pairs of associated points are respectively represented by a square, a cross, a triangle and a double cross. In the second picture (figure 9 ) we see them on the surface in the 3D space. We also present the corresponding data for the second example: figure 10 and figure 11. Here are the precise coordinates of these points and in a table the corresponding timings. 1

Figure 8: Extrema points in the domain 1

Figure 9: Extrema points on the surface 1

Figure 10: Extrema points in the domain 2

2.0Ghz P4-M (256MB)

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produces the approximate coordinates of a critical point of the curve D, say for the projection on the v1 -axis. Let K = (U1 , V1 ) be such an approximate extrema. In order to find a point K  = (U1 , V  ) on the curve D near K, we can proceed as follows. K  is the intersection of the line (u1 = U1 ) with D. So V  is a root of the univariate polynomial h(v1 ) := P (U1 , v1 ), where P is the implicit equation of D defined in section 6 via a specific resultant. We now specialize the construction of that resultant to (u := U1 ) avoiding ill-conditioned divisions. This process does not apply directly to our data: we tried but everything went wrong. First the polynomial h(v1 ) we got, had a degree much higher than the expected 44. Moreover there were no root of this h(v1 ) near the expected value. The reason seems that the norm of the matrix whose determinant gives the resultant (ours or Khetan’s one) is too large and induces ill-conditioned behavior. This numeric problem is tough and should be addressed with suited tools: e.g. SVD and generalized eigenvalues problems. This will be done in a future work. So for the moment, our first tool can only be used with exact input. While our second tool is more satisfactory for the considered application. The timings we presented in section 8 are good and show that the program can be used in a C.A.D. system. It is not only efficient but also very robust as it delivers a solution together with a (small) bound on the possible error.

Figure 11: Extrema points on the surface 2

time(s) ex 1 ex 2 extrema 1 u1 v1 l k extrema 2 u1 v1 l

u1 5.1 2.5 u 0.625524 0.681597 0.595082 0.706713 0.574816 0.744083 0.584561 0.73906 u 0.48842 0.412162 0.495816

v1 5.7 1.9 v 0.302895 0.0998794 0.310081 0.0946811 0.301324 0.113855 0.308121 0.107815 v 0.570798 0.599828 0.557963

l 5.3 2.4 u 0.273844 0.166012 0.27086 0.168988 0.260516 0.18959 0.267303 0.184838 u 0.359474 0.405339 0.361846

k 5.5 2.2 v 0.754153 0.72042 0.727953 0.736224 0.706247 0.771738 0.71729 0.765172 v 0.567007 0.599635 0.553971

• Also it would be very interesting to determine a sharp lower bound of the distance between two critical points of a projection of a curve, directly from the equation(s), and without computing them explicitly. • Here we treated the case of a bicubic B´ezier surface patch, the immediate next case to be considered is that of a rational bicubic surface patch (i.e. with a common denominator). The equations will be slightly more complicated but the same techniques could apply with better tuned algorithms, more powerful computers, even if the timings will be worse.

Let us briefly explain how the table should be read. The upper side of the table indicates for each example the time in seconds spent to compute the extrema in the directions u1 , v1 , l, k. Then we present for each example and for each direction u1 , v1 , l, k, the list of the computed solutions, given in the coordinates (u, v, u , v  ).

9.

• We defined a specific sparse resultant for our specific situation. It worked well for immediate purpose but it can be improved yet. For instance we should be able to get rid of some extra factors and possibly arrive to a factor free formula as in Khetan’s work [24]. Also the underlying ideas have to be developed further in a more general setting, with rigorous mathematical proofs and be compared with the concurrent more sophisticated approaches.

CONCLUSION

In this paper we have presented two different contributions towards the determination of the selfintersection locus of a B´ezier bicubic surface patch. Both were implemented. The first one uses a specific sparse resultant and produces an implicit equation of a plane projection of that locus. The second one uses a solver to compute accurately the coordinates of critical points on that locus, which help draw it. We plane to integrate them together with the sampling algorithm in a common framework called Axel ([19]) developed by the team Galaad at INRIA-Sophia Antipolis.

Acknowledgments: We thank L. Bus´e and M. Elkadi for fruitful discussions.

10. REFERENCES [1] L. Andersson, J. Peters, and N. Stewart, Self-intersection of composite curves and surfaces, Computer Aided Geometric Design, 15 (1998), pp. 507–527. [2] C. D’Andrea Macaulay style formulas for sparse resultants, Trans. Amer. Math. Soc. vol. 354, n. 7, pp. 2595-2629, 2002. [3] L. Bus´ e, M. Elkadi, and B. Mourrain, Using projection operators in Computer Aided Geometric

Here are several remarks and directions of research. • In theory, we should be able to check and improve the quality of the output of the semi-numeric solver, by using the specific resultant built in section 5. Indeed, the solver

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[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14] [15]

[16] [17]

[18]

Design, In Topics in Algebraic Geometry and Geometric Modeling, pp. 321-342. Contemporary Mathematics, 2003. L. Bus´ e, I.Z. Emiris and B. Mourrain, MULTIRES, http://www-sop.inria.fr/galaad/logiciels/multires. E. Cattani, D. Cox, A. Dickenstein, Residues in toric varieties. Compositio Math. 108 1 (1997), pp. 35-76. 1997. D. Cox, J. Little, D. O Shea,Using Algebraic Geometry, Springer-Verlag, New York Berlin Heidelberg (1998). E.-W. Chionh and R. N. Goldman, Degree, multiplicity, and inversion formulas for rational surfaces using u-resultants, Computer Aided Geometric Design, 9 (1992), pp. 93–108. Tor Dokken, Approximate Implicitization for surface intersection and self-intersection, ECITTT Euroconference on CAE Integration - tools, Trends and Technologies, 2001. G. Elber and M.S. Kim, Geometric constraint solver using multivariate rational spline functions, Proceedings of the sixth ACM Symposium on Solid Modeling and Applications, (2001), pp. 1–10. I. Z. Emiris, Sparse Elimination and Applications in Kinematics. Ph.D. Thesis, University of California at Berkeley, 1994. I.Z. Emiris and B. Mourrain, Matrices in Elimination Theory, J. of Symbolic Computation, *28*(1&2):3-44, 1999. A. Eisenbud, G. Fløystad, F. Schreyer, Sheaf cohomology and free resolutions over exterior algebras Available from math.AG/0104203 (preprint) (2000) G. Farin, Curves and surfaces for computer aided geometric design. A practical guide, Academic Press, Inc., Boston, MA, 1993. GAIA II, http://www.sintef.no/static/AM/gaiatwo/ A. Galligo and J. Pavone, A sampling algorithm for parametric surface self-intersection, preprint, (2004). A. Galligo and M. Stillman, The geometry of bicubic surfaces and splines, preprint, (2004). J. Garloff and A. Smith, Solution of systems of polynomial equations by using Bernstein expansion, Symbolic Algebraic Methods and Verification Methods, Springer, 2000. R. Hartshorne, Algebraic Geometry, Springer-Verlag, 1977.

[19] G. Gatelier, B. Mourrain, AXEL, http://www-sop.inria.fr/galaad/logiciels/axel [20] I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinski, Discriminants, Resultants and Multidimensional Determinants, Birkh¨ auser, Boston-Basel-Berlin, 1994. [21] M. E. Hohmeyer, A Surface Intersection Algorithm Based on Loop Detection, ACM Symposium on Solid Modeling Foundations and CAD/CAM Applications, pp 197–207, 1991. [22] M. Hosaka, Modeling of Curves and Surfaces in CAD/CAM, Springer-Verlag, New York, NY. 1992. [23] R.B. Kearfott, Chenyi Hu and M. Novoa, A review of preconditioners for the interval Gauss-Seidel method , Interval Computations (1) pp. 59-85, 1991. [24] A. Khetan,The resultant of an unmixed bivariate system, J. of Symbolic Computation, *36* pp 425-442, 2003. and http://www.math.umass.edu/ khetan/software/bires.mpl [25] Shankar Krishnan and Dinesh Manocha, An Efficient Surface Intersection Algorithm Based on Lower-Dimensional Formulation, ACM Transactions on Graphics, volume 16, number 1, pp 74–106, 1997. [26] N.M. Patrikalakis, Surface-to-surface intersections, IEEE Computer Graphics and Applications, volume 13, number 1, pp 89–95, 1993. [27] J-P. Pavone, Auto-intersection des surfaces param´etr´ees r´eelles, Th`ese d’informatique de l’Universit´e de Nice Sophia-Antipolis, D´ecembre 2004. [28] T. W. Sederberg and R. J. Meyers, Loop detection in surface patch intersections, Computer Aided Geometric Design, volume 5, number 2, pp 161–171, 1988. [29] E.C. Sherbrooke and N.M. Patrikalakis, Computation of the solutions of nonlinear polynomials systems, Computer Aided Geometric Design vol. 10 no. 5 pp. 379-405, 1993. [30] M. R. Spencer,Polynomial Real Root Finding in Bernstein Form, Ph.D Dissertation, Brigham Young University., 1994. [31] I. Shafarevitch, Basic Algebraic Geometry, New-York, Springer-Verlag, 1974. [32] J.B. Thomassen, Self-Intersection Problems and Approximate Implicitization. in Computational Methods for Algebraic Spline Surfaces, Springer pp 155–170, 2005.

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A Procedure for Proving Special Function Inequalities Involving a Discrete Parameter Stefan Gerhold

∗

Manuel Kauers

RISC-Linz Johannes Kepler Universitat ¨ A-4040 Linz, Austria

RISC-Linz Johannes Kepler Universitat ¨ A-4040 Linz, Austria

[email protected]

[email protected]

ABSTRACT

concerning the algorithmic treatment of inequalities. The importance of studying inequalities hardly needs to be emphasized. For instance, virtually any proof in analysis contains an estimation of some sort. There are classical textbooks [12, 15, 16, 23] on this broad subject, but, of course, the possibility of proving a given inequality automatically has some obvious advantages in comparison to table lookup. For some classical inequalities there is an underlying identity that makes the truth of the inequality obvious. For instance, Lagrange’s identity

We define a class of special function inequalities that contains many classical examples, such as the Cauchy-Schwarz inequality, and introduce a proving procedure based on induction and Cylindrical Algebraic Decomposition. We present an array of non-trivial examples that can be done by our method. Most of them have not been proven automatically before. Some difficult well-known inequalities such as the Askey-Gasper inequality and Vietoris’s inequality lie in our class as well, but we do not know if our proving procedure terminates for them.

n

n

n

x2k k=1

Categories and Subject Descriptors

k=1

yk2 −

x k yk


2 

=

k=1

1≤k
(xk yi − xi yk )2

immediately implies the Cauchy-Schwarz inequality

I.1.2 [Computing Methodologies]: Symbolic and Algebraic Manipulation—Algorithms; G.2.1 [Discrete Mathematics]: Combinatorics—Recurrences and difference equations

n

x k yk


2 

k=1

n

≤

n

x2k k=1

yk2 k=1

(n ≥ 0).

(1)

Askey and Gasper [2] found an identity that establishes the inequality

General Terms

n

Algorithms

(α,0)

Pk

(α > −1, −1 < x ≤ 1)

(x) > 0

k=0

Keywords

(2)

for a sum of Jacobi polynomials, which was applied in the first proof of the Bieberbach conjecture [9]. Regardless of whether such an identity is algorithmically provable [14, 10], finding a suitable one by human insight is indispensable for this line of attack. The inequality

Inequalities, Special functions, Computer Proofs, Cylindrical Algebraic Decomposition

1.

†

INTRODUCTION

While there is a panoply of algorithms for proving combinatorial and special function identities, not much is known

∞

∗Supported by the Austrian Science Foundation (FWF) grant F1305 †Partially supported by the Austrian Science Foundation (FWF) grant F1305 and the German Academic Exchange Service (DAAD) grant D/03/40515

k=n

1 k2 

n+k k 

<

2 n 

2n n

(n ≥ 1)

(3)



arose in work by Knopp and Schur [22]. Paule’s proof [19] of (3) contains an application of the extended Gosper algorithm [21]. Also, Paule [18] has applied several computer algebra tools in a proof of another inequality, which implies a conjecture of Knuth. Although computer algebra assisted, both proofs are altogether by no means mechanical. For some inequalities, computer algebra systems are able to simplify the functions that occur in it and verify the resulting inequality automatically, but this functionalities are rather limited. The present paper is an attempt at proving inequalities entirely automatically. All inequalities under consideration must depend on a discrete parameter, such as n in (1), (2)

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. ISSAC’05, July 24–27, 2005, Beijing, China. Copyright 2005 ACM 1-59593-095-7/05/0007 ...$5.00.

156

and (3). Our method is based on using Cylindrical Algebraic Decomposition (CAD) for constructing an induction proof for the conjectured inequality, given recurrence relations defining the involved quantities as specified in Section 2. The proving procedure, presented in Section 3, is not backed by a termination theorem, but it works surprisingly well on concrete examples. Section 4 is devoted to a collection of nontrivial examples that we were able to tackle with our method, and in Section 5 we show how our procedure can be applied to the problem of analyzing the sign pattern of oscillating sequences.

2.

It is possible to define quite a few interesting sequences by means of difference fields, including, e.g., all holonomic sequences. Many examples and useful closure properties for the class of expressible sequences are given in [13, 14].

2.2 Inequalities and Formulas We consider formulas involving inequalities for sequences over C := ∩ ¯ , the field of real algebraic numbers. Suppose a difference field K = C{t1 , . . . , tm } and a suitable choice of initial values is given. An atomic formula over K is of the form k1 ♦ k2 with k1 , k2 ∈ K and ♦ ∈ {=, 6=, ≤, ≥, <, >}. A formula over K is a boolean combination of true, false, and atomic formulas over K. For example, 



PRELIMINARIES

The inequalities in our consideration involve sequences that can be expressed by means of difference fields. A short introduction to the theory of difference fields, from which we need hardly more than the basic definitions, is given in this section. Later, we will introduce the set of formulas to which our method can be applied. These are essentially boolean combinations of inequalities on both sides of which there are sequences which can be expressed in terms of difference fields.

φ = t1 > 0 ∧ ¬(s2 t1 < 1 ∨ (st 2 + 1)3 ≤ t2 ) is a formula over C{t1 , t2 }. The definition of s is extended to formulas in the obvious way. Formulas over K give rise to formulas for sequences. The meaning of square brackets introduced in the previous section is extended to formulas. Continuing the example, we have

2.1 Difference Fields and Sequences

[φ]n = [t1 ]n > 0 ∧ ¬([t1 ]n+2 < 1 ∨ ([t2 ]n+1 + 1)3 ≤ [t2 ]n ).

A field K together with a distinguished endomorphism s : K → K is called a difference field. The map s is called the shift operator of the difference field. The theory of difference fields [6] allows an algebraic treatment of difference equations and sequences, quite in analogy to the treatment of differential equations and analytic functions by differential algebra. If K = (K, s) is a difference field, then for any r ∈ and any rat ∈ K 0 := K(t(0) , t(1) , . . . , t(r−1) ), we can turn K 0 into a difference field by extending s to s0 : K 0 → K 0 via

If φ is a formula then [φ] is a sequence of truth values. We say that φ is valid if [φ]1 , [φ]2 , . . . are all true. Note that for any specific number n, the truth of [φ]n can easily checked by inspection.

3. THE PROVING PROCEDURE Similarly to algorithms for proving identities, which reduce questions about recurrences to questions about algebraic sets that can be answered by Gr¨ obner basis techniques, we will reduce the question whether a formula φ is valid to questions about semi-algebraic sets which can be answered by Cylindrical Algebraic Decomposition [7, 5]. As a subroutine for the proving procedure given below, we will use a function Reduce which transforms a formula φ over C{t1 , . . . , tm } into an equivalent formula φ0 := Reduce(φ) such that φ0 = false whenever φ is unsatisfiable. Here we regard C{t1 , . . . , tm } just as a field of rational functions over C, ignoring its difference field structure, and equivalence and satisfiability are meant in this sense: the formula φ0 ⇐⇒ φ is true if it is true for any choice of real numbers in place of the si tj , and φ is called unsatisfiable if it is not true for any choice of real numbers in place of the si tj . For sets of formulas, we define



s0 (t(i) ) := t(i+1)

0

s (t

(r−1)

) := rat(t (0)

(0)

(1)

,t

(1)

,...,t

(r−1)

i = 0, . . . , r − 2, ).

(r−1)

The objects t , t , . . . , t are understood as algebraically independent indeterminates over K, and we will often write t, st , . . . , sr−1 t for them. We say that K 0 is obtained from K by adjoining a difference variable t to K. The number r is called the order of t. We write K 0 := K{t} and use the same symbol, s, for s and s0 . Difference fields can be constructed by starting with a field C of constants (i.e., s(c) = c for all c ∈ C) and successively adjoining difference variables t1 , t2 , . . . , tm as above. The resulting difference field is denoted by C{t1 , . . . , tm }. More general constructions are possible [6], but not of interest in this paper. A difference variable may serve as an algebraic formalization of a sequence. Let K = C{t1 , . . . , tm } be a difference field and r1 , . . . , rm be the respective orders of t1 , . . . , tm . Then K uniquely defines sequences [t1 ], . . . , [tm ] over C as soon as initial values [ti ]1 , . . . , [ti ]ri ∈ C are chosen for each i. The value [ti ]n is defined recursively via [ti ]n+1 = [st i ]n . For bad choices of initial values, the sequences may not be well-defined, because iterated application of the recurrence may eventually lead to a division by zero. The notation [·] is not only used for the sequences induced by the difference variables t1 , . . . , tm , but for any element of the difference field. Observe that the (possibly partial) mapping [·] : C{t1 , . . . , tm } → C only depends on the choice of some finitely many initial values.

Reduce({φ1 , . . . , φn }) := Reduce(φ1 ∧ · · · ∧ φn ). Evidently, a function Reduce can be implemented by a standard application of CAD. The basic idea of our procedure for proving a formula φ for sequences defined by difference fields is as follows: Like in the algorithm introduced in [13, 14] for identities, we want to construct a proof by complete induction on n. To this end, we first choose a number r0 and try to refute ψ := φ ∧ · · · ∧ sr0 −1 (φ) ∧ ¬sr0 (φ)



by Reduce. If ψ is indeed inconsistent, we have obtained the induction step for proving the desired inequality. If the refutation does not succeed, we choose another number r1



157

and try to refute 0

ψ := φ ∧ · · · ∧ s

r0 −1



(φ) ∧ ¬ψ ∧ · · · ∧ ¬s

r1 −1



for k = 1, . . . , ` + 1. If the left hand side of (4) is true, then in particular each χk and each

r1

(ψ) ∧ s (ψ),

k−1 ri −1

and, if necessary, iterate. The input of the procedure may include a knowledge base Φ of known valid formulas.



is true (k = 1, . . . , `), so by (5) each srk−1 (χk−1 ) is true (k = 1, . . . , `). Furthermore, χ`+1 = ¬ψ` = false by the termination criterion of the while loop, hence, again by (5) and the left hand side of (4), also sr` (χ` ) is true. Altogether we have shown that the right hand side of (4) is true. This establishes correctness if the procedure returns true. Now consider the case where the procedure returns false. Then [χk ]n = false for some k and some n. Repeated usage of (5) shows that then [χ0 ]r0 +r1 +···+rk = false, and hence φ is not valid.

1. i := 0

Of course the method can be easily adapted to prove the validity of formulas φ for n ≥ n0 instead of n ≥ 0 for any given n0 ∈ . Furthermore, it is also possible to allow continuous parameters x in the formulas, and constraints can also be specified for those parameters. Unfortunately there is no guarantee that the procedure will terminate. In fact, there are examples where it does not. The simplest example where nontermination occurs might be the inequality 3n > 2n : Define s(t1 ) = 2t1 , s(t2 ) = 3t2 and consider φ := t1 < t2 . Then an easy induction shows

3. If [φ]n is false for some n < r0 , return false



r0 −1

4. Θ := Φ ∪ {φ, . . . , s (φ)} ψ0 := Reduce(Θ ∪ {¬sr0 (φ)}) 5. While ψi 6= false repeat i := i + 1 Choose a number ri ∈ If [ψi−1 ]n is true for some n < ri , return false Θ := Θ ∪ {¬ψi−1 , . . . , ¬sri −1 (ψi−1 )} ψi := Reduce(Θ ∪ {sri (ψi−1 )})

ψi ⇐⇒ t1 < 0 ∧ ( 32 )r0 +···+ri −1 t1 < t2 ≤ ( 32 )r0 +···+ri t1 , and this is consistent for any choice of i and r0 , . . . , ri . In order to get a proof for 3n > 2n , it is necessary to supply additional information. For example, termination can be obtained by adding 2n > 0 or 3n > 0 to the knowledge base Φ. This is typical: for most of the examples given in Section 4.1 below, we had to supply additional information as well. However, it is usually sufficient to supply only trivial constraints such as n > 0 etc. to make the proof go through. It is also worth noting that additional information, which was supplied for a proof, often can itself be proven correct by the same procedure, if desired.

6. Return true Correctness of the procedure is ensured by the following theorem. Theorem 1. If the procedure above returns true, then φ is valid. If it returns false, then [φ]n = false for at least one n ∈ 0 .

Proof. Suppose the procedure returns true after ` iterations of the while loop, i.e., ψ` = false. We show by induction on n that [φ]n is true for all n ≥ 0 and that [ψk ]n is false for all n ≥ 0 (k = 0, . . . , ` − 1). W.l.o.g. we may assume Φ = ∅. For convenience of notation define χ0 := φ and χk := ¬ψk−1 for k = 1, . . . , ` + 1. Step 3 and line 3 of step 5 assert that [χk ]n is true for n = 0, . . . , rk − 1. This serves as induction base. As for the induction step, we show that ` ri −1 

4. EXAMPLES This section contains a collection of examples. We begin with a detailed execution of the proving procedure for proving the simple inequality n 

n! k=0

`

sj (χi ) =⇒ 

i=0 j=0



sri (χi ).

According to the definitions of χi and ψi and the specification of Reduce we have

s(t1 ) = t1 + 1 s(t2 ) = (t1 + 1)t2 s(t3 ) = −t3 t3 s(t4 ) = t4 − (t1 + 1)t2

k−1 ri −1 





i=0 j=0

sj (χi ) ∧ ¬srk−1 (χk−1 ) 

k−1 ri −1 

⇐⇒ ¬



i=0 j=0

j

s (χi ) ∨ s

rk−1

(χk−1 )







sj (χi ) =⇒ srk−1 (χk−1 ) 

(n ≥ 2),

( t1 ∼ n ) ( t2 ∼ n! ) ( t3 ∼ (−1)n ) ( t4 ∼ Σ )

The construction of these recurrences is the same as in [14]. Let us assume Φ = {t1 > 0} is the set of known facts. We have to apply the procedure to φ = t2 t4 > 0. We choose r0 := 1 and have to check one initial value. As the inequality is claimed only for n ≥ 2, we check that [φ]2

k−1 ri −1

⇐⇒

(−1)k >0 k!

which asserts the positivity of the derangement numbers. A suitable difference field can be constructed from the subexpressions of the formula. Take C{t1 , t2 , t3 , t4 } with

(4)

i=0

χk = ¬ψk−1 ⇐⇒ ¬

sj (χi )

i=0 j=0

INPUT: – a difference field C{t1 , . . . , tm } and initial values for the sequences [t1 ], . . . , [tm ] – a formula φ over C{t1 , . . . , tm } to be proven – a set Φ of formulas over C{t1 , . . . , tm } s.t. ψ is valid for all ψ ∈ Φ (Φ = ∅ allowed) OUTPUT: – true if φ is valid, false otherwise.

2. Choose a number r0 ∈



(5)

i=0 j=0

158

is true (cf. the remark right after the proof of Theorem 1): 2(1 − 1 + 12 ) = 1 > 0. We set

then 



3 4

n−

ψ0 := Reduce(Φ ∪ {t2 t4 > 0, s(t2 t4 ) ≤ 0}).

1 2

≤ Rn −

≤

n



√ k



t3 ≤ t4 < 0 (1 + t1 )t2 t3 ∨ t2 > 0 ∧ t 3 > 0 ∧ 0 < t 4 ≤ . (1 + t1 )t2

ψ0 = t 1 > 0 ∧ t 2 < 0 ∧ t 3 < 0 ∧

2 

k=1



n



√ 3 k



≤

3

(n ≥ 1)



k=1

• 11.1 in [15]: 







This is not the boolean false, so we have to continue the while loop. We choose r1 = 1 and check the initial value [ψ0 ]2 = false. Next we compute



n+

(n − 1) + 

··· +

2+

√ √ 1< n+1

(n ≥ 1)

ψ1 := Reduce(Φ ∪ {t4 > 0, ¬ψ0 , s(ψ0 )}).

Another bunch of interesting examples makes use of the idea that was introduced in [14], by which it is possible to bring sequences of variables (xn )n≥1 into the scope of our approach. The idea is to regard xn , xn+1 , xn+2 , . . . as a sequence without defining recurrence, and to introduce a fresh variable upon each shift. See [14] for details. Using this idea, it is possible to prove the Cauchy-Schwarz inequality entirely automatically without resorting to an underlying identity as outlined in the introduction. The following inequalities become completely routine, too.

Again using Mathematica’s CylindricalDecomposition, we now obtain ψ1 = false, hence the proof is complete. The choice of the values ri in the procedure is arbitrary. Different choices lead to the same result, but may differ with respect to efficiency. If we had chosen r0 = 2 instead of r0 = 1, we would have found ψ0 = false and an iteration would not have been necessary. A reasonable choice for most examples is ri = 1.

4.1 Example Gallery

• Thm. 6 on p. 112 in [16]:

Our method can be successfully applied to a number of nontrivial problems from textbooks on inequalities [12, 15, 16]. We were able to prove, among others, the following examples by our method, using only trivial facts like n > 0 as additional knowledge. To our knowledge, none of these inequalities can been proven by any other symbolic computation method.

n

n 



k=1

(−1)k−1 a2k ≥

(−1)k−1 ak 

2 

k=1

if (ak )k≥1 is positive and decreasing • Variations of 7.31 and 7.32 from [15]: k

• 3.27 in [15]:

!

xi

n

1 1 < n 4n 16 

2n n

2

< 

i=1 k−1 

1 3n + 1

(n ≥ 2)

>

!

xi

k=2

1−



(1 + ak ) > 1 +

ak

k=1



k=1

k=1 n

n

"

>

n

1+

ak

k=1



(1 − ak ) > 1 −

ak k=1

k=1

then (n ≥ 3, x ∈

ak ,

!

!

Fn (x) ≤ (x − 1) (x + 2)

n , n−1

n

"

>

1

F1 (x) = 1, F2 (x) = x,

n−3

xi

k=2

n

1

• 3.3.38 in [16]: If Fn (x) denotes the nth Fibonacci polynomial, defined by

2

!

i=1

n

2

>

k−1

• The Weierstraß inequalities (3.2.37 in [16]):

(0 < x ≤ 1, n ≥ 0)

2

xk



where xk > 0, x2 > 4x1 , and n ≥ 2

1 1 + nxn+1 ≤ 1 + n(1 − x)2 x−n (1 + n)xn 2

Fn+2 (x) = xFn+1 (x) + Fn (x),

n

n2 , n−1

i=1

• Levin’s inequality (3.2.12 in [16]):

2

(n ≥ 1)

• A variation of 4.15 in [15]:

A suitable choice for Reduce is Mathematica’s command CylindricalDecomposition [24], which returns

1≤

1 4

n+

(n ≥ 1) for 0 < ak < 1 with

)

n k=1 !

ak < 1

• An inequality of Beesack [3],

There is no need to restrict the defining recurrences to rational functions. Inequalities involving algebraic functions can be treated as well, provided that the algebraic functions are simple enough so that the CAD implementation can deal with them. This makes it possible to automatize the proofs for the following inequalities:

n 

k 

xi k=1

β





i=1

n 

xα i ≤

α+β



xk k=1



(n ≥ 1)

for xk > 0 and α ≥ 1, α + β ≥ 1, can be done for specific values α ∈ , β ∈ , e.g., for α = 2, β = −1. #

$

We are also able to prove inequality 5.16 from [15], • [17] If (Rn ) is defined by n Rn+1 = 1 + (n ≥ 1), Rn

n 

R1 = 1,

k=1

159

sin(kx) ≥

1 sin(nx) 2

(0 ≤ x ≤ π, n ≥ 1),

4.2 Difficult Examples

using

There are some prominent examples of quite difficult inequalities that also fit well into the class of inequalities we consider. One example is the Askey-Gasper inequality mentioned in the introduction. This inequality reduces for α = 0 to Fej´er’s inequality [1]. Another example is the inequality of Vietoris [15, § 0.8]: if (an )n≥1 is positive, decreasing, and satisfies

sin((n + 2)x) = 2 cos x sin((n + 1)x) − sin(nx) as defining recurrence and the identity (cos2 x − 1) − sin2 (nx) − 2 cos x sin(nx) sin((n + 1)x) − sin2 ((n + 1)x) = 1

(n ∈

)

%

as well as the facts −1 ≤ cos x ≤ 1 and 0 ≤ sin(nx) ≤ 1 as additional knowledge. The figure below shows the graph of n &

fn (x) := k=1

sin(kx) −

Choosing ak = 1/k gives the Fej´er-Jackson inequality [15] n

3

k=1

2.5

1.5 1 0.5 2.5

3

Inequalities which are not amenable to our proving procedure can sometimes be rewritten in a way that makes the proving procedure applicable. As an example, consider inequality (3) of Knopp and Schur. We have the relation ∞

1

&

k2 k+n n k=1 '

The procedure of Section 3 can be modified slightly in order to analyze the sign patterns of oscillating sequences. Consider C-finite sequences (fn )n≥0 defined by linear homogeneous recurrences with constant coefficients, for example √ √ √ f0 = 2 + 2, f1 = 2 + 10, f2 = −2 + 5 2 √ √ √ fn = (4 + 5)fn−1 − (5 + 4 5)fn−2 + 5 5fn−3 (n > 3).

(

where the first identity was provided by Mathematica, and (2) 2 the second follows from Thm. 1.2.7 in [1]. Hn = n k=1 1/k denotes the harmonic number of second order and ψ the digamma function [1]. Zeilberger’s algorithm [20, 21] delivers )

n−1

n−1

1

&

k2 k+n n k=1

&

=

'

k=1 (

3k2 + 3k + 1 2(k + 1)(2k + 1)k 2 '

2k−1 k−1

−

The initial values and recurrence coefficients are chosen such that (fn )n≥1 has the closed form √ (n ≥ 0) fn = 2 5n/2 (1 − 2 sin(nθ − π4 ))

1 (k + 1)2

with θ = arctan 12 . It is well known [4] that the numbers (nθ − π4 ) mod 2π lie dense in the interval [0, 2π], hence fn clearly has infinitely many positive and infinitely many negative values. Our goal is to obtain finer information on the sign of fn . As additional knowledge, we use the identity √ √ 10 25fn2 − 11 (14 + 13 5)fn fn+1 − 20 (2 − 6 5)fn fn+2 11 √ √ 2 2 2 (14 − 13 5)fn+1 fn+2 = 0, + (6 + 4 5)fn+1 − 11 + fn+2

(

by which (3) simplifies to π2 −1− 6

n−1 &

k=1

3k2 + 3k + 1 k2 (k + 1)(2k + 1) '

2k k (

<

2 n '

2n n

(0 < x < π, n ≥ 1).

5. APPLICATION: SIGN PATTERNS OF C-FINITE SEQUENCES

π2 − H(2) n 6

= ψ 0 (n + 1) =

sin(kx) ≥0 k

As opposed to the examples of Section 4.1, which could all be completed with a negligible amount of CPU time, we did not succeed in proving any of these outstanding inequalities. Already in the second iteration of the proving procedure, the CAD computations exceed our memory capacity (1Gb). We have tried both Mathematica’s CAD implementation and the QEPCAD system [8]. It is not clear whether the procedure terminates on any of these examples.

2

2

(0 < x < π).

k=1

&

1.5

ak sin(kx) > 0

n &

3.5

1

(1 ≤ k ≤ n/2),

then

1 sin(nx) 2

on 0 ≤ x ≤ π for n = 2, 3, 4, 5, suggesting the validity of the inequality for those special values of n.

0.5

2k − 1 a2k−1 2k

a2k ≤

, (

and this is indeed in the scope of our method. A minor obstacle is the fact that π is not algebraic and might cause difficulties in the CAD computations, but these are easily circumvented by regarding π as a parameter with the restriction 3 < π < 4. In this setup the proof is successful. The examples presented in this section are only a small selection of a large number of inequalities, mainly from [15, 16], that we were able to verify by means of our method. A collection of further examples is provided in form of a Mathematica notebook which is available on the world wide web [11]. Also details about the automated proofs for all the inequalities mentioned in this paper can be found in this file.

which was found by an ansatz with undetermined coefficients and verified by the algorithm proposed in [13]. In order to study the sign pattern of (fn )n≥1 , we use the procedure of Section 3 to prove that a certain sequence of sign changes determines the sign of the next value. Indeed, if C{t} is a difference field where t represents the sequence (fn )n≥1 , then Reduce({id}, {t ≥ 0, s(t) < 0}, {s2 (t) ≥ 0}) = false,

160

7. REFERENCES

where id is the identity above and Mathematica’s CylindricalDecomposition is again used for Reduce. Hence, if fn ≥ 0 and fn+1 < 0, then also fn+2 < 0. Let us write this as (+|0) − ⇒ −. In the same way, we can show that

[1] G. E. Andrews, R. Askey, and R. Roy. Special Functions, volume 71 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, 1999. [2] R. Askey and G. Gasper. Positive Jacobi polynomial sums II. American Journal of Mathematics, 98:709–737, 1976. [3] P. R. Beesack. On certain discrete inequalities involving partial sums. Canadian Journal of Mathematics, 21:222–234, 1969. [4] J. W. S. Cassels. An Introduction to Diophantine Approximation. Cambridge University Press, 1957. [5] B. Caviness and J. R. Johnson, editors. Quantifier Elimination and Cylindrical Algebraic Decomposition, Texts and Monographs in Symbolic Computation. Springer, 1998. [6] R. M. Cohn. Difference Algebra. Interscience Publishers, John Wiley & Sons, 1965. [7] G. E. Collins. Quantifier elimination for the elementary theory of real closed fields by cylindrical algebraic decomposition. Lecture Notes in Computer Science, 33:134–183, 1975. [8] G. E. Collins and H. Hong. Partial cylindrical algebraic decomposition for quantifier elimination. Journal of Symbolic Computation, 12(3):299–328, 1991. [9] L. de Branges. A proof of the Bieberbach conjecture. Acta Mathematica, 154:137–152, 1985. [10] S. B. Ekhad. A short, elementary, and easy, WZ proof of the Askey-Gasper inequality that was used by de Branges in his proof of the Bieberbach conjecture. Theoretical Computer Science, 117:199–202, 1993. [11] S. Gerhold and M. Kauers. A procedure for proving special function inequalities involving a discrete parameter – Mathematica notebook with examples. Technical Report 2005-02, SFB F013, Johannes Kepler Universit¨ at, 2005. http://www.risc.unilinz.ac.at/research/combinat/publications/. [12] G. Hardy, J. E. Littlewood, and G. P´ olya. Inequalities. Cambridge Mathematical Library. Cambridge University Press, second edition, 1952. [13] M. Kauers. An algorithm for deciding zero equivalence of nested polynomially recurrent sequences. Technical Report 2003-48, SFB F013, Johannes Kepler Universit¨ at, 2003. (submitted). [14] M. Kauers. Computer proofs for polynomial identities in arbitrary many variables. In Proceedings of ISSAC ’04, pages 199–204, July 2004. [15] D. S. Mitrinovi´c. Elementary Inequalities. P. Noordhoff Ltd., 1964. [16] D. S. Mitrinovi´c. Analytic Inequalities. Springer, 1970. [17] T. S. Nanjundiah. Problem 10347. The American Mathematical Monthly, 100(10), 1993. [18] P. Paule. A proof of a conjecture of Knuth. Experimental Mathematics, 5:83–89, 1996. [19] P. Paule. A computerized proof of ζ(2) = π 2 /6. In preparation, 2005. [20] P. Paule and M. Schorn. A Mathematica version of Zeilberger’s algorithm for proving binomial coefficient identities. Journal of Symbolic Computation, 20(5–6):673–698, 1995.

(+|0) − − ⇒ −, (+|0) − − − ⇒ −, + + + + + (+|0) − − − − ∗ ⇒ +, (+|0) − − − − ∗ + ⇒ +, (+|0) − − − − ∗ + + ⇒ +, (+|0) − − − − ∗ + + + ⇒ +, (+|0) − − − − ∗ + + + + ⇒ +,

(+|0) − − − − ∗ + + + + + ⇒ +, (+|0) − − − − ∗ + + + + + + ⇒ +, (+|0) − − − − ∗ + + + + + + + ⇒ +, (+|0) − − − −(−|0) + + + + + + + + ⇒ +, (+|0) − − − − ∗ + + + + + + + + ∗ ⇒ − where ∗ may be +, 0, or −. Putting all these rules together, we find that the sign pattern of (fn )n≥1 may be written in the form +9 [+ | 0] −4 [− | 0], i.e., there are alternating runs of nine positive and four negative values, but each run may be prolonged by one, or there might be a zero between two runs. (Following standard notation, we write [a] for “a or nothing,” a | b for “a or b,” and an for n repetitions of a.) In fact, the first values of the sequence have the signs +3 −5 +9 −4 +9 −5 +9 −4 +10 −4 +9 −5 +9 −4 +9 · · · which is in accordance with the predicted pattern. What can be said about the optional prolongations [+ | 0] and [− | 0] in the pattern? When do they occur? By a more careful application of the same technique, we obtained the following refined description of the sign pattern: (0 | +) +9 P 5 −5 +9 P 4 −4 − +9 P 4 P (−5 +9 P 4 )2 −4 *

*

+

+

where P stands for the subpattern −4 +9 −5 +9 . This gives much more detail compared to the first pattern, even though it still doesn’t describe the pattern entirely. In this pattern, there are only two uncertain signs per 674 values left. The computations we did for the sign pattern analysis can also be found in the Mathematica notebook [11].

6.

CONCLUSION

Though inequalities are considered much harder than identities from a computer algebra viewpoint, inequalities are not entirely out of the scope of symbolic methods. In this paper, we have contributed a simple but powerful method for proving special function inequalities, based on a connection of CAD with difference algebra. The method is strong enough to succeed on a wide range of interesting examples using only little – if any – support by the user. However, termination is not guaranteed in general, and it would be interesting to have a description of the class of inequalities where the procedure terminates. There might be manageable conditions on the input recurrences that ensure termination, and the class might enjoy some useful closure properties.

161

[21] M. Petkovˇsek, H. Wilf, and D. Zeilberger. A = B. AK Peters, Ltd., 1997. ¨ [22] I. Schur and K. Knopp. Uber die Herleitung der 2 Gleichung ∞ 1/n = π 2 /6. Archiv der Mathematik n=1 und Physik, 3(27):174–176, 1918.

[23] J. M. Steele. The Cauchy-Schwarz Master Class. An Introduction to the Art of Mathematical Inequalities. Cambridge University Press, 2004. [24] A. Strzebo´ nski. Solving systems of strict polynomial inequalities. Journal of Symbolic Computation, 29:471–480, 2000.

,

162

Generalized Loewy-Decomposition of D-Modules ∗

Dima Grigoriev

Fritz Schwarz

CNRS, IRMAR Universite´ de Rennes, Beaulieu, 35042, Rennes, France

FhG, Institut SCAI 53754 Sankt Augustin, Germany

[email protected] www.scai.fraunhofer.de/schwarz.html

[email protected] http://name.math.univrennes1.fr/dimitri.grigoriev ABSTRACT

multiple of irreducible right factors. Similar as in the algebraic case, if such a nontrivial decomposition may be found, the solution procedure is faciliated. Algorithms for factoring a lode have also been described by Schwarz [23] and, with improved complexity bounds, by Grigoriev [7]. A survey may be found in the book by Singer and van der Put [17]. Factoring linear partial differential equations (lpde’s) is much more difficult. So far there has been no common agreement on what to understand by factoring lpde’s in general. A first attempt to generalize the above theory by Li et al. [13], see also Tsarev [29], has been restricted to those lpde’s which have a finite-dimensional solution space. This is achieved by a fairly straightforward extension of the factorization of lode’s. Recently in [9] the problem of factoring a single lpde was studied. An algorithm was designed for factoring so-called separable lpde’s, but the general factorization problem remained open. Here an algebraic approach is suggested which subsumes the conventional factorizations and its corresponding decompositions as special cases. Any given linear differential equation is considered as the result of applying a differential operator to a differential indeterminate. This operator or, if a system of equations is involved, this set of operators, are considered as generators of a left D-module over an appropriate ring of differential operators. Some background on D-modules may be found e. g. in the book by Sabbah [20] or the article by Quadrat [18]. In our algebraic approach decomposing a D-module means finding overmodules which describe various parts of the solution of the original problem. There are two possibilites for constructing these overmodules. - A set of new generators is searched for such that the original module may be reduced to zero wrt. to them. This stands for the conventional factorization like factoring linear ode’s [23], factoring linear pde’s with a finite-dimensional solution space [13], or the factorizations that have been described in [9].

Starting from the well-known factorization of linear ordinary differential equations, we define the generalized Loewy decomposition for a D-module. To this end, for any module I, overmodules J ⊇ I are constructed. They subsume the conventional factorization as special cases. Furthermore, the new concept of the module of relative syzygies Syz(I, J) is introduced. The invariance of this module and its solution space w.r.t. the set of generators is shown. We design an algorithm which constructs the Loewy-decomposition for finite-dimensional and some kinds of general D-modules. These results are applied for solving various second- and third-order linear partial differential equations.

Categories and Subject Descriptors I.1 [Symbolic and Algebraic Manipulation]: Applications

General Terms Algorithms

Keywords D-module, Loewy decomposition, Janet basis

Introduction The concept of factorization of a linear ordinary differential equation (lode) originally goes back to Beke [1] and Schlesinger [21]. Loewy [14] extended it and introduced a unique decomposition of any lode into largest completely reducible factors, i. e. factors which are the least common ∗Partial support by a Humboldt Forschungspreis is gratefully acknowledged.

- It may be possible to construct new generators forming a Janet base of an overmodule in combination with the given ones, which are not necessarily of lower order.

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In either case, the result is a set of operators generating an overmodule of the given one. The further proceeding depends on the result of this construction. It may occur that several over-modules have been obtained such that their intersection is identical to the given one. If this is true,

163

1. INVARIANCE OF THE SPACE OF SOLUTIONS OF A D-MODULE

solving the original problem is reduced to solving several, possibly simpler problems, each of which describes some part of the desired solution. In Loewy’s terminology [14] such a module is called completely reducible. If this case does not apply, for each over-module the module of relative syzygies is constructed as defined in Section 2 below. Then the same procedure is applied to it as for the originally given module. This process terminates until no further over-modules may be constructed. The result is the natural generalization of Loewy’s decomposition of ordinary differential operators. From this decomposition the solution of the originally given equation may be obtained iteratively. At first all homogeneous problems have to be solved. The solutions of the rightmost factors are already part of the solution of the full problem. In the next step the solutions of the module of relative syzygies are taken as inhomogeneity of the respective rightmost factor. Solving these problems yields additional parts of the solution of the full problem. This process is repeated until the last module of relative syzygies has been reached. If all equations that occur in this decomposition may be solved, the general solution of the original problem has been obtained or, if this is not true, at least some part of it. In the first part of this article the algebraic background which is the basis of the above scheme is outlined. In Section 1 we show that the space of solutions of a module is determined by its class of isomorphisms (Proposition 1.1), up to an equivalence D which is called D-isomorphism. In Section 2 we introduce the new concept of the module of relative syzygies Syz(I, J) of two modules I and J with I ⊆ J. It extends the one given in [13] for finite-dimensional modules. It is shown that it is essentially invariant w.r.t. to the set of generators. We also show that for the space of solutions of Syz(I, J) there holds VSyz(I,J ) D VI /VJ (Lemma 2.4), this provides a bijective correspondence between classes of isomorphisms of the factors I/J and classes of D-isomorphisms of the solutions spaces VSyz(I,J ) (Corollary 2.5). In addition we describe a procedure to calculate the module of relative syzygies. Finally, the relation aτ (Syz(I, J)) = aτ (I) − aτ (J) (Theorem 2.7) is proved for the leading coefficients aτ of the Hilbert-Kolchin polynomials; τ is the differential type of the module I, see [11] and [12]. In Section 3, at first we define a unique Loewy decomposition of a finite-dimensional module I. The crucial role here plays the intersection R(I) of all maximal overmodules of I. Instead of I the modules R(I) and Syz(I, R(I)) with smaller differential type or smaller typical differential dimension (see e.g., [11], [12]) are considered in the inductive definition. After that the Loewy decomposition is generalized to infinite-dimensional modules I of differential type τ > 0. It relies on the intersection Rτ (I) of the classes of maximal overmodules of I with differential type τ , considered up to modules of differential types less than τ . Section 4 deals with parametric-algebraic families of D-modules. They are applied in Section 5 for the discussion of algorithms. In particular the theory outlined in the preceding sections is applied to certain classes of second- and third-order linear pde’s with rational function coefficients. An algorithm is presented that accomplishes its Loewy decomposition whenever possible. If it succeeds the solution may be obtained from it.

Let F be a universal differential field [11] with commuting derivatives d1 , . . . , dm and D = F [d1 , . . . , dm ] be the ring of partial differential operators. Denote by C ⊂ F its subfield of constants. Introduce differential indeterminates y1 , . . . , yn over F . By Θ denote the commutative monoid generated by d1 , . . . , dm and by Γ the set of all derivatives θyi for θ ∈ Θ, 1 ≤ i ≤ n. We fix also an admissible total ordering ≺ on the derivatives [12]. A background in differential algebra may be found in [11, 2, 26, 27]. Let I ⊂ Dn be a left D-module. For vectors g = (g1 , . . . , gn ), v = (v1 , .P . . , vn ) ∈ F n we denote the inner product gv = T gi vi ∈ F . By VI = {v ∈ F n : Iv = 0} ⊂ F n we (g, v ) = denote the space of solutions of I being a C-vector space. A priori VI depends on the imbedding I ⊂ Dn . The purpose of this section is to show that actually VI depends up to an isomorphism just on the factor Dn /I, considered as well up to an isomorphism. Now let I1 ⊂ Dn1 , I2 ⊂ Dn2 . We say that a n1 ×n2 matrix A = (aij ) with aij ∈ D provides a D-homomorphism from Dn1 /I1 to Dn2 /I2 if (Dn1 /I1 )A ⊂ (Dn2 /I2 ), i.e. I1 A ⊂ I2 . Clearly one gets a homomorphism of D-modules. We call Dn1 /I1 and Dn2 /I2 to be D-isomorphic if in addition there exists a n2 × n1 matrix B = (bij ) with bij ∈ D such that (Dn2 /I2 )B ⊂ Dn1 /I1 and AB|(Dn1 /I1 ) = id,

BA|(Dn2 /I2 ) = id.

(1)

For the spaces of solutions VI1 ⊂ F , VI2 ⊂ F we say that a matrix A provides a D-homomorphism if A(VI2 )T ⊂ (VI1 )T (more precisely, one should talk about a D-homomorphism of the imbeddings VI1 ⊂ F n1 , VI2 ⊂ F n2 ). In a similar way, if there exists a n2 × n1 matrix B such that B(VI1 )T ⊂ (VI2 )T and n1

AB|V T = id, I1

BA|V T = id I2

n2

(2)

we call VI1 , VI2 to be D-isomorphic and denote this by VI1 D VI2 . The following proposition extends Lemma 2.5 [25] (established for the ordinary case m = 1) to finitedimensional modules. Proposition 1.1. i) A matrix A provides a D-homomorphism of Dn1 /I1 to Dn2 /I2 if and only if it provides Dhomomorphisms of VI2 to VI1 . ii) Dn1 /I1 and Dn2 /I2 are D-isomorphic if and only if VI1 and VI2 are D-isomorphic. Proof. i) Assume that (Dn1 /I1 )A ⊂ (Dn2 /I2 ). We need to verify that A(VI2 )T ⊂ (VI1 )T . The latter is equivalent to the equality I1 A(VI2 )T = 0 which holds because of the inclusion I1 A ⊂ I2 . Conversely, assume that A(VI1 )T ⊂ (VI1 )T , then as above I1 A(VI2 )T = 0 which implies I1 A ⊂ I2 due to the duality in the differential Zariski topology (see Corollary 1, page 148 in [11], also [26]). Hence (Dn1 /I1 )A ⊂ (Dn2 /I2 ). ii) Assume that (1) holds. One has to verify (2), i. e. for any v ∈ VI1 to show that ABv T = v T . The latter holds if and only if for any g ∈ Dn1 the equality gABv T = gv T is true. Equation (1) entails that gABv T = (g + g0 )v T = gv T for a certain vector g0 ∈ I1 . We mention that D-isomorphism of D-modules implies isomorphism of the spaces of their solutions in a more general setting, see e.g. [16], [18] (while the converse essentially uses that we deal with a universal differential field).

164

2.

RELATIVE SYZYGIES OF D-MODULES

We mention also that relying on the algorithm from [8] one can produce a basis of Syz(I, J) starting with arbitrary, not necessarily Janet bases, of I and J, with double-exponential complexity. Let us denote by HI the Hilbert-Kolchin polynomial of I w.r.t. the usual filtration by order of derivatives, so (Dn )r = {f ∈ Dn : ord f ≤ r} (cf. page 223 of [12]). The degree deg(HI ) of HI is called the differential type of I [11], page 130 and [12], page 229, and its leading coefficient lc(HI ) is called the typical differential dimension of I ibid. The differential type denotes the largest number of arguments occuring in any undetermined function of the general solution. The typical differential dimension means the number of functions depending on this maximal number of arguments. The next theorem can be deduced directly from Theorem 5.2.9 of [12], cf. also Theorem 4.1 in [26].

In Loewy’s original decomposition scheme, the largest completely reducible right factors are removed by exact division. In the ring of partial differential operators this is not valid any more. In addition to the relations following from the exact division the syzygies of the right factor have to be taken into account. The proper generalization of the exact quotient is given by the following Definition 2.1. (Relative syzygies module) Let I ⊆ J ⊆ Dn be two D-modules, and let J =< g1 , . . . , gt >. The relative syzygies D-module P Syz(I, J) of I and J is Syz(I, J) = {(h1 , . . . , ht ) ∈ Dt | hi gi ∈ I}. This definition is more general than the definition of the quotient of D-modules in [13] because we do not require g1 , . . . , gt to be a Janet basis of J (for a background on Janet basis see e.g. [11, 12, 22]) and in addition it takes into account all relations among g1 , . . . , gt which put them in I. We notice that in case when I = 0 the module Syz(0, J) coincides with the usual syzygies module Syz(J). Our next goal is to show that Definition 2.1 does not depend on the choice of generators g1 , . . . , gt .

Theorem 2.7. Let again I ⊆ J ⊆ Dn . Then deg (HJ ) ≤ deg (HI ), deg (HSyz(I,J )) ≤ deg (HI ) and deg (HSyz(I,J )) = deg (HI − HJ ), lc (HSyz(I,J )) = lc (HI − HJ ). Definition 2.8. (Gauge of a module) Let I be a D-module. We call the pair (deg(HI ), lc(HI )) the gauge of I. We say that a module I1 is of lower gauge than another one I2 if the pair (deg(HI1 ), lc(HI1 )) is less than (deg(HI2 ), lc(HI2 )) in the lexicographic ordering. Taking into account Corollary 2.5 one can talk also about the gauges of the corresponding spaces of solutions VI1 and VI2 .

Lemma 2.2. Let I ⊆ I1 ⊆ J be D-modules. Then Syz(I1 , J)/Syz(I, J)  I1 /I. Corollary 2.3. i) Dt /Syz(I, J)  J/I; ii) Syz(I, J)/Syz(J)  I. The main goal for introducing the relative syzygies module is the following statement proved in [13] when g1 , . . . , gt is a Janet basis of J, one can find in [19] another proof.

The construction of the relative syzygies allows to reduce finding a basis of VI to finding a basis of VJ and joining it with any solution y of the system gi y = wi , 1 ≤ i ≤ t for each element (w1 , . . . , wt ) of a basis of VSyz(I,J ). An algorithm for solving the inhomogeneous system gi y = wi may be obtained by a proper generalization of Lagrange’s variation of constants, see e. g. the textbook [28], page 193-195 if the homogeneous system is known to have a finite-dimensional solution space which will be the case in our applications. Theorem 2.7 implies that both J and Syz(I, J) have gauges not greater than the gauge of I. In the applications in the next section, the gauges of J and Syz(I, J) will be actually lower than the gauge of I. In case of a finite-dimensional ideal I this reduction was exploited in [13].

Lemma 2.4. With the notation above there holds VSyz(I,J ) D VI /VJ . The following corollary claims that the space of solutions VSyz(I,J ) of a relative syzygies module depends just on the factor of D-modules J/I. Corollary 2.5. Let I1 ⊆ J1 ⊆ Dn1 , I2 ⊆ J2 ⊆ Dn2 . Then J1 /I1  J2 /I2 if and only if VSyz(I1 ,J1 ) D VI1 /VJ1 D VI2 /VJ2 D VSyz(I2 ,J2 ) . Proof. Corollary 2.3 implies that J1 /I1  Dq1 /Syz(I1 , J1 ) Both D-isomorphisms and J2 /I2  Dq2 /Syz(I2 , J2 ). VSyz(I1 ,J1 ) D VI1 /VJ1 and VSyz(I2 ,J2 ) D VI2 /VJ2 follow from Lemma 2.4. Proposition 1.1 entails that VSyz(I1 ,J1 ) D VSyz(I2 ,J2 ) if and only if Dq1 /Syz(I1 , J1 )  Dq2 /Syz(I2 , J2 ) ✷ Remark 2.6. Having Janet bases of I =< f1 , . . . , fs > and of J =< g1 , . . . , gt > one can construct a Janet basis of Syz(I, J), e. g. cf. Theorem 5.3.7 in [12], P also [13]. Briefly hi,j gi , 1 ≤ j ≤ s to remind, for each fj there holds fj = for certain hi,j ∈ D. Furthermore, for each pair (k, j) with 1 ≤ k < j ≤ t we representP the ∆-polynomial of gk and gj as lc(gj )θ1 gk −lc(gk )θ2 gj = hijk gi such that the operators lc(gj )θ1 gk and lc(gk )θ2 gj have the same leading terms with the minimal possible leading derivative w.r.t. the applied term ordering ≺. Then the basis of Syz(I, J) consists of the vectors (h1,j , . . . , ht,j ), 1 ≤ j ≤ s, and of the vectors

3. LOEWY DECOMPOSITIONS Let us first study the case of a finite-dimensional module I ⊂ Dn of differential type 0. Consider the intersection R(I) = J (0) = ∩J of all maximal modules J ⊇ I. Any intersection of maximal modules will be called a complete intersection. R(I) plays a role similar to the role of the radical of two-sided ideals in a ring. Note that there exists a finite number of maximal modules J1 , . . . , Jq for which J1 ∩ · · · ∩ Jq = R(I). Indeed, keep taking J1 , J2 , . . . while it is possible to have dimC VJ1 ∩···∩Ji+1 > dimC VJ1 ∩···∩Ji for every i ≥ 1. Since dimC VI < ∞ we arrive finally at J1 , . . . , Jq such that dimC VJ1 ∩···∩Jq ∩J = dimC VJ1 ∩···∩Jq for any maximal module J ⊇ I. Then J1 ∩ · · · ∩ Jq = R(I). Applying this procedure to the relative syzygies module I (1) = Syz(I, J (0) ), replacing the role of I, which one can compute making use of Remark 2.6, this yields a complete intersection J (1) such that J (1) = R(I (1) ) ⊇ I (1) . Continuing this way, one obtains successively the complete intersections J (0) , J (1) , . . . , J (s) and the modules I (1) , . . . , I (s) such that J (l) = R(I (l) ) and I (l+1) = Syz(I (l) , J (l) ) for

(h1jk , . . . , hkjk − lc(gj )θ1 , . . . , hjjk − lc(gk )θ2 , . . . , htjk ) (3) for 1 ≤ k < j ≤ t. In the special case I = 0, the relative syzygies module Syz(0, J) reduces to the syzygies module of J. Then as in Schreyer’s theorem, page 212 of [3], one can show that the constructed basis of Syz(0, J) which consists of vectors of the form (3) constitutes a Janet basis.

165

0 ≤ l ≤ s − 1, defining I (0) = I. In the last step there holds J (s) = I (s) . We have dimC VI (l) − dimC VI (l+1) = dimC VJ (l) for 0 ≤ l ≤ s, defining I (s+1) = {0}. Thus, dimC VI = P 0≤l≤s dimC VJ (l) , which provides an upper bound s < dimC VI on the number of steps of the described procedure. The uniquely defined sequences J (0) , J (1) , . . . , J (s) and I (1) , . . . , I (s) can be viewed as a Loewy decomposition of I. To get the spaces of solutions VJ (l) , 0 ≤ l ≤ s of the complete (l) (l) intersections J (l) = ∩q Jq where Jq are maximal modules, we apply proposition 3.1 [26] (see also the beginning of the proof ofP theorem 4.1 [26], p.483 and [2]) which entails that VJ (l) = q VJ (l) . q Now we proceed to a Loewy decomposition of an infinitedimensional module I ⊂ Dn of differential type τ > 0. To this end, we introduce another concept first. Definition 3.1. (Gauge-equivalence) We say that two modules J1 , J2 ⊂ Dn are gauge-equivalent if J1 , J2 and J1 ∩ J2 are of the same gauge. If J1 and J2 are gauge-equivalent, then by Theorem 4.1 in [26] also J1 + J2 is of the same gauge. Gauge equivalence is an equivalence relation. The equivalence class of gaugeequivalent modules of a module J is denoted by [J]. If the actual value of the differential type of the elements of a class [J] equals to τ , any two members of it are called τ -equivalent (below τ is fixed and |J| means a class of τ -equivalence). Example 3.2. Let J1 =< ∂x >, J2 =< ∂xx , ∂xy > and J3 =< ∂y >. Then J1 ∩ J2 = J2 , J1 + J2 = J1 all of which are of gauge (1, 1). Consequently J1 and J2 are gaugeequivalent. The general solution of zx = 0 is F (y), whereas zxx = zxy = 0 has general solution Cx + F (y), C being a constant and F an undetermined function of y. Although J3 is also of gauge (1,1), it is not gauge-equivalent to J1 because J1 ∩ J3 =< ∂xy > which is of gauge (1,2).

Clearly, p ≤ lc(HI ). Then for any maximal class [J] for which [I] ✂ [J], we obtain [J (0) ] ✂ [J]. Hence for any

finite family [J1 ], . . . , [Jq ] of τ -maximal classes for which


[I] ✂ [Jl ], 1 ≤ l ≤ q, we conclude that [J (0) ] ✂ [J1 ∩ · · · ∩ Jq ]. Therefore, the class [J (0) ] is defined uniquely and in addition I ⊆ J (0) holds. We say that J (0) = J1 ∩ J2 ∩ · · · ∩ Jp is completely τ -reducible. We define a Loewy decomposition of I by induction on the gauge of I. As a base of induction when the τ -class [I] is maximal then I provides a Loewy decomposition of itself. When [I] is not maximal one can further apply the described inductive definition of a Loewy decomposition (thereby, replacing the role of I) to the relative syzygies module I (1) = Syz(I, J (0) ) (see Section 2) taking into account that either deg(HI (1) ) < τ or deg(HI (1) ) = τ , and in the latter case lc(HI (1) ) = lc(HI )−lc(HJ (0) ) < lc(HJ ) due to Theorem 2.7; in other words, I (1) is of a lower gauge than I. In case when deg(HI (1) ) < τ we have [I] = [J (0) ] again due to Theorem 2.7 and [I] being completely τ -reducible. Continuing this way we arrive at a sequence of modules J (0) , J (1) , . . . , J (q) with non-decreasing differential types such that each module J (l) , 0 ≤ l ≤ q is completely deg(HJ (l) )reducible. We notice that this sequence is not necessarily unique unlike the Loewy decomposition of a finite-dimensional module. The obtained sequence could be called a generalized Loewy decomposition of I. At present we don’t possess an algorithm to construct it in general.

4. PARAMETRIC-ALGEBRAIC FAMILIES OF D-MODULES For the rest of the paper, dealing with the design of algorithms, we assume that the coefficients of the input operators belong to the differential field F0 = Q(X1 , . . . , Xm ) with derivatives dk = ∂/∂Xk , 1 ≤ k ≤ m and D0 = F0 [d1 , . . . , dm ], D = F [d1 , . . . , dm ] where F is a universal extension of F0 . In the sequel we suppose that all the considered algebraic (affine) varieties N W ⊂ Q are given in an efficient way, say as in [6]. Namely, W = ∪Wj where Wj are irreducible over Q components of W , and the algorithms from [6] represent each Wj (of dimension s) in two following ways. First, we represent Wj by means of a generic point, i.e. an isomorphism Q(t1 , . . . , ts )[α]  Q(Wj ) where Q(Wj ) is the field of rational functions on Wj . The elements t1 , . . . , ts ⊂ {Z1 , . . . , ZN } constitute a basis of transcendency of Q(Wj ) over Q which can be taken among the coordinates Z1 , . . . , ZN P N of the affine space Q . The element α = 1≤l≤N αl Zl for suitable integers αl is algebraic over the field Q(t1 , . . . , ts ) with a minimal polynomial φ ∈ Q(t1 , . . . , ts )[Z]. The algorithms from [6] yield the ingredients of a generic point explicitly, in other words, t1 , . . . , ts ; α1 , . . . , αN ; φ and the rational expressions of Zl via t1 , . . . , ts , α, i.e. the rational functions of the form gl (t1 , . . . , ts , Z)/g(t1 , . . . , ts ), the polynomials g(t1 , . . . , ts ), gl (t1 , . . . , ts , Z) ∈ Q[t1 , . . . , ts , Z] being such that Zl = gl (t1 , . . . , ts , Z)/g(t1 , . . . , ts ) holds everywhere on Wj . Second, the algorithms from [6] yield polynomials h1 , . . . , hM ∈ Q[Z1 , . . . , ZN ] such that Wj coincides with the variety N of all points from Q satisfying h1 = · · · = hM = 0. The algorithms from [6] allow to produce the union, in-

We say that [J1 ] is subordinated to [J2 ] if J1 ∩ J2 is τ equivalent to J1 . One can verify that this relation does not depend on representatives J1 and J2 of the classes. We denote this relation by [J1 ] ✂ [J2 ]. Then lc(HJ1 ) ≥ lc(HJ2 ). If in addition [J1 ] = [J2 ] (we denote this by [J1 ] ✁ [J2 ]) then lc(HJ1 ) > lc(HJ2 ). Hence any increasing chain of τ equivalence classes stops and one can consider maximal τ equivalence classes. For any τ -equivalence classes [J1 ], [J2 ] satisfying [J] ✂ [J1 ], [J]✂[J2 ] one can uniquely define the class [J1 ∩J2 ] such that [J]✂[J1 ∩J2 ]. One can verify that deg(HJ1 ∩J2 ) = τ and the class [J1 ∩ J2 ] does not depend on the representatives J1 , J2 . Example 3.3. Let J =< ∂xyy > with gauge (1, 3), J1 =< ∂x > and J2 =< ∂y >, both with gauge (1, 1). Because J ∩ J1 = J ∩ J2 = J there holds [J] ✂ [J1 ] and [J] ✂ [J2 ]. Furthermore J1 ∩ J2 =< ∂xy >≡ J3 with gauge (1, 2) and [J] ✂ [J3 ]. Because lc(HJ ) = 3, lc(HJ3 ) = 2 and lc(HJ1 ) = lc(HJ2 ) = 1, both [J1 ] and [J2 ] are maximal. Now take all τ -maximal classes [J] such that [I] ✂ [J]. Since J + I is τ -equivalent to J (again due to Theorem 4.1 [26]) we can assume without loss of generality that the representatives are chosen in such a way that I ⊆ J. We choose consecutively such classes [J1 ], [J2 ], . . . , [Jp ] while it is possible to have [J1 ] ✄ [J1 ∩ J2 ] ✄ · · · ✄ [J (0) = J1 ∩ J2 ∩ · · · ∩ Jp ].

166

5. CONSTRUCTING LOEWY-DECOMPOSITIONS. ALGORITHMS

tersection, complement of varieties, to get the dimension of Wj , to project a variety (in other words, to eliminate quantifiers), to find all points of Wj if it is finite (zero-dimensional) or to yield any number of points if Wj is infinite (positivedimensional). Moreover, one extends these algorithms from varieties to constructive sets , i.e. the unions of the sets





of the form W \ W where W , W are varieties (in other terms, constructive sets constitute the boolean algebra generated by all the varieties). Definition 4.1. (Parametric-algebraic D-modules) We say that a family of D-modules J = {J} ⊂ Dn is parametricN algebraic if there is a constructive set V = ∪Vj ⊂ Q for an appropriate N such that J = ∪Jj and for any fixed j the following holds. A Janet basis of any J ∈ Jj has fixed leading derivatives lder(J) = lderj and the parametric derivatives pder(J) = pderj , see [13]. Moreover, any element of the Janet basis of J has the form X γ0 + Aγ (Z1 , . . . , ZN )γ (4)

Now we are able to construct the Loewy decomposition for any finite-dimensional D-module I ⊂ D0n . According to Corollary 4.4 we determine the intersection R(I) of all maximal modules from Imax . To this end we conduct the internal recursion on dimC VR(I) . Assume that a complete intersection J0 of several maximal modules from Imax has already been constructed. Applying Lemma 4.3 we test whether there exists a maximal module J ∈ Imax which does not contain J0 . Then we replace J0 by the complete intersection J ∩ J0 and continue the internal recursion. Finally, we arrive at R(I) and, by external recursion, proceed to the relative syzygies module Syz(I, R(I)), provided that the latter is not zero, else halt. Thus, we have shown the following Corollary 5.1. For a finite-dimensional D-module I ⊂ D0n one can construct its Loewy decomposition. This construction is the basis in [13] for decomposing finitedimensional modules. An algorithm has been given there which applies these steps. An implementation may be found in the ALLTYPES system [24]. For general modules the answer is less complete. In [9] proper factorizations and the corresponding decompositions have been considered for second- and third-order operators. Here this approach is extended to the case where genuine factors of such operators do not exist. Most of the research on finding closed-form solutions of lpde’s has been restricted to second-order equations for an unknown function z depending on two arguments x and y. The general linear equation of this kind may be written as

γ∈pderj

where γ0 ∈ lderj and Aγ ∈ Q(Z1 , . . . , ZN )(X1 , . . . , Xm ). When (Z1 , . . . , ZN ) ranges over the constructive set Vj , the set of linear differential operators of the form (4) for all γ0 ∈ lderj ranges over the Janet basis for all modules J from Jj . Thus, we have a bijective correspondance between the points of Vj and the modules, or rather their Janet basis) from Jj . We rephrase in our terms the following proposition which was actually proved in [13]. Proposition 4.2. ([13]). One can design an algorithm which for any finite-dimensional D-module I ⊂ Dn finds a parametric-algebraic family of all the factors of I, i.e. the modules J ⊂ Dn such that I ⊂ J.

Rzxx + Szxy + T zyy + U zx + V zy + W z = 0

Lemma 4.3. One can design an algorithm which for a pair of parametric-algebraic families I, J of D-modules yields the parametric-algebraic family of all the pairs (I, J) where I ∈ I, J ∈ J such that I ⊆ J. P Proof. Let {γ0 + γ∈pderj Aγ (Z1 , . . . , ZN )γ}γ0 ∈lderj be a P Janet basis of Jj and {λ0 + λ∈pders Bλ (Z1 , . . . , ZN )λ}λ0 ∈lders be a Janet basis of Is . Then the condition that I ⊆ J for I ∈ Is , J ∈ Jj can be expressed asPthe existence for each λ0 ∈ lders of operators of the form θ Cθ,γ0 ,λ0 θ ∈ D where θ ≺ θ0 and λ0 = θ0 yi for a certain 1 ≤ i ≤ n such that P λ0 + λ∈pders Bλ (Z1 , . . . , ZN )λ = P P P γ0 ∈lderj ( θ Cθ,γ0 ,λ0 θ)(γ0 + γ∈pderj Aγ (Z1 , . . . , ZN )γ) (5) where the external summation in the right-hand side ranges over the elements of the Janet basis of Jj . One can rewrite (5) as a system of linear algebraic equations in the unknowns Cθ,γ0 ,λ0 , while the entries of this system are rational functions from Q(X1 , . . . , Xm ) (Z1 , . . . , ZN ). One can N find the constructive set U = Uj,s ⊂ Q such that for (Z1 , . . . , ZN ) ∈ U this linear system is solvable. Combining this for all pairs l, s completes the proof. ✷ Corollary 4.4. For a finite-dimensional D-module I ⊂ Dn one can find a parametric-algebraic family Imax of all maximal D-modules J which contain I. Proof. Among the family of all the factors J of I produced in proposition 4.2 one can relying on Lemma 4.3 distinguish ✷ all J0 such that if J0 ⊆ J then J0 = J holds.

167

(6)

where R, S, . . . , W are from some function differential field which is usually called the base field. Under fairly general constraints for its coefficients it can be shown that it may be transformed either of the following two forms. zxy + A1 zx + A2 zy + A3 z = 0,

(7)

zxx + A1 zx + A2 zy + A3 z = 0.

(8)

In this section it is always assumed that all Ak ∈ Q(x, y). Any solution scheme is closely related to the question what type of solutions are searched for. For linear ode’s the answer is well known. The general solution is a linear combination of a fundamental system over the constants. For pde’s the answer is much more involved. Equations of the form (7) may allow solutions of either of the two forms f0 (x, y)F (x) + f1 (x, y)F  (x) + . . . + fm (x, y)F (m) (x), (9) g0 (x, y)G(y) + g1 (x, y)G (y) + . . . + gn (x, y)G(n) (y) (10) where the fk , gk are determined by the given equation, and F (x) and G(y) are undetermined functions of the respective argument. The existence of either type of solution, or of both types, depends on the values of the coefficients Ak . To decide their existence is already highly nontrivial. Moreover there may be solutions with integrals involving the undetermined elements. An algorithm is described now which performes these steps for certain pde’s of second or third order. Equation (7) is written as Dxy z = 0 where Dxy ≡ ∂xy + A1 ∂x + A2 ∂y + A3 .

(11)

a Janet base. The resulting system of equations is

This case has been studied most thorougly in the literature. It will be discussed first. The principal ideal < Dxy > is of gauge (1, 2). There may exist operators forming a Janet base in combination with (11) which are of the form Dxm ≡ ∂xm + a1 ∂xm−1 + . . . + am−1 ∂x + am or Dy n ≡ ∂y n + b1 ∂y n−1 + . . . + bn−1 ∂y + bn

b1,x + (Pn,n − Pn−1,n−1 )b1 − Pn,n−1 = 0, b2,x − Pn−1,n−2 b1 + (Pn,n − Pn−2,n−2 )b2 − Pn,n−2 = 0, .. .. . . bn−1,x − Pn−1,1 b1 − . . . + (Pn,n − A2 )bn−1 − Pn,1 = 0, bn,x − Pn−1,0 b1 − . . . − A3 bn−1 + Pn,n bn − Pn,0 = 0, bn − Rn−1 b1 − Rn−2 b2 − . . . − R2 bn−2 − A1 bn−1 = 0.

(12) (13)

with m and n positive integers. Usually it is a difficult problem to construct new operators which extend a set of given ones to form the Janet base of a larger ideal. However, due to the special structure of the problem, the auxiliary systems for the unknown coefficients aj and bj in (12) and (13) may always be solved as is shown next.

The last equation may be solved for bn . Substituting it into the equation with leading term bn,x , and eliminating the first derivatives bj,x for j = 1, . . . , n−1 by means of the preceding equations, it may be solved for bn−1 . Proceeding in this way, due to the triangular structure, finally b1 is obtained from the equation with leading term b2,x . Backsubstituting these results, all bk are explicitly known. Substituting them into the first equation, a constraint for the coefficients Ai expressing the condition for the existence of a Janet base comprising (7) and (13) is obtained. The proof for case b) is similar and is therefore omitted. ✷ Goursat [5], Section 110, describes a method for constructing a linear ode which is in involution with a given second order equation zxy + azx + bzy + cz = 0. The advantage of the method given above is that it may be applied to many other problems, e. g. exactly the same strategy works for the third-order equations discussed below. It is not obvious how to generalize Goursat’s scheme to any other case beyond the second-order equation considered by him. Case a), n = 1 and case b), m = 1, have been discussed in detail in [9]. The corresponding ideals are maximal and principal, because they are generated by ∂y + a1 and ∂x + b1 respectively. The term factorization is applied in these cases in the proper sense because the obvious analogy to ordinary differential operators where all ideals are principal. For any value m > 1 or n > 1 the overideals are Jm =< Dxy , Dxm > or Jn =< Dxy , Dy n >. For any fixed values m1 < m2 , the corresponding ideals obey Jm2 ⊂ Jm1 , and similary for values of n. This situation becomes particularly clear from the following graph. y ❛ .. ✻ n .

Proposition 5.2. Let an operator of the form (11) be given. The following types of overideals may be constructed. a) If n ≥ 2 is a natural number, it may be decided whether there exists an operator (13) such that (11) and (13) combined form a Janet base. If the answer is affirmative, the operator (13) may be constructed explicitly with coefficients bi ∈ Q(x, y), the ideal < Dxy , Dy n > is of gauge (1,1). b) If m ≥ 2 is a natural number, it may be decided whether there exists an operator (12) such that (11) and (12) combined form a Janet base. If the answer is affirmative, the operator (12) may be constructed explicitly with coefficients ai ∈ Q(x, y), the ideal < Dxy , Dxm > is of gauge (1,1). Proof. The proof will be given for case a). If the operator (11) is derived repeatedly wrt. y, and the reductum is reduced in each step wrt. (11), n − 2 equations of the form ∂xy k + Rk ∂x + Pk,k ∂y k + Pk,k−1 ∂y k−1 + . . . + Pk,0

(14)

for 2 ≤ k ≤ n − 1 may be obtained. All coefficients Rk and Pi,j are differential polynomials in the ring Q{A1 , A2 , A3 }. There is no reduction wrt. (13) possible. Deriving the last expression once more wrt. y and reducing the reductum wrt. both (7) and (13) yields ∂xy n + Rn ∂x + (Pn,n−1 − Pn,n b1 )∂y n−1 +(Pn,n−2 − Pn,n b2 )∂y n−2 + ... +(Pn,1 − Pn,n bn−1 )∂y + Pn,0 − Pn,n bn .

3 2 1

(15)

In the first derivative of (13) wrt. x ∂xy n + b1,x ∂y n−1 + b2,x ∂y n−2 + . . . + bn−1,x ∂y + bn,x +b1 ∂xy n−1 + b2 ∂xy n−2 + . . . + bn−1 ∂xy + bn ∂x the terms containing derivatives of the form ∂xy k for 1 ≤ k ≤ n − 1 may be reduced wrt. (14) or (7) with the result ∂xy n + (b1,x − Pn−1,n−1 b1 )∂y n−1 +(b2,x − Pn−1,n−2 b1 − Pn−2,n−2 b2 )∂y n−2 .. .. . . +(bn−1,x − Pn−1,1 b1 − Pn−2,1 b2 . . . − P2,1 bn−2 − A2 bn−1 )∂y +bn,x − Pn−1,0 b1 − Pn−2,0 b2 − . . . − P2,0 bn−2 − A3 bn−1 +(bn − Rn−1 b1 − Rn−2 b2 − . . . − R2 bn−2 − A1 bn−1 )∂x . If this expression is subtracted from (15), the coefficients of the derivatives must vanish in order that (7) and (13) form

168



✲

❛ 1 2 3 ..m . x

The heavy dot at (1, 1) represents the leading derivative ∂xy of the given equation. If a second equation with leading derivative ∂xm represented by the circle at (m, 0) exists, the ideal is enlarged by the corresponding operator. For m = 1 this ideal contains the original operator with leading derivative ∂xy , i. e. this operator is redundant. This shows clearly how the conventional factorization corresponding to a firstorder operator is obtained as a special case for any m. A similar discussion applies to additional equations with leading derivative ∂xn . Next the algebraic approach will be applied third to order equations of the form Dxyy z = 0 where Dxyy ≡ ∂xyy + A1 ∂xy + A2 ∂yy + A3 ∂x + A4 ∂y + A5 . (16) The ideal < Dxyy > is of gauge (1, 3). Proper right factors of differential type 1 and of first or second order may be obtained by Corollary 4.3 of [9]. For completeness they are given next without proof.

Proposition 5.3. An operator of the form (16) generates an ideal < Dxyy > of gauge (1, 3). It may have the following proper right factors of order two or one.

This algorithm has been implemented in ALLTYPES which may be accessed over website www.alltypes.de [24]. From this decomposition large classes of solutions of an equation Lz = 0 may be obtained. In the completely reducible case, from the operators returned in step S3 solutions may be constructed as described in [9]. If L is not completely reducible, the result of step S4 is applied as follows. From F or G a partial solution is obtained similar as in the previous case. Solving the equations corresponding to S and taking the result as inhomogeneity for F or G respectively yields an additional part of the solution. This proceeding may fail if not all of the equations which occur can be solved. In these cases only a partial solution is obtained. The following examples have been treated according to this proceeding. The first one which is due to Forsyth. It shows how complete reducibility has its straightforward generalization if there are no proper factors. Example 5.5. (Forsyth 1906) Define

a) If 2A2,y + A1 A2 − A4 = 0, b1,y − b21 + A1 b1 − A3 = 0, b1 = 2A + A1 A − A (A2,yy + 2A2,y A1 2,y 1 2 4 +A2 A1,y − A4,y − A1 A4 − A2 A3 + A21 A2 ) a right factor ∂xy + b1 ∂x + b2 ∂y + b3 exists, b2 = A2 , b3 = A2 b1 + A4 − A2,y − A1 A2 . b) If 2A2,y + A1 A2 − A4 = 0 and A5 − A2,yy − A2,y A1 − A2 A3 = 0, a right factor ∂xy + b1 ∂x + b2 ∂y + b3 exists where b1 is a solution of b1,y − b21 + A1 b1 − A3 = 0, and b2 = A2 , b3 = A2 b1 + A2,y . c) If A4 − 2A2,y − A1 A2 = 0 and A5 − A2,yy − A2,y A1 − A2 A3 = 0, a right factor ∂x + b exists with b = A2 . d) If A4 − A1 A2 − A1,x = 0, by − b2 + A1 b − A3 = 0 where A −A A −A b = A5 − A2 A3 − A3,x , a right factor ∂y + b exists. 4 1 2 1,x

Dxy ≡ ∂xy +

e) If A4 − A1 A2 − A1,x = 0 and A5 − A2 A3 − A3,x = 0, a right factor ∂y + b exists where b is a solution of b1,y − b21 + A1 b − A3 = 0.

2 2 4 ∂x − ∂y − x−y x−y (x − y)2

which generates the principal ideal I =< Dxy > of gauge (1,2). The equation Dxy z = 0 has been considered in [4], vol. VI, page 80. In step S1 no first-order factor is obtained. Step S2 shows that there exist both generators

The ideals generated in case a) and b) are of gauge (1,2), in the remaining cases they are of gauge (1,1).

2 ∂ + 2 Dxx ≡ ∂xx − x − y x (x − y)2 , 2 ∂ + 2 Dyy ≡ ∂yy + x − y y (x − y)2

If such a factor does not exist, over-ideals of the form < Dxyy , Dxm > or < Dxyy , Dy n > may be searched for. Proposition 5.4. Let an operator of the form (16) be given. The following types of overideals of differential type 1 may be constructed with coefficients ai , bi ∈ Q(x, y). a) If n ≥ 2 is a natural number, it may be decided whether there exists an operator (13) such that (16) and (13) combined form a Janet base. If the answer is affirmative, the operator (13) may be constructed explicitly. b) If m ≥ 2 is a natural number, it may be decided whether there exists an operator (12) such that (16) and (12) combined form a Janet base. If the answer is affirmative, the operator (12) may be constructed explicitly. The results obtained up to now are combined to produce the algorithm DecomposeLpde which returns the most complete decomposition for any operator of the form (7) or (16). Algorithm DecomposeLpde(L, d). Given an operator L of the form (7) or (16) generating I =< L >, its decomposition into overideals of differential type 1 and with leading derivative of order not higher than d is returned. S1 : Proper factorization. Determine right factors f1 , f2 , . . . of L as described in Corollary 3.3. If any are found, collect them as F := {f1 , f2 , . . .}. S2 : Extend ideal. If step S1 failed, apply Proposition 5.2 or 5.4 in order to construct operators g1 , g2 , . . . of the form (12) or (13) with m ≤ d and n ≤ d, beginning with m = n = 2 and increasing its value stepwise by 1 until d is reached. If any are found, assign them to G := {g1 , g2 , . . .}. If F and G are empty return L. S3 : Completely reducible? If J := Lclm(F ) =< L > return F , else if for the elements of G there holds J := Lclm(< L, g1 >, < L, g2 >, ...) =< L >, return G. S4 : Relative syzygies. Determine generators of S := Syz(I, J) and transform it into a Janet base. If F is not empty return (S, F ) else return (S, G).

169

such that the ideals J1 =< Dxy , Dxx > and J2 =< Dxy , Dyy >, each of gauge (1,1), are generated by a Janet base. In step S3 it is found that I = Lclm(J1 , J2 ), i.e. I is completely reducible, the sum ideal is J1 + J2 =< Dxy , Dxx , Dyy >. The general solution of Dxx z = 0 is C1 (x − y) + C2 x(x − y), C1,2 are undetermined functions of y. Substitution into Dxy z = 0 yields C1,y + yC2,y − C2 = 0. They may be represented as C1 = 2F (y) − yF  (y) and C2 = F  (y). Consequently the solution z1 = 2(x − y)F (y) + (x − y)2 F  (y) is obtained. The equation Dyy z = 0 has general solution C1 (y − x) + C2 y(y − x), C1,2 are undetermined functions of x now. Similar as above, the solution z2 = 2(y − x)G(x) + (y − x)2 G (x) is obtained. The general solution of Dxy z = 0 is z1 + z2 . The following example by Imschenetzky has been reproduced in many places in the literature. Example 5.6. (Imschenetzky 1872) The equation (∂xy + xy∂x − 2y)z = 0 has been considered in [10]. Step S1 shows again that there are no first-order right factors. According to step S2, an operator of the form (13) with n ≤ 3 does not exist. However, for m = 3 there is an operator ∂xxx such that the ideal < ∂xy + xy∂x − 2y, ∂xxx > of gauge (1,1) is generated by a Janet base. The equation zxxx = 0 has the general solution C1 + C2 x + C3 x2 where the Ci , i = 1, 2, 3 are constants wrt. x. Substituting it into the above equation and equating the coefficients of x to zero leads to 1 yC = 0. The C may the system C2,y − 2yC1 = 0, C3,y − 2 2 i  1 1 be represented as C1 = 2 F − 3 F  , C2 = y2 F  , C3 = y y F , F is an undetermined function of y, F  ≡ dF/dy. It 2xy 2 − 1  F (y) + 12 F  (y) yields the solution z1 = x2 F (y) + y3 y of the given equation. In step S4, from the ideals I =<

∂xy + xy∂x − 2y > and J =< ∂xy + xy∂x − 2y, ∂xxx > the relative syzygy module Syz(I, J) =< (1, 0), (∂xx , −∂y − xy) >=< (1, 0), (0, ∂y + xy) > of gauge (1,1) is constructed. 2 Its solution (0, G(x)s(x, y)) with s(x, y) = exp (− 1 2 xy ) and G(x) an undetermined function of x yields the solution 1 R G(x)s(x, y)x2 dx z2 = 2 R R 2 −x G(x)s(x, y)xdx + 1 2 x G(x)s(x, y)dx of the original equation, its general solution is z1 + z2 .

7. REFERENCES [1] E. Beke, Die Irreduzibilit¨ at der homogenen linearen Differentialgleichungen, Mathematische Annalen 45, 278–294(1894). [2] P. Cassidy, Differential Algebraic Groups, Amer. J. Math., 94, 891-954 (1972). [3] D. Cox, J. Little, D. O’Shea, Using Algebraic Geometry, Springer, 1998. [4] A. R. Forsyth, Theory of Differential Equations, vol. I,...,VI, Cambridge, At the University Press (1906). [5] E. Goursat, Le¸ con sur l’int´ egration des ´ equation aux d´ eriv´ ees partielles, vol. I and II, A. Hermann, Paris 1898. [6] D. Grigoriev, Computational complexity in polynomial algebra, Proc. Intern. Congress of Mathematicians, vol. 2, Berkeley, 1452-1460, 1986. [7] D. Grigoriev, Complexity of Factoring and Calculating the GCD of Linear Ordinary Differential Operators, J. Symbolic Computation, 7, 7-37 (1990). [8] D. Grigoriev, Complexity of Solving Systems of Linear Equations over Rings of Differential Operators, Progress in Math., Birkhauser, 94, 195-202 (1991). [9] D. Grigoriev, F. Schwarz, Factoring and solving linear partial differential equations, Computing 73, 179-197 (2004) ´ [10] V. G. Imschenetzky, Etude sur les m´ ethodes d’int´ egration des ´ equations aux d´ eriv´ ees partielles du second ordre d’une fonction de deux variables ind´ ependantes, Grunert’s Archiv LIV, 209-360 (1872). [11] E. Kolchin, Differential Algebra and Algebraic Groups, Academic Press, 1973. [12] M. Kondratieva, A. Levin, A. Mikhalev, E. Pankratiev, Differential and difference dimension polynomials, Kluwer, 1999. [13] Z. Li, F. Schwarz, S. Tsarev, Factoring systems of linear PDE’s with finite-dimensional solution space, J. Symbolic Comput., 36, 443-471 (2003). ¨ [14] A. Loewy, Uber vollst¨ andig reduzible lineare homogene Differentialgleichungen, Math. Ann. 56, 89-117 (1906). [15] J.-F. Pommaret, A. Quadrat, Generalized Bezout Identity, Appl. Algebra in Engineering, Communications and Computing 9, 91-116(1998). [16] J.-F. Pommaret, A. Quadrat, A functorial approach to the behaviour of multidimensional control systems, Appl. Math. and Comput. Sci., bf 13, 7-13 (2003). [17] M. van der Put, M. Singer, Galois theory of linear differential equations, Grundlehren der Math. Wiss., 328, Springer, 2003. [18] A. Quadrat, An introduction to the algebraic theory of linear systems of partial differential equations, to appear. [19] A. Quadrat, D. Robertz, Parametrization of all solutions of uncontrollable multidimensional linear systems, 16th IFAC World Congress, Prague. [20] C. Sabbah, D-modules coh´ erents et holonomes, Travaux en cours, 45, Hermann, 1993. [21] L. Schlesinger, Handbuch der Theorie der linearen Differentialgleichungen, Leipzig, Teubner, 1897. [22] F. Schwarz, Janet bases for symmetry groups, in: Groebner bases and applications, London Math. Soc., LNS 251, Cambridge Univ. Press, 221-234 (1998), B. Buchberger and F. Winkler, eds. [23] F. Schwarz, A Factorization Algorithm for Linear Ordinary Differential Equations, Proc. ISSAC’89, page 17-25, ACM Press, 1989, G. Gonnet, ed. [24] F. Schwarz, ALLTYPES: An ALgebraic Language and TYPE System, URL: www.alltypes.de. [25] M. Singer, Testing Reducibility of Linear Differential Operators: A Group Theoretic Perspective, Applic. Alg. Engin., Communic. Comp. 7, 77-104 (1996). [26] W. Sit, Typical Differential Dimension of the Intersection of Linear Differential Algebraic Groups, Journal of Algebra 32, 476-487 (1974). [27] W. Sit, The Ritt-Kolchin theory for differential polynomials, in Differential algebra and related topics, ed. Li Guo et al., World Scientific, 2002. [28] W. Stepanow, Lehrbuch der Differentialgleichungen, Deutscher Verlag der Wissenschaften, Berlin, 1956. [29] S. Tsarev, Factorization of linear partial differential operators and Darboux’ method for integrating nonlinear partial differential equations, Theo. Math. Phys. 122, 121-133 (2000).

The last example is a third-order equation which allows a single over-ideal generated by ∂xxx . Example 5.7. Let the third-order operator Dxyy ≡ ∂xyy + (x + y)∂xy + (x + y)∂x − 2∂y − 2 be given. It generates the principal ideal I =< Dxyy > of gauge (1,3). Step S1 does not yield any right factors of order one or two. In step S2 an operator of the form (13) and n ≤ 5, or an operator of the form (12) for m ≤ 2 is not found. However, for m = 3 there is an operator Dxxx ≡ ∂xxx such that the ideal J =< Dxyy , Dxxx > of gauge (1,1) is generated by a Janet base. The equations Dxyy z = 0 and Dxxx z = 0 yield the solution z1 = [(x + y)2 − 2(x + y) + 2]F (y) + 2(x + y − 1)F (y) + F  (y) where F is an undetermined function of y. In step S4, I and J yield the relative syzygy module of gauge (1,2) Syz(I, J) =< (1, 0), (∂xx , −∂yy − (x + y)∂y − x − y) > =< (1, 0), (0, ∂yy + (x + y)∂y + x + y) > . R dy , where Its solution is G(x)s(x, y) + H(x)s(x, y) e−y s(x, y) 2 1 s(x, y) = exp (− 2 (x + y − 2) − y) and G, H are undetermined functions of x. According to the discussion in the Introduction one finally obtains 1 R G(x)s(x, y)x2 dx z2 = 2 R R 2 −x G(x)s(x, y)xdx + 1 2 x G(x)s(x, y)dx and for z3 an identical expression with G(x) replaced by R dy H(x) and s(x, y) by s(x, y) e−y . The general sos(x, y) lution of the given equation Dxyy z = 0 is z1 + z2 + z3 .

6.

CONCLUSION

The results presented in this article allow decomposing partial differential operators of the form (7) or (16) into components of lower gauge. If such a decomposition is found, it may be applied to determine the general solution of the corresponding pde, or at least some parts of it. It is highly desirable to develop a similar scheme to large classes of modules of partial differential operators. The possible types of overmodules can always be determined. The hard part is to identify those for which generators may be constructed algorithmically. An important field of application is the symmetry analysis of nonlinear pde’s, because the determining equations of these symmetries are linear homogeneous pde’s [22]. Another problem is to find an upper bound for the order d in algorithm DecomposeLpde. It would mean that full classes of over-modules could be excluded. On the other hand, a negative answer would be an evidence that this problem could be undecidable

170

On Computing Nearest Singular Hankel Matrices




Markus A. Hitz North Georgia College & State University Dahlonega, Georgia 30597, USA

[email protected]

ABSTRACT

to distance to singularity. In the case of structured matrices, it is not generally known, whether the distance to singularity can be computed in polynomial time. There are several results, showing that preserving a generic structure may lead to an NP-complete problem. Among these problems are the “componentwise distance” to singularity ([22]), and the structured SVD ([6],[26], see also [4]). Therefore, the question has to be reformulated for special structures. The polynomial-time solution of the approximate GCD problem for two univariate polynomials ([18],[19]), also resulted in an efficient algorithm for computing nearest singular Sylvester matrices in the Frobenius norm. Hankel matrices, and - their inverses - Bezout matrices, are closely related to Sylvester matrices, as they are intimately linked to the GCD of polynomials (see Fact 2 below). However, the coefficients of the two defining polynomials are not directly exposed in the entries of Hankel matrices, as would be the case with Sylvester matrices. The problem of finding nearest singular Hankel matrices is therefore much harder, in particular, if we use componentwise norms. Bounds for the distance to singularity have been obtained for special cases before (see, e.g., [27]). A major breakthrough is a recent result [23] by S. Rump. It shows that, for Hankel (and Toeplitz) matrices, distance to singularity - measured in the matrix 2-norm - is exactly the same as the one for unstructured matrices. [23] extends the classic Eckart and Young theorem ([10]) to several types of structured matrices. The companion paper [24] also provides bounds for condition numbers, and the distance to singularity in the componentwise Infinity norm. However, not all of these bounds are sharp. We will use the proof constructs in [23] to explore a simple, and efficient algorithm for computing nearest singular Hankel matrices in the matrix 2-norm. To further analyze the problem in other norms, we propose a parametric construction of singular Hankel matrices. One application, that we have in mind, is the formulation of finding nearest singular Hankel matrices as parametric minimization problems involving symbolic norm expressions. This approach proved to be successful in the case of approximate GCD computations, and for the solution of related problems, involving “nearest” polynomials ([15] and [16]). Although (this time), we cannot provide efficient algorithms that solve the resulting minimization problems, we hope that the interesting properties of our representation will give new insights into the structure of this intriguing problem. In section 2, we introduce some notation, and review well known theorems on Hankel matrices. In section 3, we present the algorithm for the matrix 2-norm. We then derive a (symbolic) representation of a formally singular Hankel matrix, which allows us to state parametric minimization problems for norm expressions. We use the Frobenius norm, and the componentwise Infinity norm as examples.

We explore the problem of computing a nearest singular matrix to a given regular Hankel matrix while preserving the structure of the matrix. Nearness is measured in a matrix norm, or a componentwise norm. A recent result for structured condition numbers leads to an efficient algorithm in the spectral norm. We devise a parametrization of singular Hankel matrices, to discuss other norms.

Categories and Subject Descriptors I.1.2 [Computing Methodologies]: Symbolic and Algebraic Manipulation—Algebraic Algorithms; G.1.6 [Mathematics of Computing]: Numerical Analysis—Unconstrained Optimization; F.2.1 [Theory of Computation]: Analysis of Algorithms and Problem Complexity—Computations on Matrices, Computations on Polynomials

General Terms Algorithms, Experimentation, Theory

Keywords nearest singular matrix, Hankel, Toeplitz, parametric minimization, hybrid symbolic/numeric computing

1.

INTRODUCTION

Hankel matrices and their siblings, Toeplitz matrices have numerous applications in Control Theory, and other areas of Mathematics, as well as Computer Algebra. Their special structure became the base of fast algorithms for matrix inversion, and linear systems solving. These applications are either explicitly, or implicitly affected by how close a given Hankel matrix is to a singular one. In particular, the stability of numerical computations depends on the condition of matrices and linear systems. For general (unstructured) matrices, the condition number is inversely proportional
This material is based on work supported in part by the National Science Foundation under Grant No. CCR-0098175.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. ISSAC’05 July 24–27, 2005, Beijing, China. Copyright 2005 ACM 1-59593-095-7/05/0007 ...$5.00.

171

We have 2n coefficients (of p and q) that linearly depend on the 2n  1 entries of H. The coefficient a0 could be set to a given number, e.g., 0. However, in section 4, we will require the constant coefficient to be unbound.

Throughout the paper, we restrict ourselves to the case of real numbers, mainly for the sake of simplicity. Except for the subsection on the componentwise Infinity norm, all derivations could be adapted to complex numbers. All results in [23] are also shown for matrices with complex entries. Nearest singular Toeplitz matrices can be computed by the same algorithms that we use for Hankel matrices. If T is an n  n Toeplitz matrix, then both JT and TJ are Hankel matrices, where J is the n  n permutation matrix


0  ..  .   

 . .. . 0 .. 1 0

0 . .. . .. 

FACT 2 ([11],[13],[3]). The Hankel Matrix H
generated by the rational function R of Fact 1 is regular, iff the polynomials p and q are relatively prime.

  0  ..  

This theorem could be extended to any reduction in rank, which in turn, would depend on the degree of the GCD of p and q.

1

FACT 3 ([1]). Let p  x and q  x as above. Then the Bezout matrix B p
q of p and q is the product:

. 0

B p
q
T p  qC p 


(which is regular).

2.

where

PRELIMINARIES

In order to be compatible with the (control theory) literature that we cite, our notation uses companion matrices that are the transpose of what most computer algebra systems return.

2.1 Terminology

For n 

n  1 let H  n
n be a regular Hankel matrix, given by the 2n  1 real numbers s0
  
s2n2 :


s s  0 1  s1    sn1   . . H
 .. ..   sn2 sn1   

sn1    s2n3


p1  p2  T p
 ...  pn1

(1)

˜ H

min

Ênn det
H˜ 0

˜  H H

is a triangular Hankel matrix. This is the so-called Barnett factorization of B p
q  The following relation between Bezout and Hankel matrices is attributed to Hermite. FACT 4 ([20],[14],[3]). Let the Hankel matrix H be generated by the polynomials p and q as above, then

(2)




B p
q
T p  H  T p 

or δs


s˜

Ê2n

min

1
det
H ˜ 0

s˜  s


˜ s˜
hvecH

(3)

FACT 5 ([3]). Let the Hankel matrix H be generated by the polynomials p and q as above, then 1 H
qC p   T p 

2.2 Facts about Hankel Matrices

For the following, let the Hankel matrix H  n
n
with hvecH
s0
  
s2n2 T 

FACT 1 ([11],[13],[3]). There exist p  x
monic, of degree n, and q  x of degree n  1,


xn 


Rx :





 xs12

     xs2n2n21






The matrix T p is invertible for any (monic) p. Its inverse is the Hankel matrix H1 p , generated by the rational function 1 px 

This section borrows heavily from [23]. Although not explicitly stated in Rump’s paper, the proofs lead to a constructive method for computing nearest Hankel matrices in the matrix 2-norm. We state the main results in our notation, and restrict them to real matrices.

an1 xn1      a1 x  a0
bn1 xn1      b1 x  b0
s0 x

(7)

3. SPECTRAL NORM SOLUTION

such that the Laurent series of the rational function qx px

(6)

Finally, we combine (5) and (6) to arrive at the representation of a Hankel matrix by a matrix polynomial.

where δ is the distance to singularity, measured in a matrix or vector norm. In the case of (2), we use the matrix 2-norm (“spectral norm”), or the Frobenius norm. Case (3) is the componentwise distance, which is typically expressed in the vector infinity-norm.

px qx



p2    pn1 1     pn1 1  . . .. ..  1 0 

1

We define hvecH :
s
s0
  
s2n2 T to be the Hankel vector of H. Our goal is to compute the Hankel vector s˜ of a “nearest” singular ˜ We have to solve either one of the minimization Hankel matrix H. problems δH




0 1 0  0 .. ..  .. .. .. . . . . .   .. .. .. . . . 0   1  0   0  p0  p1     pn1

is the companion matrix of p, and



sn1 sn   ..  .   s2n3  s2n2


  Cp
  

(5)

L EMMA 1 ([23], L EMMA 10.1). Let x  n  Then there exists a Hankel matrix D  n
n
such that

(4)

Dx
x

where p and g are unique up to the choice of a0 .

172

and

D2  1 

(8)

(e.g., using the decomposition of the next section) would have to be conducted, to show whether other minimal solutions could exist. Steps S1 (computing the eigenvalues) and S2.2 (in particular, computing the pseudoinverse) dominate the complexity of the algorithm. For exact solutions, we have to be able to find the splitting field of the characteristic polynomial. For approximate solutions, we can use Sturm sequences, or numerical methods. Example 1: The characteristic polynomial of the 3  3 Hankel matrix

The proof relies on several theorems, we refer to [23] for details. We outline the construction of D  The vector x
x1
  
xn  is embedded in the 2n  1  2n  1 circulant


x 1     Cx
  xn    ...

 .. .

xn ..

. x1    x1    .. .. 0 . . x2    xn

    xn   xn1    ..  .  0

(9)


1 H
2

x1

Finally, it is shown that D
with T hvecD :
C x Cx e1



2 1 1 1  1 1 2

(10)

is λ3  4λ2  λ  4 

has the required properties of Lemma 1. In (10), Cx is the MoorePenrose pseudoinverse of Cx
and e1
1
0
  
0T has length 2n  1 

The eigenvalues

The main result for real Hankel matrices is

are all of multiplicity 1. Yet, we have two solutions that result in the same distance to singularity: λ1 
λ2 
1  If we select λ1
then x1
1
1
0T is an eigenvector,



λ1
1
λ2
1
λ3
4

T HEOREM 1 ([23], T HEOREM 12.1). Let the Hankel matrix H  n
n be regular. The “structured” distance to singularity, measured in the matrix 2-norm, is the same as the one for unstructured matrices, i.e., δH


1

H1 2


σmin





1  0 Cx 1
  0

1 1 0 0 0 1 0

(11)

where σmin is the smallest singular value of H 

0 1 1 0 0

0 0 1 1 0



0 0  0 
1 1

and


0  1 T  C x 1 Cx 1
 0 

0 0 1 0 0 0 0

0 0 0 1 0

0 0 0 0 1



1 0  0  0 0

˜
H  ∆H is singular, with Finally, the perturbed matrix H


0 ∆H
λ1  1

Hankel matrices are symmetric, therefore σmin
λmin 




1 0 0 0
0 0 0

where λmin is an eigenvalue whose absolute value is minimal. Furthermore, for real H
the eigenvalues are real (see [12]).

and

The second eigenvalue λ2 yields




This justifies the following algorithm.


1 ˜
1 H



1 1 1 1   1 1 2



6 3 14 3 14 24 
14 24 12

1 ∆H
  31

Algorithm S: Input: a regular Hankel matrix H  n
n  Output: a set of singular Hankel matrices ˜  n
n : H ˜  H2
δH
SH
H where δH is the distance to singularity. S1: Compute the eigenvalues λ1
  
λn of H  S1.1: Find δH
min1kn λk   S1.2: Determine the set Λm
λk : 1  k  n
λk 
δH  S2: Set SH
 For each λk  Λm : S2.1: determine an eigenvector xk
Hxk
λk xk  T S2.2: compute the Hankel vector hk
C xk Cxk e1
where Cxk is the circulant of (9). S2.3: set ∆H
λk D
with hvecD
hk
and let SH :
SH  H  ∆H 

and




˜
H



9 5  5 13 13 9 11   31 9 11 10

˜
0 One can easily verify that ∆H2
λ2 
1
and det H The example shows (in the case of λ1 ), that minimal perturbations in a matrix norm can single out particular anti-diagonals, and leave all the others unchanged. For some applications, such as sensitivity analysis, minimal perturbations in a componentwise norm would give us more meaningful results. Furthermore, Algorithm S (in its current form) does not permit us to select individual entries in the Hankel vector, in order to have only these entries perturbed, while the others would be fixed.

Discussion of Algorithm S The correctness of the algorithm follows from Lemma 1 and Theorem 1. In (8), Dx
x and D2  1 , implies D2
1 . T The choice of e1 in (10) is not arbitrary. The matrix C x Cx is a circulant. Every column is a circular rotation of the first column, and yields the Hankel vector of a Hankel matrix D with D2
1  However, only the D which is generated by the first column satisfies Dx
x  In most cases, the pseudoinverse reduces to the regular inverse. As shown in [23], the circulant Cx is diagonalized by a Fourier H  matrix F , Cx
FH dF and d
FCx FH , such that C x
F d F, allowing for the application of fast techniques. The algorithm proceeds along the axes of the hyperellipsoid given by the singular values of H (see [12]). A more detailed analysis

4. SINGULAR HANKEL MATRICES In this section, we want to characterize the space of singular Hankel matrices. We will use the facts of section 2 to construct an n  n Hankel matrix whose rank is at most n  1 

4.1 Parametrization By Fact 2, we can construct two polynomials p and q that share a common factor x  α
such that the Hankel matrix generated by the rational function Rx
qx px is rank-deficient. For n  1


173

and α  
let

T HEOREM 2. Let the (coefficient) vectors u
u0
  
un2 
v
v0
  
vn2   n1
and H  n
n
then

f x :
xn1  un2 xn2      u1 x  u0
gx :
vn2 xn2      v1 x  v0


(i) For any choice of u and v
the Hankel matrix H
defined by hvecH
Mv
is singular.

and px :
x  α f x
qx :
x  αgx  The coefficients ui and v j are either symbolic or numeric parameters. In the symbolic case, we will try to determine the coefficients by parametric minimization of a norm expression. Next, we want to use Fact 5 to generate a singular Hankel matrix, whose entries correspond to the rational function Rx
qx px  symbolic entries. Looking at the Laurent series expansion (4) of Rx, it is obvious that the so-called Markov parameters si do not depend on the choice of α in the common factor of p and q  We could carry α along; however, term simplification would eventually eliminate it from all entries of H  For our derivation, we can set α
0, leading to the companion matrix


0 ..   .  .. Cp
  .  0

1 0  0 .. .. .. .. . . . . .. .. . . 0   0 1    un2 0 u0

(ii) For any singular Hankel matrix H
there exist u and v
such that hvecH
Mv  P ROOF. (i) Every term of (12) contains the factor Cp  The first column of C p is zero; therefore C p and H are singular. (ii) follows from Facts 1 and 2.

4.2 Properties of Matrix M Next, we list - without going into details - further properties of the matrix M  1. M is the leading 2n  1  n  1 submatrix of the infinite Hankel matrix H1 f generated by the rational function 1 f x 

       

2. Due to f being monic, M has full column rank n  1 

3. If M is symbolic, its entries are multivariate polynomials in the coefficients u0
  
un2 of f 

of px
xxn1  un2 xn2      u0   The matrix polynomial qC p  becomes: qC p 
v0 C p  v1 C2p      vn2 Cnp1 

4. The rows mk of M can be expressed as



mk
Ckf  en1

(12)


qC p   Tp 1
v0 C p  v1 C2p      vn2 Cnp1   Tp 1    
v0 C p Tp 1      vn2 Cnp1 Tp 1





1p

where k
0
  
n  1  This is due to the fact that every entry in a Hankel vector, with index k rankH is “linearly generated” by the coefficients of the denominator polynomial (see [11]).

5. NORM EXPRESSIONS

(13)

where v
v0
  
vn2 T  Obviously, we have the same relationship among the Hankel vectors





1p



v

We use the parametric representation of a singular Hankel matrix of the previous section to derive norm expressions for the Frobenius (matrix) norm, and the componentwise infinity (vector) norm. Similar expressions could be constructed for other norms, and norm expressions that would exclude specific entries from being perturbed. The symbolic norm expressions allow us to formulate the problem of computing nearest Hankel matrices as a parametric minimization problem.

(14)

Finally, we recombine the column vectors hvecHxk  p  into the 2n  1  n  1 matrix


0 ..   .     0 M
 1      u.n2 ..

 . .. . .. un2

..

.

0 1 . .. ..

.

..

.

  un2   ..  .         . 1

5.1 Frobenius Norm

For n  1
we define the 2n  1  2n  1 weight matrix W :
diag1
  
n  1
n
n  1
  
1 

(15)

Then, for a Hankel matrix H with s
hvecH
we can express the Frobenius norm of H in terms of the Hankel vector:


sT Ws un2 T and v
v0

..

H2F



(19)

T be vectors with Let u
u0
  

  
vn2  symbolic entries (as in section 4). Our representation (16) lets us construct the Hankel vector of a (formally) singular Hankel ma˜
by setting hvecH ˜ 
Mu v
where the entries of Mu solely trix H depend on the entries of u 

and arrive at hvecH
Mv 

(17)

u0 mk  u1 mk1      un2 mkn2  mkn1
0
(18)



v


hvecH
hvecHx p 
  
hvecHxn

k
0
  
2n  2

5. Between n consecutive rows mk
mk1
  
mkn1 of M
the coefficients of f define the linear dependency

1 The products Ckp T p are the Hankel matrices Hxk  p which are generated by the rational functions xk  px  We can rewrite the last line of the previous equation as a vector of matrices times vector product

H
Hx p
Hx2  p
  
Hxn




where C f is the n  1  n  1 companion matrix of f
C0f is the identity matrix In1
and en1
0
  
0
1T  n1 

Therefore, H

T

(16)

The following theorem states the main property of our representation.

174

For a regular Hankel matrix H  n
n
the distance to singularity δH in the Frobenius norm can be parametrized in the entries of u and v as ˜  HF
δH
min H

where

u
v

˜ 
Mu v  hvecH

and the perturbed matrix ˜ H

(20)

:


˜  H2F H


vT MTu WMu v  vT MTu Ws  sT WMu v  sT Ws
vT Qv  vT r  rT v  sT Ws Q :
MTu WMu and r :
MTu Ws

(21)

linear constraints:






with



s
hvecH
1
2
3T

is regular. We have two scalars u and v that parametrize our minimization problem. The matrix Mu consists of the single column 1
u
u2 T  The weight matrix W is diag1
2
1
while Q is a 1  1 matrix with the single entry u4  2u2  1  Therefore,

N m u
and

N m u
8

3u2  4u  12 u4  2u2  1

uv  δ

u2 v  δ

(22)

0  k  2n  2 0  k  2n  2






1 2 3

We have to determine u
v
and δ for all sign combinations, although we can narrow down our computations to half of the cases. Patterns that are complements in terms of  and  (e.g.,    and   ) result in identical values for u and v
while the ones for δ are the negative of each other. Eventually, we have to compare the δ values, and select the (or a) minimum. For the given example, we have a unique solution, namely



The roots of the numerator are 1 1 1 1 1
1
5 
and  5  3 2 2 2 2

1 1 The last value u
 2 5  2 minimizes N m  Finally, we compute v


δ sk  mk  v δ sk  mk  v

vδ




3u2  4u  1u2  u  1 u2  1u4  2u2  1



where mk is a row of Mu  We are not aware of efficient methods (e.g., inner-point techniques) for solving linear programs containing symbolic expressions, which - in our case - are multivariate polynomials in u0
  
un2  In [16], we used the first step of Stiefel’s [25] “exchange” algorithm, which in turn is based on the method of de la Vall´ee-Poussin [7]. At that time, our problem had a single parameter, resulting in two cases which had to be examined. We could try to apply the remaining steps of the algorithm to our current problem. However, the algorithm involves linear systems that contain “sign vectors” which clearly lead to exponential running time. On the other hand, expression swell is less severe than in the case of the Frobenius norm. For one thing, we are directly using (16), without squaring or weighting expressions. Then, the driving factor are the linear dependencies among rows. Because of property 5 (of M), these are relatively simple. Example 3: We use the Hankel matrix from the previous example, so that we can compare the perturbations in the Frobenius norm with the ones obtained for the componentwise norm. We continue to use the scalars u and v as the variables in our parametric representation. The given Hankel vector is s
hvecH
1
2
3T  We examine the vertices, defined by the system

Exact solutions can only be obtained in special cases, and for small values of n  Nonetheless, we give a low-degree example to illustrate the steps that are involved. Example 2: The 2  2 Hankel matrix 2 3



u
v

minimize: δ

3. We can use methods of geometric reasoning to find “nearest” intersection points with the manifold of singular Hankel vectors, as defined by (16).

1 2

189 307

As shown in [15], [16], and [25] the minimization problem can be expressed as a linear program. We use the distance to singularity in the (vector) infinity norm δ :
δH  as an additional slack variable.

of N m u
find all real solutions for the parameters ui
and finally select the ones that minimize N m u 



117 189

˜ H

2. We can examine the critical points of N m u
which leads to a nonlinear system of n  1 equations in n  1 variables. For that, we have to derive the gradient

H




˜   hvecH∞
min Mu v  s∞  δH
min hvecH

1. In a hybrid approach, we can apply numeric techniques of unconstrained optimization to directly minimize N as a function of 2n  2 parameters.

∂N m ∂N m

∂u0 ∂un2



Continuing with the notation of the previous subsection, we state the minimization problem for the componentwise infinity norm, which is applied to the Hankel vectors. Here, we want to find values for the entries of u and v that result in

This result is mainly of theoretical interest. To actually find values for the symbolic entries of u and v
we have several choices, all of which lead to rather expensive computations:



u u2

5.2 Componentwise Infinity Norm

 where The quadratic form N is parametrized in the 2n  2 entries of u and v  Because Mu has full rank (property 2), the matrix Q is invertible. We can derive the symbolic minimum N m as a function of u :

N m u
r Q1 r  sT Ws
where v
Q1 r 

1 u

˜  HF  0236  δH
H


Mu v  sT WMu v  s

T



˜ 
v 1  u2  u2 
0  The distance to singuEvidently, detH larity in the Frobenius norm is

With s :
hvecH
we derive the norm expression

N


v



 1  15  7 5 3  5
20

u


175

5 9
v

3 8

and

δ


1  8

The perturbed matrix ˜ H




  15

9 8 8 15 25 8 8




1125 1875 1875 3125

[10] C. Eckart and G. Young. The approximation of one matrix by another of lower rank. Psychometrika, 1(3):211–218, 1936. [11] F. R. Gantmacher. The Theory of Matrices. Chelsea, 1959. (Volume I and II). [12] G. H. Golub and C. Van Loan. Matrix Computations, 3rd ed. Johns Hopkins University, 1996. [13] G. Heinig and K. Rost. Algebraic Methods for Toeplitz-like Matrices and Operators. Birkh¨auser, 1984. [14] U. Helmke and P. A. Fuhrmann. Bezoutians. Linear Algebra Appl., 122/123/124:1039–1097, 1989. [15] M. A. Hitz and E. Kaltofen. Efficient algorithms for computing the nearest polynomial with constrained roots. In Proceedings of the 1998 International Symposium on Symbolic and Algebraic Computation, ISSAC’98, pages 236–243, 1998. [16] M. A. Hitz, E. Kaltofen, and Lakshman Y. N. Efficient algorithms for computing the nearest polynomial with a real root and related problems. In Proceedings of the 1999 International Symposium on Symbolic and Algebraic Computation, ISSAC’99, pages 205–212, 1999. [17] W. Kahan. Numerical linear algebra. Canadian Math. Bull., 9:757–801, 1966. [18] N. Karmarkar and Lakshman Y. N. Approximate polynomial greatest common divisors and nearest singular polynomials. In Proceedings of the 1996 International Symposium on Symbolic and Algebraic Computation, ISSAC’96, pages 35–39, 1996. [19] N. K. Karmarkar and Lakshman Y. N. On approximate GCDs of univariate polynomials. JSC, 26(6):653–666, 1998. Special issue on Symbolic Numeric Algebra for Polynomials S. M. Watt and H. J. Stetter, editors. [20] M. G. Krein and M. A. Naimark. The method of symmetric and Hermitian forms in the theory of the separation of the roots of algebraic equations. Linear and Multilinear Algebra, 10:265–308, 1981. Originally published in Russian, Kharkov (1936). [21] H. Park, L. Zhang, and J. B. Rosen. Low rank approximation of a Hankel matrix by structured total least norm. BIT, 39(4):757–779, 1999. [22] S. Poljak and J. Rohn. Checking robust nonsingularity is NP-hard. Math. Control Signals Systems, 6:1–9, 1993. [23] S. M. Rump. Structured perturbations part I: Normwise distances. SIAM J. Matrix Anal. Appl., 25(1):1–30, 2003. [24] S. M. Rump. Structured perturbations part II: Componentwise distances. SIAM J. Matrix Anal. Appl., 25(1):31–56, 2003. ¨ [25] E. Stiefel. Uber diskrete und lineare Tschebyscheff-Approximationen. Numerische Mathematik, 1(1):1–28, 1959. ¨ [26] O. Toker and H. Ozbay. On the NP-hardness of the purely complex µ computation, analysis/synthesis, and some related problems in multidimensional systems. In Proceedings of the American Control Conference, pages 447–451, 1995. [27] E. E. Tyrtyshnikov. How bad are Hankel matrices? Numerische Mathematik, 67:261–269, 1994.



is singular by construction. We notice that the size, and the sign of the perturbations agree with our solution for the Frobenius norm.

6.

CONCLUDING REMARKS

We explored the problem of computing nearest singular Hankel matrices in various norms. For the matrix 2-norm, the solution is surprisingly simple. In other norms, the problem still resists efficient solutions. We developed a representation that lets us use tools that proved to be useful for similar applications. However, unlike with our minimization problems related to “nearest polynomials,” the current one is not a rank-one (or rank-two) problem. In the case of univariate polynomials, we were able to parametrize the minimization in a single variable (the root α). Here, the common factor x  α is divided out from the generating rational function. There might be better ways to set up the parametric minimization problem. There are interesting questions that remain for future research. So far, we only considered real (and complex) numbers. We do not know whether any of the algorithms would carry over to finite fields. For general matrices, the structured SVD, and componentwise distance to singularity are both NP-hard problems. It will be interesting to study the relationship of the two problems in more detail. For Hankel matrices, we can construct a perturbed matrix in the 2-norm, and then compute the SVD “after the fact.” The question is whether this would have any implications for computing the componentwise distance to singularity.

7.

ACKNOWLEDGEMENTS

We would like to thank the anonymous reviewers for their helpful comments. Discussions with students and colleagues at North Georgia - most often unwittingly - contributed to the development.

8.

REFERENCES

[1] S. Barnett. A note on the Bezoutian matrix. SIAM J. Appl. Math., 22(1):84–87, 1972. [2] S. Barnett. Matrices in Control Theory. Robert E. Krieger, 1984. rev. ed. [3] D. Bini and V. Pan. Polynomial and Matrix Computations, Volume 1, Fundamental Algorithms. Birkh¨auser, 1994. [4] J. Chen, K. H. Fan, and C. N. Nett. Structured singular values and stability analysis of uncertain polynomials, part 1: the generalized µ. Systems & Control Letters, 23(1):53–65, 1994. [5] J. Chen, K. H. Fan, and C. N. Nett. Structured singular values and stability analysis of uncertain polynomials, part 2: a missing link. Systems & Control Letters, 23(2):97–109, 1994. [6] G. E. Coxson and C. L. DeMarco. Testing robust stability of general matrix polytopes is an NP-hard computation. In Proceedings of the 29th Allerton Conference, 1991. [7] C. J. de la Vall´ee-Poussin. Sur la m´ethode de l’approximation minimum. Soc. Scient. Bruxelles, Annales, 2i`eme partie, m´emoires, 35:1–16, 1911. [8] J. W. Demmel. The componentwise distance to the nearest singular matrix. SIAM J. Matrix Anal. Appl., 13:10–19, 1992. [9] J. Doyle. Analysis of feedback systems with structured uncertainty. IEE Proc. Part D, 129:242–250, 1982.

176

A Reliable Block Lanczos Algorithm over Small Finite Fields Bradford Hovinen

∗

Wayne Eberly

†

Department of Mathematics University of Toronto

Department of Computer Science University of Calgary

[email protected]

[email protected]

ABSTRACT

Keywords

Blocked versions of the Lanczos procedure have been successfully applied to sample nullspace elements of very large sparse matrices over small finite fields. The heuristic currently in use, namely, Montgomery’s method [10], is unreliable for certain input matrices. This paper introduces a new biconditional block Lanczos approach based on lookahead, a technique designed to improve the reliability of the scalar Lanczos algorithm. Empirical data show that the performance of the lookahead-based algorithm is competitive with that of Montgomery’s heuristic when their relative reliability is taken into account. The reliability of this new algorithm for arbitrary matrices over small finite fields is then established. In the process, some results on the ranks of certain submatrices of a randomly determined block Hankel matrix are established. These results may be applicable in other contexts, such as Coppersmith’s block Wiedemann algorithm [3].

Lanczos algorithm, black box matrix, finite field, parallel computing, linear system solution, randomized algorithm

1. INTRODUCTION Random sampling of elements of the nullspace of a large, sparse matrix defined over a small finite field, such as GF (2), is a key step in the number field sieve algorithm for factoring integers and computing discrete logarithms. Structured Gaussian elimination has been used to do this, but it typically results in the input matrix filling in, using more memory than is available. Krylov-based algorithms that treat the input matrix as a black box, allowing only the application of the matrix or its transpose to a vector, typically have far more predictable memory requirements. One such approach is a variant of the Lanczos algorithm originally proposed in [9] and adapted to the finite field setting in [11]. Let A ∈ Kn×n be a symmetric matrix over a field K and suppose that we wish to solve a consistent linear system Ax = b. The Lanczos process operates by constructing an orthogonal basis {v0 , . . . , vr } of the Krylov space span{b, Ab, A2 b, . . . } iteratively, setting v0 := b and, for i ≥ 0,

Categories and Subject Descriptors F.2.1 [Analysis of Algorithms and Problem Complexity]: Numerical Algorithms and Problems—computations in finite fields, computations on matrices; I.1.2 [Symbolic and Algebraic Manipulation]: Algorithms—algebraic algorithms, analysis of algorithms

vi+1 := Avi −

General Terms

i X (Avi )T Avk vk . vkT Avk k=i−1

(1)

In (1), vi is produced by orthogonalizing Avi with respect to vi and vi−1 . It turns out that Avi is already orthogonal to vi−2 , vi−3 , . . . , v0 so that vi+1 is as well. The process terminates when vi+1 = 0, and the solution x to the system Ax = b can be recovered easily from the iterates vi as the algorithm progresses. Over a finite field, it can never be guaranteed that vkT Avk 6= 0. This problem is especially pronounced on a small finite field GF (q), where vkT Avk = 0 on approximately 1/q of the iterations. The technique of lookahead, first applied in the finite field setting in [8] and adapted from a numerical technique introduced in [12], solves this problem for the scalar Lanczos process. We refer to [8] for a complete description of the solution. We would like to take advantage of parallelization to improve the algorithm’s performance. This is particularly true over GF (2), where field arithmetic operations can be performed in parallel with bit operations. The opportunities to parallelize the Lanczos process described above are few, and for that reason we introduce a blocked variant. The key idea is that we select a parameter N > 0, called the blocking

Algorithms, Performance, Reliability, Theory

∗Research was supported in part by the Natural Sciences and Engineering Research Council of Canada. †Research was supported in part by the Natural Sciences and Engineering Research Council of Canada research grant OGP0089756.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. ISSAC’05, July 24–27, 2005, Beijing, China. Copyright 2005 ACM 1-59593-095-7/05/0007 ...$5.00.

177

factor, and let the iterates v0 , v1 , . . . be matrices in Kn×N . A straightforward generalization of the Lanczos procedure is described in [6], where the method’s reliability is proved for fields whose cardinality exceed n. Over smaller fields, this na¨ıve blocked algorithm encounters difficulty similar to the na¨ıve scalar algorithm: it is necessary to invert the matrix vkT Avk , but it can never be guaranteed that that matrix is invertible, even with high probability. The block Lanczos heuristics of Coppersmith [2] and Montgomery [10] address this problem using different techniques, and the latter has been used to tackle nullspace-sampling problems in practise. However, for neither approach exists an analysis showing it to be reliable for arbitrary input matrices. Indeed, it is known that both methods successfully generate nontrivial nullspace vectors of an input matrix with only exponentially small probability when the matrix has too many invariant factors divisible by x2 . They must symmetrize the input by working with AT A or AAT instead of the given matrix A. There is no guarantee over small fields that rank AT A and rank A are the same, or even very close together. Thus the vectors they produce do not necessarily belong to the nullspace of A. Another algorithm that may be used for nullspace sampling is Coppersmith’s [3] blocked version of the Wiedemann algorithm [15]. As compared with Coppersmith’s algorithm, block Lanczos methods may solve nullspace-sampling problems substantially more quickly, especially on input matrices with low rank. The randomized behaviour of block Wiedemann was fully analyzed in [14], with improvements given in [1]. The Wiedemann and Lanczos algorithms are closely related, as highlighted in [8], so the results proved in [14] can be adapted to help to establish the reliability of a suitable block Lanczos approach. In both algorithms appear certain randomly determined matrices with a so-called block Hankel structure. The ranks of these matrices and submatrices thereof are of interest. However, the results in [14] do not address the ranks of the relevant submatrices adequately for our purposes. They also appear to be inadequate for the analysis of a block Wiedemann algorithm that incorporates “early termination.” This paper makes two contributions in this area. The first contribution is in Section 2, where we introduce a new blocked variant of the Lanczos procedure which overcomes the deficiencies of the methods of Montgomery and Coppersmith. In particular, the algorithm is developed in such a way that the input matrix need not be symmetric. In Sections 3 and 4 we establish the reliability of this algorithm. That is, we prove that, for any singular input matrix, the algorithm returns nontrivial elements of the matrix’s nullspace with a probability bounded away from zero by at least a constant. Thus the algorithm is randomized of the Las Vegas type. The second contribution is in Section 4, where we give an analysis of the ranks of various submatrices of the associated block Hankel matrix. These results may be applied in the context of the block Wiedemann algorithm, as suggested in Section 6 below, as well as other blocked iterative algorithms.

2. A LOOKAHEAD-BASED BLOCK LANCZOS ALGORITHM We describe an approach for nullspace sampling that combines the lookahead method in [8] with blocking to achieve both provable reliability and a high degree of parallelizability. The algorithm is a biconditional variant of Lanczos. That is, it produces two sets u0 , . . . , um−1 and v0 , . . . , vm−1 of matrices in Kn×N with the property that uTi Avj is invertible for i = j and 0 otherwise. To begin, let us define a simple (but unworkable) block generalization of the scalar Lanczos process. We choose u0 and v0 uniformly at random from Kn×N . Now suppose that we have computed iterates u0 , . . . , ui and v0 , . . . , vi so that uTj Avk is invertible if j = k and zero otherwise. We next compute iterates ui+1 and vi+1 as follows. Let Ui and Vi be matrices given by 9 8 (2) Ui := : u0 u1 · · · ui ; , 9 8 (3) Vi := : v0 v1 · · · vi ; .

Then UiT AVi has a block diagonal structure. We augment Ui by adjoining the columns of the matrix AT ui to form U˜i and we augment Vi by adjoining the columns of Avi to form ˜iT AV˜i has nearly a block diagonal V˜i . The resulting matrix U structure, except that the blocks corresponding to uTi A2 vi are off the diagonal but are nonzero. Assuming that uTi Avi is invertible, we perform one block row operation and one block column operation to zero out these off-diagonal blocks. Let Li+1 and Ri+1 be matrices representing these elimination ˜iT AV˜i Ri+1 is block diagonal. Then we steps, so that LTi+1 U can define Ui+1 := Ui Li+1 and Vi+1 := Vi Ri+1 . Finally, we let ui+1 ∈ Kn×N be the matrix whose columns are the last N columns of Ui+1 and vi+1 ∈ Kn×N be the matrix whose columns are the last N columns of Vi+1 . Clearly, this process reduces to a biconditional variant of the scalar Lanczos process if N = 1. The problem with this approach is that uTi Avi is generally not invertible. Our algorithm resolves this problem with a block analogue of the process of lookahead. In the scalar algorithm, lookahead operates by adjoining rows AT ui , (AT )2 ui , . . . , (AT )j ui to Ui and columns Avi , A2 vi , . . . , Aj vi

to Vi , where j > 0 is the minimal value such that the re˜iT AV˜i has full rank. If no such j exists, then ansulting U other technique is required. Extending this approach to the blocked case is more complicated, and we restrict our attention to lookahead for one side only, as the other side is entirely symmetric. Algorithm 2 given below looks for a maximal invertible submatrix of uTi AVi and works to compute the values described in this paragraph. The size of the submatrix menu tioned above is denoted ρui . We define v˙ i ∈ Kn×ρi as the matrix whose columns are the columns of Vi corresponding to this submatrix. Let σ be an invertible transform such that the first ρui rows of (ui σ)T Av˙ i are the desired submatrix and the remaining rows are zero. Let u ˇi := ui σ, let u ¯i be the first ρui columns of u ˇi , and let u ˆi be the last N − ρui columns of u ˇi . Then the desired invertible submatrix is u ¯Ti Av˙ i and T u ˆi Av˙ i = 0. Elimination steps are now performed on u ¯Ti AVi

178

so that all entries outside of the invertible submatrix u ¯ Ti Av˙ i are zero. The matrix u ¯Ti Av˙ i is then used to zero as much of the off-diagonal portion of ui AV˜i as possible. Of course, not all of the latter can be zeroed if ρui < N , so there will generally be nonzero entries off the block diagonal left over for the next iteration. This is the situation that “lookahead” addresses. As long as those entries are nonzero, the corresponding columns of uTi AVi+1 are independent of uTi Av˙ i . Another invocation of Algorithm 2 finds a new maximal submatrix by extending v˙ i with a maximal independent set of columns of uTi AVi+1 . In this manner, Algorithm 2 operates incrementally: it extends the existing invertible submatrix as far as possible on each invocation. We denote by u ˜i the columns of u ˇi that are added to u ¯i in the current iteration. The procedure itself takes two arguments: i, the current iteration and l, the iterate for which u ¯l and v˙ l are to be extended. The nonzero entries are concentrated in u ˆTl Aˆ vi , so a Gauss-Jordan decomposition is performed on that matrix. The independent columns are appended to v˙ l and u ˇl is updated with the computed transform. The nonzero entries of UiT AVi below the columns added to v˙ l are zeroed with elimination steps. The invertible submatrix of uTl Avi is also used to eliminate ˜iT Avi , so u˙ l and vˇi are updated apprononzero portions of U priately. The above steps establish the invariant that u ¯Tj Av˙ k is invertible if j = k and zero otherwise, and the same holds for u˙ Tj A¯ vk . Termination occurs at iteration m where um = vm = 0. The solution x is then recovered from the formula x=

m−1 X j=0

“ ”−1 v˙ j u ¯Tj Av˙ j (¯ uj )T b,

gorithm 2.8 in [13]. Its signature is (B, I, S, T, r) ← GJDecomp(C) where C ∈ Kk×l is the matrix on which to perform the Gauss-Jordan decomposition. The values returned are • r, the rank of C; • B ∈ Kr×r , the inverse of a maximal invertible minor of C; • I, the indices of columns of C corresponding to B −1 ; and • S ∈ Kk×k and T ∈ Kl×l such that 8 −1 0 > B SCT = > : 0 0

9 > > ;.

For simplicity of presentation, it is assumed that the various iterates {ui } and {vi } as well as the ranks {ρui } and {ρvi } are shared between Algorithms 1 and 2. Algorithm 4 given below is the main driver for sampling the nullspace of A. It first determines matrices y and v uniformly at random and invokes Algorithm 1 with the inputs b := Ay and v0 := Av. Hoping that Ax − Ay is rankdeficient, it then invokes GJDecomp on the input Ax − Ay to find the column dependencies, which are placed in the matrix T . Then the columns of (x − y)T are in the right nullspace of A. Not all return values of GJDecomp are required for this, so the unused ones are marked with a hyphen. When the algorithms are implemented in practise, the values given in brackets are not recomputed each time but are stored and carefully updated on each iteration, yielding a significant performance benefit. For simplicity, the process is omitted from this exposition. See [7] for details on how this is done. The blocking allows the algorithm to be parallelized by computating the columns of Avi , AT ui , uTi Avi , and uTi A2 vi in parallel. Furthermore, each of the columns of ui+1 and vi+1 can be updated in parallel in lines 10 and 11 of Algorithm 1. In practice, these computations use by far the most CPU time of the algorithm; see [7], Chapter 4, for a breakdown. Each iteration of Algorithm 1 has complexity O(N ω(A) + nN θ−1 + N θ ), where ω(A) is the time required to multiply A or AT by a single n-vector and θ is a constant such that two N × N matrices may be multiplied in time O(N θ ). The expected total number of iterations is shown in Section 4 to be n/N , deviating from that mean by a constant ∆ with probability that diminishes exponentially with ∆. Thus the total expected complexity of Algorithm 1 is O(nω(A) + n2 N θ−2 + nN θ−1 ), which is consistent with other iterative system solvers.

(4)

which may be accumulated as the iteration progresses. Upon termination, x has the property that uT (Ax − b) = 0 for all u ∈ span{u0 , . . . , um−1 }. This observation forms the basis of our analysis of reliability in Section 3. If after some iteration k ≥ i, ρui = N , then it is guaranteed that the elimination steps will zero the block uTi A2 vk . It follows by a straightforward modification of the analysis of the scalar algorithm that uTi−1 A2 vk+1 = 0. Thus the iterate ui−1 is no longer needed and can be removed from memory, provided that the estimate of the solution is also updated as suggested by (4). The purpose of Algorithm 3 given below is to find the largest index l0 such that ρuk = ρvk = N for all k < l0 . It then updates the solution x and throws out all iterates uk and vk with k < l0 . It is possible that, at some point while running the algorithm, one of the matrices ui or vi becomes nonzero and rank-deficient. When this happens, neither the iterates ui and vi nor any subsequent iterates may be thrown out of memory, as ρuk < N or ρvk < N for all k ≥ i. For this reason, Algorithm 3 has a mode, determined by the input f , in which it updates the solution using all remaining iterates. When the termination condition is reached, Algorithm 3 is invoked in this mode. Algorithm 1 is the main block Lanczos algorithm, which takes A, b, and v0 as input and produces a candidate solution x with the property that Ax − b is orthogonal to all u ∈ span{u0 , . . . , um−1 }. The listings below refer to an algorithm GJDecomp, which performs a Gauss-Jordan transform on the input matrix. It is a slight modification of Al-

3. ANALYSIS OF RELIABILITY The method described in Section 2 for sampling the nullspace of a matrix is randomized and can fail to identify any nontrivial nullspace vectors in four ways. We would like to bound the probability that, for uniformly randomly selected y, v0 , and u0 , any of the following events occur: 1. For some iterate i, uTi Aj vi = 0 for all j > 0 whilst neither ui = 0 nor vi = 0. This is known as incurable breakdown.

179

Algorithm 4 x ←SampleNullspace (A)

Algorithm 1 x ← BlockLanczos (A, b, v0 )

1: 2: 3: 4: 5:

1: Choose u0 uniformly at random from Kn×N 2: i ← 0; l0 ← 0; x ← 0 3: while ui 6= 0 and vi 6= 0 do 4: u ˇi ← ui ; vˇi ← vi 5: ρui ← 0; ρvi ← 0 6: ui+1 ← AT ui 7: vi+1 ← Avi 8: for l from l0 to i do 9: DecomposeIterates (i, l) h` ´−1 iT ˆ T ˜T vl ui+1 A¯ vl 10: ui+1 ← ui+1 − u˙ l u˙ Tl A¯ h` i ´−1 ˆ T ˜ 11: vi+1 ← vi+1 − v˙ l u ¯Tl Av˙ l u ¯l Avi+1 12: end for 13: (l0 , x) ←Cleanup (l0 , x, b, false) 14: i←i+1 15: end while 16: (l0 , x) ←Cleanup (l0 , x, b, true) 17: return x

2. The matrix A(x − y) has full rank. 3. The matrix (x−y)T has rank zero. More generally, we would like to bound the probability that the k sampled nullspace vectors span a vector space of dimension less than r for 0 ≤ r ≤ k. 4. A sufficient number of iterates of lookahead are required that available memory is exhausted. We address conditions (1), (2), and (3) in this section. Section 4 addresses condition (4). We shall see that, to ensure reliability, the difference between the blocking factor N and the number of nontrivial invariant factors of the input matrix A should be at least log q n. We will then apply a method from the literature [6] to ensure that the number of invariant factors of the matrix A is less than 2 log q n with high probability, so that a blocking factor of 3 log q n is sufficient to establish reliability. Condition (1) is, in fact, one of the events that cause Coppersmith’s block Wiedemann algorithm in [3] to fail. We use the analysis of block Wiedemann in [14], improved in [1], to bound its probability. The following theorem is a trivial application of their results.

Algorithm 2 DecomposeIterates (i, l) `ˆ T ˜´ 1: (µ, ζ, σuT , τv , ru ) ← GJDecomp “ u ˆl Aˆ vi ” ˆ T ˜T 2: (ν, η, σvT , τu , rv ) ← GJDecomp u ˆi Aˆ vl 3: if l < i then 9 8 ; 4: u˙ l ← : u˙ l u(ζ) i 8 9 5: v˙ l ← : v˙ l vi(η) ; 6: σ ˜u ← Diag(Iρui , σu ); u ˇl ← u ˇl σ ˜u ˜v 7: σ ˜v ← Diag(Iρvi , σv ); vˇl ← vˇl σ 8: end if 9: for j from l + 1 to i do iT h (η) ˆTj Avi 10: u ˆj ← u ˆj − u ˜ l µT u »“ – ”T (ζ) 11: vˆj ← vˆj − v˜l ν T ui Aˆ vj

Theorem 3.1. Let φ be the number of nontrivial invariant factors in the Frobenius normal form of A. Then the probability of failure condition (1) is at most 6/q N −φ . In particular, if N − φ > log q n, then the aforementioned probability is at most 6/n. Proof. See Theorem 3.5 in [7].

12: end for 9 8 ˜l ; 13: u˙ i ← : u˙ i u 8 9 14: v˙ i ← : v˙ i v˜l ; 15: 16: 17: 18: 19: 20:

Choose y ∈ Kn×N uniformly at random Choose v ∈ Kn×N uniformly at random x ← BlockLanczos (A, Ay, Av) (−, −, −, T, ρ) ← GJDecomp (A(x − y)) return rightmost N − ρ columns of (x − y)T

We now consider condition (2). Assume that, at iteration m, um = vm = 0. Then the algorithm ensures that uT A(x− y) = 0 for all u ∈ span{u0 , . . . , um−1 }. Suppose the span of the columns of u0 , . . . , um−1 , which is the same as the span of the columns of u0 , AT u0 , (AT )2 u0 , . . . , contains the rows of A. It follows from Proposition 1 in [14] that this holds as long as failure condition 1 does not occur. Then A(x − y) is orthogonal to the row span of A and thus A2 (x−y) = 0. The rank of A(x − y) is then bounded by the number of invariant factors of A that are divisible by x2 . Thus, provided that the number of invariant factors of A is small, then failure condition (2) is also ruled out and need not be considered further. Now we move to condition (3). The required probability distribution is obtained by elementary methods and given in the following result.

τ˜u ← Diag(Iρui , τu ); u ˇi ← u ˇi τ˜u τ˜v ← Diag(Iρvi , τv ); vˇi ← vˇi τ˜v if l < i then ρul ← ρul + ru ; ρvl ← ρvl + rv end if ρui ← ρui + ru ; ρvi ← ρvi + rv

Algorithm 3 (l, x) ←Cleanup (l, x, b, f ) 1: if ρul < N or ρvl < N then 2: return (l, x) 3: end if u = N and ρvl+1 = N ) do 4: while f is true h` or (ρl+1 ´−1 i T T ¯l Av˙ l u ¯l b 5: x ← x + v˙ l u 6: Throw out iterates ul and vl 7: l ← l+1 8: end while 9: return (l, x)

Theorem 3.2. Let k be such that, in Algorithm 4, the computed matrix T has dimension N × k where k > 0, and let K := n − rank A. Then the probability that the rank of (x − y)T is at least r for 0 ≤ r ≤ min(k, K) is at least ` ´ `K ´ r−1 N 1 X j q j q `j ´ , (5) 1 − KN q j j=0 q

180

1. the matrix Aˆ = LAR, where LAR is the Wiedemann sparse preconditioner, has rank less than that of A, or

` ´ where Nj , etc. are the q-binomial coefficients given by `N ´ Qq N i := j−1 i=0 (q − q ). In particular, assuming that k > 0, j q

ˆ has more than 2 log q n nontrivial invari2. the matrix A ant factors.

failure condition (3) occurs with probability at most q −KN . Proof. The matrix T ∈ KN ×k is chosen to have full column rank k and thus the rank of (x − y)T is equal to the minimum of k and the rank of x − y. We therefore seek to estimate the probability that the rank of x − y is at least r, for 0 < r < min{k, K}. Since x =BlockLanczos(A, b, v0 ) and v0 is in the column span of A, x is in the column span of A as well. Furthermore, y ∈ Kn×N is selected uniformly at random. Let V := hyi ⊕ hAi, where hyi is the span of the columns of y and hAi is the span of the columns of A. Let us decompose V into the spaces W ⊕ hAi, where W ⊆ Kn×N is chosen so that W ∩ hAi = {0}. Let K 0 := dim W and 0 let w ∈ Kn×K be a matrix whose columns form a basis of 0 W . Then we have y = wα + Aβ, where α ∈ KN ×K has full row rank and β ∈ KN ×n . If we write x = Aγ where γ ∈ KN ×n , then x − y = −wα + A(γ − β). The columns of w are independent of the columns of A, so the rank of x − y is at least the rank of wα, and, since α has full row rank, the rank of wα is equal to K 0 . Therefore, the probability in question is at least the probability that K 0 ≥ r. Consider the images of the columns of y under the canonical projection of Kn onto Kn /hAi ≡ KK . These vectors are randomly, uniformly, and independently-determined elements of KK , which we may view as forming the columns of a uniformly and randomly selected matrix y 0 ∈ KK×N . The problem now reduces to computing the probability that a uniformly randomly selected matrix in KK×N has rank j. This probability is classical: `N ´ `K ´ j q j q 1 0 `j ´ Pr(rank y = j) = KN . (6) q j

The claim is trivial in the case that n ≤ 6, since the statement gives a negative number as a lower bound for a probability. The failure condition that the matrix Aˆ has at least 3 logq n invariant factors divisible by x2 is subsumed by the second condition above and so it is not considered. It follows from Theorem 3.6 in [6] that the probability of the first condition above is at most 1/n. Theorem 3.13 in [6] then states that the probability of the second condition above is at most (r − 2 log q n + 2)q 2−2 logq n + 4q −3 logq n q−1 +(r − 2 log q n + 4)n

ˆ 1− lnc q

.

(7)

We have chosen cˆ in such a way that lncˆq ≥ 3. Observing also that r − 2 log q n + 2 ≤ n and r − 2 log q n + 4 ≤ n, we may conclude that (7) is at most 2

q 4 1 + 2 + . n n n

(8)

The sum of the probabilities of failure conditions (1) and (3) is n6 + q N (rank A−n) . The claim now follows. Remark 1. The bound here can be made more general if it is desired that more than one linearly independent nullspace vector be retrieved by replacing the term q N (rank A−n) by an appropriate term from Theorem 3.2.

4. MEMORY REQUIREMENTS

q

Consider the matrices 9 8 Kju := : u0 AT u0 · · · (AT )j u0 ; , 8 9 Kkv := : v0 Av0 · · · Ak v0 ;

The sum of (6) as j varies from 0 to r−1 yields the probability that the rank of y 0 is less than r. The desired probability is the difference between one and this sum, as claimed.

The reliability of Algorithm 4 is dependent on the structure of the invariant factors of the input matrix A. If A has a number of invariant factors larger than the blocking factor, the above bounds do not provide any useful information. If A has a large number of invariant factors that are divisible by x2 , then the algorithm in fact becomes provably unreliable. It becomes necessary to apply a preconditioner : we select suitable matrices L and R so that rank LAR = rank A and LAR has a small number of nontrivial invariant factors. We then run Algorithm 4 on the input LAR. The required preconditioner is a variation on a sparse preconditioner of Wiedemann [15] that is defined in Section 3.4.1 of [6], where the required properties are established. Putting together Theorems 3.1 and 3.2 with Theorem 3.13 in [6], we obtain the following description of the reliability of Algorithm 4.

(9) (10)

and the block Hankel matrix Hj,k := (Kju )T AKkv , for j, k ≥ 0. Let i ≥ 0 and suppose that, for some ∆ > 0, both of the matrices Hl,i and Hi,l have full rank for all 0 ≤ l ≤ i−∆. We claim that, in this case, ρul = ρvl = N for all 0 ≤ l ≤ i − ∆ at the end of the i-th execution of the while loop in Algorithm 1. If this is true, then the invocation of Algorithm 3 during this iteration removes the iterates with indices less than i−∆ from memory and sets l0 to be at least i − ∆. Let us prove this claim. The statement of Algorithm 1 shows that, for l ≥ 0, there exist ψl,k , ϕl,k ∈ KN ×N , 0 ≤ P P k < l, so ul = lk=0 (AT )k u0 ψl,k and vl = lk=0 Ak v0 ϕl,k at the end of the i-th execution of the loop of Algorithm 1. Let 8 9 I ψ2,1 · · · ψl,1 > > > > > I · · · ψl,2 > > > > > > >, Ψl,l := > (11) .. > > > .. > > > > . . > > : ; I 8 9 I ϕ · · · ϕ 2,1 l,1 > > > > > I · · · ϕl,2 > > > > > > > > Φl,l := > (12) . > >. .. > > . > > . . > > : ; I

Theorem 3.3. The nullspace sampling algorithm 4 will find at least one nontrivial nullspace vector on an input A preconditioned with the Wiedemann sparse preconditioner and using blocking factor at least N := 3 log q n with probability at least 1 − (8 + 2q)/n − 4/n2 − q N (rank A−n) . Proof. We estimate the probability of failure by taking the sum of the probabilities of failure conditions (1) and (3) and two additional conditions:

181

We have that Ul = Klu Ψl,l and Vl = Klv Φl,l . If l < i and uTl AVi has rank N , then the invocation of Algorithm 2 in the i-th iteration of the while loop of Algorithm 1 sets ρul = N . Similarly, if l < i and UiT Avl has rank N , then the same invocation sets ρvl = N . Clearly Ψi−∆,i−∆ and Φi,i are nonT singular and thus rank Ui−∆ AVi = rank Hi−∆,i . Similarly, rank UiT AVi−∆ = rank Hi,i−∆ . Therefore, if Hi+∆,i and Hi,i+∆ each have maximal rank for all i ≥ 0, then l0 ≥ i − ∆ at each iteration of the loop in Algorithm 1 so that at most ∆ + 1 iterates must be kept in memory at any one time. It will be useful to consider properties of sequences. Given ` ´N a sequence c ∈ KK×N and integers i, j ≥ 0, we define the block Hankel matrix associated to this sequence, denoted Hi,j (c), by the following. Write c = {ci }∞ i=0 . Then 8 > > > > > > Hi,j (c) = > > > > > :

c1 c2 .. . ci

c2 c3 .. . ci+1

··· ··· .. . ···

cj cj+1 .. . ci+j−1

9 > > > > > > > . > > > > ;

selected choice of vector u ∈ Kn yields a uniformly selected linearly recurrent sequence {uT Ai v0 }∞ i=1 . Proof. See Lemma 3.13 in [7]. To compute the probability that c ∈ ker ϕi+∆ , we comj pute dimK ker ϕi+∆ . The next results, which generalize and g were motived by Lemma 4.1 in [5], accomplish this. For the following lemma and the subsequent theorem, we refer to the determinant of a submodule M ⊆ KN [x] with N generators. This determinant is obtained by taking a set of generators of M and computing the determinant of the matrix whose columns are those generators. Such a determinant is defined only up to multiplication by a nonzero scalar in K, but that is sufficient for our purposes. Lemma 4.3. Let M ⊆ KN [x] be a submodule and g ∈ ` ´N KN [x]. With ϕg : KN → KN and MM as defined above, the image of ϕg |MM is M(h) where h := det M/ det(M + hgi) ∈ K[x] and M + hgi := {m + gv ∈ KN [x] | m ∈ M and v ∈ K[x]}.

(13)

Proof. See Lemmas 3.16 and 3.17 in [7].

u T v In particular, Hi,j ({uT0 Ai v0 }∞ i=1 ) = (Ki ) AKj . N A polynomial g ∈ K [x] of degree d induces a linear map ` ´N ` ´N ϕg : KK×N → KK given by

(ϕg (c))i :=

d X

ci+j gj

Remark 2. The fact that det M/ det(M + hgi) is a polynomial in K[x] follows from the fact that M ⊆ M + hgi. The probability that a randomly-determined sequence is in the kernel of ϕi+∆ is an easy corollary. g

(14)

j=0

Pd

Theorem 4.4. Let M ⊆ KN [x] be a submodule, g ∈ KN [x], and δ > 0. Let h = det M/ det(M + hgi). The probability that a uniformly and randomly chosen linearly recurrent sequence c ∈ MM is in the kernel of ϕδg is 1/q δ if δ < deg h and 1/q deg h if δ ≥ deg h. Furthermore, if δ ≥ deg h and c ∈ ker ϕδg , then c ∈ ker ϕg as well.

j

where g = j=0 gj x . The set of polynomials that annihilate a given sequence is a submodule of KN [x]. If M ⊆ KN [x] is a submodule, then we denote by MM the set of sequences in (KN )N that are annihilated by all elements of M . Let M ⊆ KN [x] be the module of annihilators of the se(1) (N ) quence {Ai v0 }∞ i=0 . If we split u0 by columns u0 , . . . , u0 (k) T i and consider the resulting sequences {(u0 ) A v0 }∞ i=0 , we obtain N randomly and independently determined sequences in (KN )N , all annihilated by M . Given an integer j > 0, we ´N ` define the linear map ϕjg : KN → Kj by ϕg followed by projection onto the first j entries of the image. We may identify g ∈ KN [x] of degree at most i with a vector in KN i in the usual way. The following proposition relates ϕjg (c) with Hi+∆,i (c).

Proof. See Theorem 3.18 in [7]. From Theorem 4.4, we see that the candidate nullspace vectors of Hi+∆,i (c) fall into two categories. To a given vector g, we may assign a polynomial hg ∈ K[x] by the M . If deg hg < i+∆, then g annihilates formula hg = detdet M +hgi c if and only if c ∈ ker ϕi+∆ . Otherwise, the probability g −(i+∆) that c ∈ ker ϕi+∆ is q . Now we wish to bound the g probability that any nonzero vector g is in the nullspace of Hi+∆,i , for any iteration i before termination. Let us suppose that the block Lanczos algorithm terminates at iteration m and fix some ∆ > 0, which shall be the number of “excess” iterates required by the lookahead process. We consider separately the following two events:

` ´N Proposition 4.1. Let c ∈ KN be linearly recurrent. Let i ≥ 0 and ∆ ≥ 0. Then, identifying a vector g ∈ KN i with a polynomial in KN [x], g ∈ ker Hi+∆,i (c) if and only if c ∈ ker ϕi+∆ . g

1. There exists a polynomial g ∈ KN [x] of degree at most (1) (N ) m − ∆ that annihilates the sequences u0 , . . . , u0 simultaneously.

Proof. See Proposition 3.14 in [7]. To determine the probability that c ∈ ker ϕi+∆ , we must j first establish that a uniform random selection of u ∈ Kn [x], (k) as is the case with u0 , yields a uniform random choice T i of sequence {u A v0 }∞ i=0 among sequences in MM . The following lemma generalizes of Lemma 3 in [5].

2. For some i = 0, . . . , m − 1, there exists a polynomial g ∈ KN [x] such that deg hg ≥ i+∆ with ck ∈ ker ϕi+∆ . g The probability of the first event has been analyzed in [14]. It is obtained by observing the differences between the column degrees of the module of annihilators of the sequence {Ai v0 }∞ i=0 . The probability that the column degrees differ by more than ∆ diminishes exponentially with ∆. We estimate the probability of the second event by assuming that all polynomials g ∈ KN [x] have the property that

Lemma 4.2. Let v0 ∈ Kn×N . Let M ⊆ KN [x] be the n module of annihilators of {Ai v0 }∞ i=0 . Let u ∈ K be chosen uniformly at random and let s ∈ MM . Then the probability that s = {uT Ai v0 }∞ i=1 is 1/|MM |. In particular, a uniformly

182

6. FUTURE WORK

deg hg ≥ i + ∆. While this gives a rough upper bound for the probability, it is sufficient for our purposes. We then compute the probability of the event by adding the probabilities given in Theorem 4.4 for all possible g ∈ KN [x] of degree at most i and for all 0 ≤ i < m. The iteration of termination m is also randomly determined, but its probability distribution is given by the analysis in [14] by the same method as described above. With this, we obtain the following bound:

Some potential enhancements to the work presented here are apparent: • Eliminate the need for the preconditioner mentioned in Section 3 in certain cases. When the number of invariant factors of the matrix divisible by x2 is sufficiently small, the preconditioner may be dropped by using different blocking factors for the left and right blocks. That is, we choose k, l ≥ 0, not necessarily equal, and let ui ∈ Kn×k and vi ∈ Kn×l , i ≥ 0. The techniques used in the analysis still apply in this context, but the algorithm itself must be altered.

Theorem 4.5. Suppose q 2 /n + q 3 /n2 < 1. Let N > 0 be chosen so that N − φ ≥ log q n, where φ is the number of nontrivial invariant factors of A. Then the probability that either Hi,i+∆ or Hi+∆,i is rank-deficient for some index i ≥ 0 is at most „l m « 6 q 8 n 8q −∆/2 + + + q −N (∆−1) . (15) n q−1 N q−1

• After Algorithm 1 has been modified to reduce the need for a preconditioner, use it as a basis for reliable system solving. The reliability of such an algorithm to solve linear systems can be deduced using the same methods as in Section 3, but analyzing its memory requirements requires a significant refinement of the techniques in Section 4.

It follows that (15) also bounds the probability that Algorithm 1 requires at least ∆ iterates of lookahead.

• Investigate the block Wiedemann algorithm introduced in [3] that was proved reliable over small finite fields in [14], in order to introduce a reliable “early termination” mechanism that will improve the performance of this algorithm when its input has low rank. It seems likely that the results in this paper will be directly applicable, but a bit more work does need to be done.

Proof. See Theorem 3.22 in [7]. Putting together Theorem 3.11 in [6] and Theorem 4.5, we obtain the following as an immediate corollary. Theorem 4.6. Suppose q 2 /n + q 3 /n2 < 1 and let N ≥ 3 logq n. Suppose also that the input matrix A has been preconditioned with the Wiedemann sparse preconditioner. Then the expected memory requirements of Algorithm 1 with blocking factor N are O(nN ).

5.

7. ACKNOWLEDGEMENTS The authors would like to thank Mark Giesbrecht, Arne Storjohann, Justin Wan, and Gilles Villard for their valuable discussions. The computations mentioned herein were performed at the University of Waterloo.

IMPLEMENTATION

The block Lanczos algorithm described in Section 2 has been implemented in LinBox [4] and a version is available for download at http://www.linalg.org/ The block Lanczos heuristic of Montgomery [10] has also been implemented in LinBox using the same primitives. We compare the performances of Montgomery’s heuristic and the new lookaheadbased algorithm when sampling nullspace vectors from uniformly random 10% nonzero matrices of various sizes over GF (2). The blocking factor was set to 128 and no preconditioner was used. Montgomery’s heuristic consistently terminates in approximately 10% less time than the lookaheadbased approach. This difference is entirely due to the fact that the former is a symmetric variant in which only one iterate must be updated in each iteration. See [7], Chapter 4, for a more detailed exposition of the performance differences. To compare reliability, the heuristic of Montgomery and the algorithm described here were run ten times each on the same randomly-generated 8000 × 8000 matrix with 10% nonzero entries, using a blocking factor of 32. We applied a modification of Algorithm 4 in which the vectors of x − y that were either zero or not in the nullspace of A were simply thrown out. The lookahead-based block Lanczos algorithm found 32 vectors on all invocations, while Montgomery’s heuristic found an average of 15 for each invocation. The unreliability of Montgomery’s heuristic here is due to its discovering many vectors that are in the nullspace of AT A but not of A. Here it is also apparent that the reliability bounds given in Section 3 are somewhat weak.

8. REFERENCES [1] R. P. Brent, S. Gao, and A. G. B. Lauder, Random Krylov spaces over finite fields, SIAM J. Discret. Math. 16 (2003), no. 2, 276–287. [2] D. Coppersmith, Solving linear equations over GF(2): block Lanczos algorithm, Linear Algebra and its Applications 192 (1993), 33–60. [3] , Solving homogeneous linear equations over GF(2) via block Wiedemann algorithm, Math. Comput. 62 (1994), no. 205, 333–350. [4] J.-G. Dumas, T. Gautier, M. Giesbrecht, P. Giorgi, B. Hovinen, E. Kaltofen, B. D. Saunders, W. J. Turner, and G. Villard, LinBox: A generic library for exact linear algebra, Proc. First Internat. Congress Math. Software ICMS 2002, Beijing, China, pp. 40–50. [5] W. Eberly, Early termination over small fields, Proc. of the 2003 International Symposium on Symbolic and Algebraic Computation (Rafael Sendra, ed.), ACM Press, 2003, pp. 80–87. Full version (with proofs) available at http://www.cpsc.ucalgary.ca/~eberly/publications , Reliable Krylov-based algorithms for matrix [6] null space and rank, Proc. of the 2004 International Symposium on Symbolic and Algebraic Computation (Jaime Guitierrez, ed.), ACM Press, 2004, pp. 127–134. Full version (with proofs) available at http://www.cpsc.ucalgary.ca/~eberly/publications

183

[12] B. N. Parlett, D. R. Taylor, and Z. A. Liu, A look-ahead Lanczos algorithm for unsymmetric matrices, Math. Comp. 44 (1985), 105 – 124. [13] A. Storjohann, Algorithms for matrix canonical forms, Ph.D. thesis, Swiss Federal Institute of Technology, 2000. [14] G. Villard, Further analysis of Coppersmith’s block Wiedemann algorithm for the solution of sparse linear systems (extended abstract), Proc. of the 1997 International Symposium on Symbolic and Algebraic Computation (Wolfgang W. K¨ uchlin, ed.), ACM Press, 1997, pp. 32–39. [15] D. H. Wiedemann, Solving sparse linear equations over finite fields, IEEE Trans. Inf. Theor. 32 (1986), no. 1, 54–62.

[7] B. Hovinen, Blocked Lanczos-style algorithms over small finite fields, Master’s thesis, University of Waterloo, 2004. Available at http://www.math.utoronto.ca/hovinen/ [8] R. Lambert, Computational aspects of discrete logarithms, Ph.D. thesis, University of Waterloo, 1996. [9] C. Lanczos, An iteration method for the solution of the eigenvalue problem of linear differential and integral operators, J. Res. Nat. Bur. Standards 45 (1950), 255–282. [10] P. Montgomery, A block Lanczos algorithm for finding dependencies over GF(2), Lecture Notes in Computer Science, vol. 921, Springer-Verlag, 1995, pp. 106–120. [11] A. M. Odlyzko, Discrete logarithms in finite fields and their cryptographic significance, Proc. of the EUROCRYPT 84 Workshop on Advances in Cryptology: Theory and Application of Cryptographic Techniques, Springer-Verlag New York, Inc., 1985, pp. 224–314.

184

Schur Partition for Symmetric Ternary Forms and Readable Proof to Inequalities ∗

Fangjian Huang

Shengli Chen

Chengdu Institute of Computer Application Chinese Academy of Science Chengdu, Sichuan, China

Chengdu Institute of Computer Application Chinese Academy of Science Chengdu, Sichuan, China

[email protected]

[email protected]

ABSTRACT

[15] found the necessary and sufficient conditions for ternary forms of degree 4 and 5. There is a common point in these results. The forms are all under degree 6. Timofte in [14] found that to prove whether a n-ary d(d > 2)-form f is psd suffices to prove f is psd for n-ary elements in Rn + with at most bd/2c distinct non-zero components, which may have multiplicities. This is a nice result, but it becomes not so useful when the degree is larger than 5. From 1993, Shengli Chen([1]P.486) has been working on the ternary forms. We’ll show in this paper that Schur partition can deal with the ternary forms in arbitrary degree which become zero when all the variables are substituted by 1. In Section 3.3, we give an algorithm for computing the Schur partition. It’s easy to see that the algorithm terminates in polynomial time. In Section 4, we deduce the necessary and sufficient condition for positivity of ternary forms in degree 3 and 4 as corollaries of our main result. We also test our algorithm on several ternary forms of degree larger than 5.

In this paper, we give a way to partition the ternary symmetric forms. Based on this method, we get a sufficient condition for ternary form to be positive semi-definite. At the end of the paper, we will show the application of the partition. Categories and Subject Descriptors: G.0 [GENERAL]. General Terms: ALGORITHMS. Keywords: Schur partition, ternary symmetric form, positive semi-definite, symmetric inequality.

1.

INTRODUCTION

In 1900, David Hilbert published his famous 23 problems. The 17th problem is about partitioning a form into some sum of squares. A related problem is to indicate whether a form is positive semidefinite(psd) on Rn + . Many people have been working on this problem and got a lot of interesting results. In 1950, Taski,A. published his famous result([13]): all the propositions in elementary algebra and geometry are solved by mechanizable algorithm. After 25 years, Collins discovered cylindrical algebraic decomposition(CAD) and used it to deal with the quantifier elimination problems. Nevertheless, the computational complexn ities of these two methods are O(22 ), in which n is the number of the variables. In the 1990s, based on dimensiondecreasing algorithm and partial cylindrical algebraic decomposition(PCAD), Yang L.([16],[17],[18],[19]) wrote an efficient program named Bottema to prove rational functions and irrational functions systems and discover inequalities automatically. Using this program, it costs only three seconds to prove 100 theorems in [2]. Another remarkable thing is to decrease the complexity of the algorithm. An involuntary way is to think about some specific problems. Choi et al. in [12] published a necessary and sufficient condition for the real symmetric n-ary 3-forms is psd. Harris in ∗ This work is 2004CB318003.

supported

in

part

by

2. NOTATION First, we shall give an announcement that the polynomials mentioned in this paper are all in real closed field, such as the real number field. Then we will specify some definitions and notations mentioned in this paper. Definition 2.1. A function, f (x, y, z) of three variables is homogeneous of degree n if f (tx, ty, tz) ≡ tn f (x, y, z). It’s very easy to see that a polynomial is homogeneous of degree n if and only if each term has degree n.(The degree of a term is the sum of the exponents of the variables appearing in that term.) Definition 2.2. A homogeneous polynomial in degree k is called k-form.

NKBRPC-

Definition 2.3. A function of three variables is symmetric if f (x, y, z) = f (y, z, x) = f (z, x, y) = f (x, z, y) = f (z, y, x) = f (y, x, z).

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. ISSAC’05, July 24–27, 2005, Beijing, China. Copyright 2005 ACM 1-59593-095-7/05/0007 ...$5.00.

Notation 2.4. We will use the sum notation X f (x, y, z) = f (x, y, z) + f (y, z, x) + f (z, x, y) without any expression above or below the sum symbol to denote the cyclic sum of the specified expression. And we

185

3.2 Relevant Theorems

will use the product notation Y f (x, y, z) = f (x, y, z) · f (y, z, x) · f (z, x, y)

Theorem 3.2.1. Assuming that

without any expression above or below the product symbol to represent the cyclic product of the specified expression. And we have

(n)

(n−3)

(n)

Proposition 2.6. Given a symmetric polynomial f (x, y, z), there is an unique polynomial g(α1 , α2 , α3 ), s.t.

f

f (x, y, z) = g(σ1 , σ3 , σ3 )

That is to say x can divide the left hand of (3.2.4). Using the same method, we have that y, z can divide the left hand of (3.2.4) too. So σ3 can divide the left hand of (3.2.4). From (3.2.1), we can get that (3.2.2) holds when j = b n+2 c. 2 c and n Using the method, we can get that when j = b n+2 2 is even, (3.2.2) holds. ¤

h(i) = 0,

SCHUR PARTITION AND THE RELEVANT THEOREMS

Remark 3.2.2. We introduce some lexicographic-type ordering on εn = {σ1n−2j−3i σ2j σ3i }2j+3i6n

Schur Partition

Given a ternary symmetric n-form Pn (x, y, z), we can partition it into such form that

i=0

σ1n−2j−3i σ2j σ3i  σ1n−2j

(n)

λi,j+1 σ1n−3i−2j σ2j σ3i

⇐⇒i < i0

(3.1.1)

j=0

≡ =

(n) f0,j

f

(n)

X

(3.1.2)

(n > 3)

n−3

x

(y + z)(x − y)(x − z) n ≡ (3 6 j 6 b c) 2 Y = σ1n−2j σ2j−3 (y − z)2

(n)

(3.1.3)

(n)

=

i = i0 , j < j 0 .

of (3.1.4)

(n) fi,j

(n)

 the one of

(n) fi0 ,j 0

in the lexicographic-type ordering.

Theorem 3.2.3. If x > 0, y > 0, z > 0we have (n)

fi,j (x, y, z) > 0

(3.2.5)

Proof: From (3.1.2)-(3.1.6), we will get the result from the following: ∀k > 0 X k x (x − y)(x − z) > 0 (3.2.6) X k (yz) (x − y)(x − z) > 0 (3.2.7) X k x (y + z)(x − y)(x − z) > 0 (3.2.8) X (yz)k (y + z)(x − y)(x − z) > 0 (3.2.9)

(5 6 3i + 2 6 n, i ∈ N)

(n−3i) i f0,j σ3

(n)

(n)

≡ f n+2 (x, y, z) (n > 4) 0,b 2 c X n−2 = (yz)b 2 c (y + z)nmod2 (x − y)(x − z) (3.1.5)

(n)

or

That is to say fi,j  fi0 ,j 0 is equivalent to the leading term

(n) f0,j (x, y, z)

fi,j ≡ fi,j (x, y, z)

0

fi,j  fi0 ,j 0 ⇐⇒ (i, j)
(n)

0,b n+2 c 2

0

σ2j σ3i

From the above theorem, the leading term of fi,j is σ1n−2j−3i+2 σ2j−1 σ3i in the lexicographic-type ordering we just (n) introduced. And the leading term of fi,j is n−2i−j+1 i+j−1 i x y z in the usual lexicographic order. Introduce some lexicographic-type ordering on Fn

(n)

(n) f0,2 (x, y, z)

−3i0

(n)

(n > 2) f0,1 ≡ f0,1 (x, y, z) X n−2 = x (x − y)(x − z) (n) f0,2

0

⇐⇒(i, j)
Now we give out some more notations: (n)

1

1

i=m

Pn (x, y, z) =

− σ1 σ22

= (yz) 2 (n−3) (y + z)yz − (y + z)(yz) 2 (n−1) =0 (3.2.4)

Notation 2.7. If m > k,

X

(3.2.2)

1 (n−1)

(n)

0,b n+2 c 2

This is a well-known theory in linear algebra.

n−3i c 2

ai,j = 0, (1 6 i < j)

We can substitute the right hand of (3.2.3) for its left hand into (3.1.4), then we have (3.2.2) holds when j = 3, 4, · · · , c. First, b n2 c. Now let’s prove (3.2.2) holds when j = b n+2 2 assuming n is odd, we can get from (3.1.5) that when x = 0

We call them ternary basic symmetric polynomials.

X

(3.2.1)

Proof: Let’s expand (3.1.2) and (3.1.3), we get that (3.2.2) holds when j = 1 or 2. And we know Y (y −z)2 = σ12 σ22 −4σ23 −(4σ13 −18σ1 σ2 )σ3 −27σ32 . (3.2.3)

σ3 = xyz.

bn/3c b

(n)

aj,j = 1;

σ1 = x + y + z, σ2 = xy + yz + zx,

3.1

σ1n−3 + · · · )σ3 + · · ·

Then we have

Notation 2.5. Additionally, we have the following three notations:

3.

(n)

+ (a1,j

bxc = max{k|k ∈ N, k 6 x} x ∈ R

k X

(n)

f0,j = (a1,j σ1n + a2,j σ1n−2 σ2 + · · · )

(3.1.6) (n) fi,j .

Given some n, let Fn denote the set of the In this partition, we mainly use the inequalities that look like the Schur inequality(see[10]). So we name it Schur Partition.

186

(n)

(3.2.6) is a part of the famous Schur inequality. The full Schur inequality is that ∀k ∈ R, (3.2.6) holds. If x 6= 0, y 6= 0, z 6= 0, we have X X −k (yz)k (x − y)(x − z) = (xyz)k x (x − y)(x − z) (3.2.10) From the full Schur inequality, we get that (3.2.7) holds(the left hand of (3.2.7) vanishes if and only if x = y = z). If there is a ’0’ in x, y and z,we assume without loss of generality that z = 0. Then we have X (yz)k (x − y)(x − z) = (xy)k xy > 0.

fi,j in (3.2.12), x1 , x2 , · · · , xkn in the lexicographic-type (n)

ordering of fi,j introduced in Remark 3.2.2 and the coefficients

in (3.1.1) µ1 , µ2 , · · · , µkn , µkn +1 . Substitute (n)

(n)

the σ1 , σ2 , σ3 form of fi,j (see (3.1.6)and (3.2.1)) for fi,j in (3.2.12) and call the result temp. Then we match the coefficient of σ1r1 σ2r2 σ3r3 in temp and in (3.1.1). From the uniqueness of the σ1 , σ2 , σ3 form of Pn (x, y, z), we can get kn + 1 equations. And from condition (3.2.11), we could omit the relation between µkn +1 and x1 , x2 , · · · , xkn . Then we can see the problem is equivalent to the existence and the uniqueness of the solution of polynomial system about x1 , x2 , · · · , xkn

If there are two or more ’0’ in x, y and z, the above is zero. So (3.2.7) holds. In order to prove (3.2.8) and (3.2.9), we assume without loss of generality x > y > z > 0. Since that, we have X k x (y + z)(x − y)(x − z)

A · (x1 , x2 , · · · , xkn )T = (µ1 , µ2 , · · · , µkn )T In which A is a unit lower triangular matrix(whose every entry on principle diagonal is 1) since Theorem 3.2.1 and Remark 3.2.2. Obviously, the answer is true. So we just proved the theorem. ¤

> y k (y + z)(x − y)(x − z) + y k (z + x)(y − z)(y − x)

3.3 Partition Algorithm

= 2zy k (x − y)2 > 0. and

(n) λi,j

Algorithm 3.3.1. Schp:(exp,var)→ X (yz)k (y + z)(x − y)(x − z)

P i,j

(n) (n)

αi,j fi,j .

Given a form exp which satisfies (3.2.11) and its variables (n) list var, this algorithm will return the sum of the fi,j sumf which is equivalent to exp.

> (zx)k (z + x)(y − z)(y − x) + (xz)k (x + z)(z − x)(z − y)

S 1: Check whether exp vanishes at (1, 1, 1). If not, return ”Error, the expression doesn’t match (3.2.11)” and end the algorithm.

= (zx)k (z + x)(y − z)2 > 0 Since above, we get the correctness of (3.2.8) and (3.2.9).¤

S 2: Assign the value of exp to temp, 0 to sumf, and the degree of exp in var to n;

(n)

Remark 3.2.4. From the above proof, every fi,j vanishes at (1, 1, 1).

S 3: repeat the following steps until temp = 0:

Now we need a condition (3.2.11)

S 3.1: Assign to temp1 the highest degree of temp in var1 , and assign its coefficient to temp11 ;

Theorem 3.2.5. Let Gn denote the R-vector space consisting of all symmetric ternary n-forms over R with the restriction (3.2.11).K = b n3 c. Fn is a basis of Gn . That (n) means there is an unique {αi,j } such that

S 3.2: Assign to temp2 the highest degree of temp11 in var2 , and assign its coefficient to temp21 ;

Pn (1, 1, 1) = 0

(2)

S 3.3: Assign to temp3 the highest degree of temp21 in var3 , and assign its coefficient to temp31 ;

(2)

S 3.4: Assign the value of temp3 to m, the value of temp2 − temp3 + 1 to l;

P2 = α0,1 f0,1 Pn =

K−1 X

b n+2−3i c 2

i=0

j=1

X

b n−3K 2

+

X

(n)

(n)

(n)

S 3.5: Subtract from temp the value of temp31 ∗ fm,l (

(3.2.12)

αi,j · fi,j

(n)

the value of fm,l is defined in (3.1.2)–(3.1.6), and just substitute var1 for x, var2 for y and var3 for z). And assign the difference back to temp;

c (n)

(n)

αK,j · fK,j

(n > 3)

(n)

S 3.6: sumf := sumf + temp31 ∗ fm,l .

j=1

Proof: n = 2.From (3.1.1) and (3.2.11), we have P2 =

(2) λ0,1 σ12

=

(2) λ0,1 σ12

=

(2) λ0,1 (2)

+

S 4: Return the value of sumf and end the algorithm.

(2) λ0,2 σ2

In this algorithm, we need to find the highest degree of some variables in the expression. If the variable(e.g. var2 ) doesn’t appear in this expression(e.g. exp), we say that the highest degree of var2 in exp is 0 and the leading coefficient of var2 in exp is exp itself. And from the proof of Theorem 3.2.5, we know that this algorithm can deal with forms in arbitrary degree. Now let’s prove the termination and the correctness of the algorithm.

(2) 3λ0,1 σ2

− X (x − y)(x − z) (2)

= λ0,1 f0,1

(n)

So (3.2.12) holds. Let kn denote the number of αi,j in (n)

(3.2.12). When n > 3, we name αi,j , the coefficients of

187

First, the termination of the algorithm is equivalent to whether the polynomial exp is equivalent to some linear combination (n) of fi,j . From Theorem 3.2.5, we get the answer ’Yes’. Second, in order to explain the correctness of the algorithm, let’s review the Property 2.6 in Section 2 and the proof of Theorem 3.2.1, 3.2.5. Then we can get the answer is ’true’.

in which α0,1 = λ0,1 , α0,2 = 5λ0,1 + λ0,2 , α0,3 = 16λ0,1 + 4λ0,2 + λ0,3 . From (3.1.2)-(3.1.4), we know that P 2 (4) f0,1 = x (x − y)(x − z) > 0, P (4) f0,2 = x(y + z)(x − y)(x − z) > 0, P (4) f0,3 = yz(x − y)(x − z) > 0. So, if (4.1.4) holds, we have

4.

APPLICATIONS

4.1

P4 (x, 1, 1) =(x − 1)2 (α0,1 x2 + 2α0,2 x + α0,3 ) > 0.

Corollary

From now on, let’s assume x, y, z are all non-negative real numbers.

Then we get

Corollary 4.1.1. If one polynomial Pn is in the form (n) of (3.2.12) and the coefficients αi,j are all non-negative, we can say that Pn > 0.

Since that, we have

Proof: From Theorem 3.2.3, we know that all Since

(n) αi,j

(n) fi,j

√ α0,1 > 0, α0,3 > 0, α0,2 > − α0,1 α0,3 .

P4 (x, y, z) q √ (4) (4) (4) >2 α0,1 α0,3 f0,1 f0,3 − α0,1 α0,3 f0,2  q  √ (4) (4) (4) = α0,1 α0,3 2 f0,1 f0,3 − f0,2 .

> 0.

> 0, we get the result immediately.

¤

Remark 4.1.2. This is a sufficient condition for ternary forms are psd. For example, F4 (x, y, z) = (σ12 − 3σ2 )2 > 0, (4) (4) (4) but we can only get F4 (x, y, z) = f0,1 − f0,2 + f0,3 in which

And we have  q  q  (4) (4) (4) (4) (4) (4) 2 f0,1 f0,3 − f0,2 2 f0,1 f0,3 + f0,2

(4)

the coefficient of f0,2 is −1.

2  (4) (4) (4) =4f0,1 f0,3 − f0,2

Corollary 4.1.3. The ternary symmetric inequality P3 (x, y, z) = λ0,1 σ13 + λ0,2 σ1 σ2 + λ1,1 σ3 > 0


=3σ12 σ22 (σ12 − 4σ2 ) − 2σ1 σ3 (2σ12 − 9σ3 ) − 27σ32 Y (y − z)2 > 0. (4.1.5) =3σ12

(4.1.1)

holds if and only if P3 (1, 0, 0) = λ0,1 > 0, P3 (1, 1, 0) = 2(4λ0,1 + λ0,2 ) > 0, P3 (1, 1, 1) = 27λ0,1 + 9λ0,2 + λ1,1 = λ > 0.

As a consequence, we get P4 > 0. That’s our goal.

Corollary 4.1.5. A necessary and sufficient condition for the polynomial of degree 6, which satisfies P6 (1, 1, 1) = 0 and like the form

Proof: It is obvious to see that (4.1.2) is a necessary condition for (4.1.1). What we need to do is only to prove the sufficiency. Let P30 (x, y, z) = P3 (x, y, z) − λσ3 . Since P30 (1, 1, 1) = P3 (1, 1, 1) − λ = 0, we use algorithm Schp to get: (3)

P6 (x, y, z) = λ0,3 σ12 σ22 + λ0,4 σ23 + σ3 (λ1,1 σ13 + λ1,2 σ1 σ2 ) + λ2,1 σ32 > 0

(3)

P30 (x, y, z) = α0,1 f0,1 + α0,2 f0,2 (3)

P6 (x, 1, 0) > 0, P6 (x, 1, 1) > 0

If (4.1.2) holds, P30 (x, y, z) > 0 since Corollary 4.1.1. And x > 0, y > 0, z > 0, λ > 0, P3 (x, y, z) = P30 (x, y, z) + λxyz > 0.That is to say (4.1.1) holds. ¤

(6)

(4.1.3)

(4.1.4)

Proof: It is obvious to see that (4.1.4) is the necessary condition for (4.1.3). We only need to prove the sufficiency. Use algorithm Schp to get P4 (x, y, z) =

+

(4) α0,2 f0,2

+

(6)

(6)

(6)

in which α0,3 = λ0,3 , α0,4 = 4λ0,3 + λ0,4 , α1,1 = 4λ0,3 + λ1,1 , α1,2 = 14λ0,3 +4λ1,1 +4λ0,4 +λ1,2 . From (3.1.2)-(3.1.6) and Theorem 3.2.3, we have Q (6) f0,3 = (y − z)2 > 0 P 2 2 (6) f0,4 = y z (x − y)(x − z) > 0 P 2 (6) f1,1 = x yz(x − y)(x − z) > 0 P (6) f1,2 = xyz(y + z)(x − y)(x − z) > 0

is

(4) α0,1 f0,1

(4.1.7)

P6 = α0,3 f0,3 + α0,4 f0,4 + α1,1 f1,1 + α1,2 f1,2

P4 (x, y, z)

(x > 0)

(x > 0)

Proof: Use algorithm Schp, we can get

Corollary 4.1.4. If P4 (1, 1, 1) = 0, a necessary and sufficient condition for quartic inequality

P4 (x, 1, 1) > 0

(4.1.6)

is

(3)

= λ0,1 f0,1 + (4λ0,1 + λ0,2 )f0,2

=λ0,1 σ14 + λ0,2 σ12 σ2 + λ0,3 σ22 + λ1,1 σ1 σ3 > 0

¤

(4.1.2)

So, if (4.1.7) holds, from P6 (1, 1, 1) = 0 we can get P6 (x, 1, 0) = x2 (α0,3 (x − 1)2 + α0,4 x) > 0, P6 (x, 1, 1) = (x − 1)2 (α1,1 x2 + 2α1,2 x + α0,4 ) > 0

(4) α0,3 f0,3 ,

188

Then we have α1,1

The left inequality of (4.2.4) is equivalent to α0,3 > 0, α0,4 > 0, √ > 0, α0,4 > 0, α1,2 > − α1,1 α0,4 .

L2,1 (x, y, z) (5)

and

(5)

(4.2.5)

The right one of (4.2.4) is equivalent to P6 (x, y, z) (6)

(6)

(6)

L2,2 (x, y, z)

(6)

=α0,3 f0,3 + α0,4 f0,4 + α1,1 f1,1 + α1,2 f1,2  q  √ (6) (6) (6) (6) >α0,3 f0,3 + α0,4 α1,1 2 f0,4 f1,1 − f1,2

(5)

 2 (6) (6) (6) =4f0,4 f1,1 − f1,2   (3) (6) (9) =σ3 f0,2 f0,3 + 3f0,4 > 0

(5)

(5)

=(1 − 2k0 )f0,2 + (1 − k0 )f0,3 + (4 − 6k0 )f1,1 > 0. (4.2.6) Because we have y > 0 and limz&0 L2,1 (1, 1, z) = 2(k − 1) > 0, L2,3 (y) = limz&0 L2,2 (1, y, z)

Because of  q   q (6) (6) (6) (6) (6) (6) 2 f0,4 f1,1 + f1,2 2 f0,4 f1,1 − f1,2

=y(1 − 2k0 + k0 y + k0 y 2 + y 3 − 2k0 y 3 ) > 0, limy&0 (4.1.8)

L2,3 (y) = 1 − 2k0 > 0, y

we get k > 1, k 0 6 12 . Substitute the right hand of these two inequalities for the left hand into (4.2.5) and (4.2.6). We get

we get P6 (x, y, z) > 0. So we just prove the sufficiency. And it’s easy to see that (4.1.7) is a necessary condition for (4.1.6). ¤

L2,1 (x, y, z) > 0, L2,2 (x, y, z) > 0. Now we can see the answer is k > 1, 0 6 k0 6 12 .

Remark 4.1.6. Corollary 4.1.3 is a particular case (three variables) of Theorem (3.7) in [12].Corollary 4.1.4 is a particular case (three variables) of Theorem (2.3) in [15].

4.2

(5)

=(2k − 1)f0,2 + (k − 1)f0,3 + (6k − 4)f1,1 > 0.

Expamle 4.2.3 (Baoqian Liu). If k > 1, x > 0, y > 0, z > 0, we have X xyz 3 6 (4.2.7) y 3 + z 3 + kxyz 2+k

Example

Expamle 4.2.1 (Gangsong Leng). Prove p √ √ 3 3 abc 6 a2 b2 c2 − (a − b)2 (b − c)2 (c − a)2 S6 (4.2.1) 4 in which a, b, c are the lengths of three sides of one triangle and S is the area of the triangle.

Proof:(4.2.7) is equivalent to (9)

(9)

L3 = 3f0,4 + 3f0,5

(9)

(9)

+ 2(5k + 1)f1,4 + 2(k2 − k + 3)f2,1

Proof: First let’s rationalize (4.2.1), and do some substitution a = y + z, b = x + z, c = x + y into the result. Then we can get Y Y 27 (x + y)2 (x + y)2  Y − (x − y)2 − 4096σ13 σ33 > 0 (4.2.2)

(9)

+ (2k2 + 2k − 1)f2,2 (9)

It’s obvious to see that the coefficient of every fi,j is nonnegative if k > 1. From Corollary 4.1.1, we get (4.2.7) holds. ¤ Expamle 4.2.4. If k > 4, x > 0, y > 0, z > 0, we have X 3 xyz 6 (4.2.8) (y + z)3 + kxyz 8+k

which is equivalent to (4.2.1). Assign L1 (x, y, z) to the left side of (4.2.2). It’s easy to find L1 (1, 1, 1) = 0. So,we can use algorithm Schp to partition L1 . (12)

Proof: After do some simple computation, we get

(12)

L1 (x, y, z) = 108f0,6 + 432f0,7 (12)

(12)

+

(12) 6372f2,3

(12) 20304f2,4

+

(12) 16208f3,1

(12)

(12)

+ 108f1,3 + 324f1,4 + 2160f1,5 + 432f2,1 +

(12) 2160f2,2

(9)

(9)

(9)

+ 2(k − 1)f1,1 + (2k + 1)f1,2 + (2k + 1)f1,3

+

+

(12) 13856f3,2 .

(9)

(9)

(4.2.8) ⇐⇒ 3f0,4 + 12f0,5 (9)

(9)

(9)

+ (−8 + 2k)f1,1 + (−20 + 8k)f1,2 + (1 + 14k)f1,3 (4.2.3)

(9)

(9)

+ (56 + 34k)f1,4 + (−48 + 28k + 2k2 )f2,1

Then we can get L1 (x, y, z) > 0 from Corollary 4.1.1. That is to say (4.2.1) holds. ¤

(9)

+ (8 + 8k + 5k2 )f2,2 Because k > 4, we get

Expamle 4.2.2. Find out the range of the non-negative real number k, k 0 to satisfy if x > 0, y > 0, z > 0 P P X x (y − z)2 (y − z)2 3 P P 6 − 6 4k yz y+z 2 4k0 yz

2k − 8 > 0, 8k − 20 > 0, 1 + 14k > 0, 56 + 34k > 0, 2k2 + 28k − 48 > 0, 5k2 + 8k + 8 > 0

(4.2.4)

Since that, (4.2.8) holds.

189

¤

Expamle 4.2.5 ([1]BW27(c)). If k > 3 + y > 0, z > 0, we have X X x 9 x > x2 + kyz 1+k

√

√ √ Proof:Let’s do some substitution s − a = x, s − b = √ y, s − c = z into (4.2.12). Then we have

7, x > 0,

(4.2.12) ⇐⇒L7 (x, y, z)

(4.2.9)

(8)

(8)

(8)

(8)

(8)

=f0,3 − 2f0,4 + 2f0,5 + 2f1,1 − 2f1,2 (8)

Proof:We can get that(4.2.9) is equivalent to

(8)

− 2f1,3 + 4f2,1 > 0

(4.2.13)

Because

L5 (x, y, z) (6)

(6)

(6)

L7 (x, y, z)     (8) (8) (8) (8) (8) = f0,3 − 2f0,4 + 2 f0,5 + f2,1 − f1,3   (8) (8) (8) + 2 f1,1 + f2,1 − f1,2   (8) (8) (2) (6) f0,3 − 2f0,4 = f0,1 + σ2 f0,3 > 0,

= (k2 + k)f0,3 + (4k2 − 5k)f0,4 + k(k2 − 6k + 2)f1,1 (6)

+ (2k3 − 2k2 − 3k + 1)f1,2 > 0 √ Because k > 3 + 7, we have

(4.2.10)

k2 + k > 0, 4k2 − 5k > 0, k(k2 − 6k + 2) > 0, 3

(8)

(8)

(8)

(8)

(8)

(8)

f0,5 + f2,1 − f1,3 X 2 2 = y z (x − y)2 (x − z)2 > 0,

2

2k − 2k − 3k + 1 =(2k + 10)(k2 − 6k + 2) + 53k − 19 > 0

f1,1 + f2,1 − f1,2 X =σ3 x(x − y)2 (x − z)2 > 0,

Then from Corollary 4.1.1, we get L5 (x, y, z) > 0. Since that, (4.2.9) holds. ¤

we know that the right hand of (4.2.13) holds. So (4.2.12) holds. ¤

Expamle 4.2.6. Assume that x > 0, y > 0, z > 0, then we have 0 6 k 6 3, k ∈ N X  y + z − x k X  y + z − x k+1 6 (4.2.11) x x

Expamle 4.2.8. a, b, c are the lengths of the three sides of a triangle. s is the semiperimeter. R and r is the radii of the circumcircle and the incircle of the triangle. And t is an arbitrary real number. Then we have

Proof:

ts2 6 (t + 1)2 R2 − 2(t2 − 5t + 2)Rr + (4 − t)r2

(3)

1. k = 0: (4.2.11) ⇐⇒ L6,1 = f0,2 > 0, (6)

(6)

(6)

(9)

(9)

Proof: First, as usual, let’s do some substitution x = s − a, y = s − b, z = s − c. Then we have

(6)

2. k = 1: (4.2.11) ⇐⇒ L6,2 = f0,3 +4f0,4 +2f1,1 +f1,2 > 0, (9)

σ3 σ1 σ2 − σ3 2 ,r = , 4σ1 σ1 2 (σ1 σ2 − σ3 ) R2 = 16σ1 σ3

s = σ1 , R · r =

(9)

3. k = 2: (4.2.11) ⇐⇒ L6,3 = f0,4 + 4f0,5 + f1,2 + 2f2,1 + (9)

f2,2 > 0, (12)

(12)

(12)

4. k = 3: (4.2.11) ⇐⇒ L6,4 = f0,5 + 4f0,6 + 16f0,7 − (12) (12) 11f1,4 − 4f1,5 (12) (12) 2f3,1 + f3,2 >

+

(12) 2f2,1

+

(12) f2,2

+

(12) 6f2,3

+

(12) 4f2,4

And

+

(6)

(4.2.14) ⇐⇒ L8 =(t + 1)2 f0,3

0

(6)

(12)

(6)

+ 4(t2 − 1)f1,2 > 0. From (4.1.8), we have q (6) (6) (6) 2 f0,4 f1,1 − f1,2 > 0

(4) (6)

f0,6 − 3f1,4 = σ2 f0,3 f0,3 > 0, (12)

(12)

(6)

+ 4(t + 1)2 f0,4 + 4(t − 1)2 f1,1

From Corollary 4.1.1, we know that the right hand of 1.,2.,3. are all true. Now let’s prove the right hand of 4. We have     (12) (12) (12) (12) (12) L6,4 > 4 f0,6 − 3f1,4 + 4 f0,7 + f2,4 − f1,5 , (12)

(4.2.14)

Since that, we get q (6) (6) (6) L8 > 4 4(t2 − 1)2 f0,4 f1,1 + 4(t2 − 1)f1,2

(12)

f0,7 + f2,4 − f1,5 X = (yz)4 (x − y)2 (x − z)2 > 0.

>

So we just got L6,4 > 0. From above all, we proved that (4.2.11) holds if k = 0, 1, 2 or 3. ¤

(6)

4(|t2 − 1| + t2 − 1)f1,2 > 0

From above all, we know that (4.2.14) holds. ¤ (4.2.14) is a generalization of Gerretsen inequality([2]).

Expamle 4.2.7 ([1] BW107(b)). a, b, c are the lengths of the three sides of a triangle. s is the semiperimeter. R and r is the radii of the circumcircle and the incircle of the triangle. Then we have P X √ 2  bc r P s−a 6 4+ (4.2.12) R a

Remark 4.2.1. Example 4.2.8 is a generalization of Gerretsen inequality([2]). Expamle 4.2.9 (Yumin Huang). Assume x, y, z > 0, x2 + y 2 + z 2 = 1, g = x + y + z − xyz 1. Prove min{g(x, y, z)} = 1

190

2. Predicate whether g 6 your answer.

8

√

3

9

Q If −3σ1 (x−y) 6 0, we Q will have the correctness of (4.2.18). Now we assume −3σ1 (x − y) > 0 holds.

is true or false, and prove

√ 1. x2 + y 2 + z 2 = 1 =⇒ x + y + z > 3 3 xyz > 3xyz and g = x+y +z −xyz > 0. From σ12 −2σ2 = x2 +y 2 +z 2 = 1, we have g = σ1 (σ12 − 2σ2 ) − σ3 and

(4.2.18) ⇐⇒ 2σ14 − 11σ12 σ2 + 14σ22 + 3σ1 σ3  2 Y > −3σ1 (x − y)  2 (4) (4) (4) ⇐⇒ 2f0,1 − f0,2 + 2f0,3   2  (4) (4) (4) > 3 4f0,1 f0,3 − f0,2

g > 1 ⇐⇒(σ13 − 2σ1 σ2 − σ3 )2 > (σ12 − 2σ2 )3 ⇐⇒2σ14 σ2 −

−

8σ12 σ22

2σ1 σ3 (σ12

(6)

(4.2.15) 8σ23

+

 2  2  2 (4) (4) (4) ⇐⇒4 f0,1 + 4 f0,2 + 4 f0,3

− 2σ2 ) + σ32 > 0

(6)

(6)

(4) (4)

⇐⇒2f0,2 + 2f0,3 + 8f0,4 (6)

(6)

+ 8f1,1 + 12f1,2 + 37σ32 > 0

(4.2.16)

Expamle 4.2.11. In 4ABC,a, b, c is the length of the sids correspondent to vertex A, B, C. ha , hb , hc is the altitute through A, B, C. ra , rb , rc is the radii of the excircles.

(6)

⇐⇒L9 = 37f0,1 − 17f0,2 + 94f0,3 (6)

(6)

X

(6)

+80f0,4 + 117f1,1 − 28f1,2 > 0

h2b

and

(6)

ra = (4.2.17)

From (4.1.8),√we get the correctness of (4.2.17). So ¤ L9 > 0, g 6 8 9 3 holds.

X

1X 2 (y − z 2 )2 3

X  (4.2.18)

Proof: (4.2.18) ⇐⇒σ14 − 4σ12 σ2 + 7σ22 − 3σ1 σ3 X 2 − 3σ1 x y>0 ⇐⇒2σ14

−

11σ12 σ2 > −3σ1

14σ22

+ Y

−1 4z 2 4z 2 + (y + z)2 (z + x)2  −1 ! 4z 2 − + 1 >0 (x + y)2

(4.2.20)

And (4.2.20) is equivalent to L11 = L11,1 + L11,2 > 0.

+ 3σ1 σ3

(4.2.21)

in which

(x − y)

(18)

(18)

L11,1 = 8f0,5 − 8f0,6

From (4.1.5), we have

(18)

(4) f0,3

(18)

(18)

(18)

+ 54f0,7 + 150f0,8 + 625f0,9 + 16f1,3

2σ14 − 11σ12 σ2 + 14σ22 + 3σ1 σ3 (4) =f0,1

2S 2S 2S , rb = , rc = , b+c−a c+a−b a+b−c 2S 2S 2S ha = , hb = , hc = , a b c p S = (s − a)(s − b)(s − c)s,

we can get that (4.2.19) is equivalent to

([1] BL179(b)). Prove

x3 (x − y) >

(4.2.19)

In fact, (4.2.19) holds. First let’s do some substitution a = x + y, b = y + z, c = z + x, x > 0, y > 0, z > 0. Since

(6)

f0,1 +f1,1 − f0,2 X 2 = x (x − y)2 (x − z)2 > 0, q (6) (6) (6) (6) (6) (6) f0,4 +f1,1 − f1,2 > 2 f0,4 f1,1 − f1,2 > 0.

Expamle 4.2.10

X ra2 ra2 > + h2c h2a + ra2

Predicate whether (4.2.19) is true or false and prove your answer.

  (6) (6) (6) L9 >17 f0,1 + f1,1 − f0,2   (6) (6) (6) + 28 f0,4 + f1,1 − f1,2 (6)

(4) (4)

It’s obvious to see that the last inequality holds. So (4.2.18) holds. ¤

2. Using the same method, we have √ 8 3 g6 ⇐⇒81(σ13 − 2σ1 σ2 − σ3 )2 9 6 192(σ12 − 2σ2 )3 (6)

(4) (4)

> 4f0,1 f0,2 + 4f0,2 f0,3 + 4f0,3 f0,1  2  2 (4) (4) (4) (4) ⇐⇒ f0,1 − f0,2 + f0,2 − f0,3  2 (4) (4) + f0,3 − f0,1 >0

It’s easy to see that (4.2.16) holds. So (4.2.15) holds, and we have g(1, 0, 0) = min{g(x, y, z)} = 1.

(6)


2

(18)

(18)

(18)

(18)

(18)

(18)

− 48f1,4 + 60f1,5 − 248f1,6 − 1050f1,7

(4) f0,2

+ − q (4) (4) (4) >2 f0,1 f0,3 − f0,2 > 0

(18)

(18)

+ 97f2,3 − 494f2,4 + 125f2,5 + 13576f2,6 (18)

(18)

(18)

+ 2246f3,3 − 18508f3,4 + 32743f4,3

191

(4.2.22)

(18)

(18)

L11,2 = 2500f0,10 + 9500f1,8 + +

(18) 4f2,1

(18) 1004f3,2

+

(18) 28f2,2

+

(18) 42220f3,5

+

(18)

(18) 53400f2,7

+

+

(18) 7228f4,1

(18)

(18) 408f3,1 (18)

+ 9976f4,2

(18)

+ 90364f4,4 + 42704f5,1 + 35576f5,2 > 0

(4.2.23)

and  (6) (12) (12) L11,1 = f0,3 · 8f0,3 + 24f0,4 (12)

(12)

(12)

(12)

+ 150f0,5 + 750f0,6 + 3625f0,7 + 48f1,1 (12)

(12)

(12)

(12)

+ 384f1,2 + 1668f1,3 + 1540f1,4 + 20825f1,5 (12)

(12)

(12)

(12)

+ 6841f2,1 + 21105f2,2 + 63631f2,3 + 216399f2,4  (12) (12) +201015f3,1 + 346898f3,2 + 1402336σ34 > 0. (4.2.24) So (4.2.21) holds. Then we get that (4.2.19) holds.

5.

¤

SUMMARY

In this paper, we introduce Schur partition and a sufficient positive condition for ternary forms are psd. Compared with the former methods, this partition can indicate the definition quickly and distinctly and offer a readable proof (n) when the coefficients of fi,j is positive. And this method can partition forms of arbitrary degree(larger than 1).

6.

ACKNOWLEDGEMENTS

This work is supported in part by NKBRPC-2004CB318003. We would like to thank the all for their helps. And we thank Lu Yang for his arrangement for the meeting of the two authors.

7.

REFERENCES

[1] Baoqian Liu: BOTTEMA, what we have seen–the new theory, new method and new result of the research on triangle inequalities. Lhasa: Tibet People Press, 2003.(In Chinese) [2] O. Bottema,et al., Geometric Inequalities, Wolters-Noordhoff Publishing, Groningen, The Neherlands(1969). [3] Bruce Reznick, Extremal psd forms with few terms Duke Math.J. 45, pp.363-374(1978). [4] Bruce Reznick, Some Concrete Aspects of Hilbert’s 17th Problem Contemporary Mathematics,2000 253:pp. 251C272.

192

[5] G.E. Collins, Quantifier elimination for real closed fields by cylindrical algebraic decomposition, In: Automata Theory and Formal Languages (Brakhage,H.,ed.), LNCS 33,pp.134-165.Springer, Berlin Heidelberg(1975). ¨ ber die Darstellung definiter Formen als [6] D.Hilbert, U Summe von Formenquadraten Math. Ann.32 pp.342-350(1888). ¨ ber die Zerlegung definiter Funktionen in [7] E.Artin, U Quadrate, Hamb. Abh. 5(1927), 100-115. [8] G.H.Hardy, J.E.Littlewood, and G.P´ olya, Inequalities (O.Shisha,Ed.),pp.205-224, Academic Press, New York(1967). [9] http://guestbook.nease.net/read.php? owner=zgbdsyjxz&page=1&comment ID=1100526260(in Chinese). [10] Jichang Kuang: The useful inequationsthe third edtion.Shandong Science and Technology Press (2004.1).(in Chinese) [11] M.D.Choi,T.Y.Lam and Bruce Reznick, Real zeros of positive semidefinite forms,I Math.Z. 171 pp.1-26(1980). [12] M.D.Choi,T.Y.Lam and Bruce Reznick, Even Symmetric Sextics, Math.Z. 195,559-580(1987). [13] A. Tarski, A decision method for elementary algebra and geometry, The RAND Corporation, Santa Monica(1948) [14] Vlad Timofte, On the positivity of symmetric polynomial functions. Part I: General results, J.Math.Anal.Apple. 284,174-190(2003). [15] William R. Harris, Real Even Symmetric Ternary Forms, Journal of Algebra 222,204-245(1999). [16] L. Yang, Xia Shihong, Automated proving for a class of constructive geometric inequalities, Chinese J. Computer,26(7), pp.769-778(2003). [17] L. Yang, Zhang J, A practical program of automated proving for a class of geometric inequalities, Automated Deduction in Geometry, Lecture Notes in Artificial Intelligence 2061, pp.41-57, Springer-Verlag, (2001). [18] L. Yang, Recent advances in automated theorem proving on inequalities, J. Comput. Sci. & Technol., 14:5, pp.434-446(1999). [19] L. Yang, A Dimension-Decreasing Algorithm with Generic Program for Automated Inequality Proving. High Technology Letters (Chinese ed.), 8:7, 20-25(1998)

Affine Transformations of Algebraic Numbers D.J. Jeffrey

∗

Pratibha

Department of Applied Mathematics The University of Western Ontario London, Ontario, Canada

[email protected]

[email protected]

ABSTRACT

The Symbolic Computation Group The University of Waterloo Waterloo, Ontario, Canada

[email protected]

equivalent construction. RootOf has two forms: indexed and non-indexed. The non-indexed RootOf(p(x), x) represents either one root or all of the roots of p(x), depending on the context, while the newer indexed RootOf(p(x), x, index = n) represents the nth root of p(x), as determined by the Maple indexing scheme. Maple’s RootOf will accept any polynomial p(x) ∈ Q[x] as an argument, whether p(x) is irreducible or composite over Q, and indeed it will accept nonpolynomial functions. Here, however, the argument p(x) is restricted to being irreducible, because then the indexing scheme is cleaner, there being no repeated roots [4]. The original motivation for the current investigation came from the calculation of series solutions for an ordinary differential equation about a regular singular point, using the method of Frobenius. Della Dora and Tournier [1] point out that computing such solutions requires a test to determine when two roots of the characteristic equation differ by an integer, and comment that this is non-trivial. It is natural to generalize this question to a general affine relationship. The general question is the following. Given a root r of a polynomial p(x) ∈ Q[x], and a root s of a polynomial q(x) ∈ Q[x], which may or may not equal p(x), determine whether an affine transformation exists between them, i.e., determine α, β ∈ Q such that s = αr + β. This is answered here by expressing each algebraic number r ∈ / Q in terms of a uniquely defined algebraic number t such that if r = α1 t+β1 for α1 , β1 ∈ Q, then there is an affine transformation from s ∈ / Q to r only if s is also expressed in terms of t, i.e., s = α2 t + β2 for α2 , β2 ∈ Q.

We consider algebraic numbers defined by univariate polynomials over the rationals. In the syntax of Maple, such numbers are expressed using the RootOf function. This paper defines a canonical form for RootOf with respect to affine transformations. The affine shifts of monic irreducible polynomials form a group, and the orbits of the polynomials can be used to define a canonical form. The canonical form of the polynomials then defines a canonical form for the corresponding algebraic numbers. Reducing any RootOf to its canonical form has the advantage that affine relations between algebraic numbers are readily identified. More generally, the reduction minimizes the number of algebraic numbers appearing in a computation, and also allows the Maple indexed RootOf to be used more easily.

Categories and Subject Descriptors G.1.5 [Numerical Analysis]: Roots of Nonlinear Equations—Polynomials, methods for

General Terms Algorithms

Keywords Algebraic numbers, RootOf, Affine Transformation

1.

K.B. Roach

Ontario Research Centre for Computer Algebra The University of Western Ontario London, Ontario, Canada

INTRODUCTION

We consider univariate polynomials over Q, the field of rational numbers, and the way in which their roots should be represented. When Maple computes the roots of a polynomial p(x) ∈ Q[x], it uses the RootOf function to represent any algebraic numbers required. Mathematica uses an

Example 1. The aim of this paper is not simply to repair some shortcomings in Maple’s current implementation, but it is relevant to see what the situation is at present. We illustrate the shortcomings of Maple’s current simplification of the RootOf function with respect to the questions just posed. Let

∗Present address: Information Technology Development Agency (ITDA), Government of Uttaranchal,272-B, Phase II, Vasant Vihar, DEHRADUN, INDIA 248 006

p(x) =

x10 − 10x9 + 40x8 − 78x7 + 66x6 + 14x5 − 89x4 + 116x3 − 106x2 + 76x − 27 .

(1)

The Maple 9.5 command solve(p(x)) returns the results presented in table 1. (We have abbreviated RootOf to R and omitted the underscore.) It is not obvious from these results that the roots differ by an integer. As a consequence of the properties of the indexing scheme, the indexes must be selected differently for the two families of roots. In other words, starting from index=1 for the first RootOf, one must search all indexes of the other RootOf to find the one that

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193

R(Z^5+Z^2+1,index = 1), R(Z^5+Z^2+1,index = 2), R(Z^5+Z^2+1,index = 3), R(Z^5+Z^2+1,index = 4), R(Z^5+Z^2+1,index = 5), R(Z^5-10*Z^4+40*Z^3-79*Z^2+76*Z-27,index R(Z^5-10*Z^4+40*Z^3-79*Z^2+76*Z-27,index R(Z^5-10*Z^4+40*Z^3-79*Z^2+76*Z-27,index R(Z^5-10*Z^4+40*Z^3-79*Z^2+76*Z-27,index R(Z^5-10*Z^4+40*Z^3-79*Z^2+76*Z-27,index

is monic. Second, suppose M (α, β)p(x) = a(x)b(x), where a(x) ∈ Q[x] and b(x) ∈ Q[x]. Then p(x) = αdeg(p) a(α−1 x − α−1 β)b(α−1 x − α−1 β) , = = = = =

where we must have that a(α−1 x − α−1 β) ∈ Q[x] and also that b(α−1 x−α−1 β) ∈ Q[x]. If neither a(x) nor b(x) is a
unit of Q[x] (an element of Q), then neither a α−1 x − α−1 β nor
b α−1 x − α−1 β is a unit of Q[x].

1), 2), 3), 4), 5)

We can in fact make the stronger statement that if p ∈ Pn then M (α, β)p ∈ Pn .

Table 1: The roots of the polynomial p(x) defined in equation 1 as expressed in RootOf notation by Maple 9.5. The function name RootOf has been abbreviated to R to save space.

Theorem 2. The set of monic affine shifts M (α, β) with α ∈ Q\{0}, β ∈ Q is a group action on P. Proof. The action is well defined because of theorem 1, and the verification of the group laws is straightforward.

differs from it by an integer. Once this is done, the following statement is true.

Now we establish the correspondence between affine shifts of the polynomials and affine transformations of algebraic numbers.

R(_Z^5-10*_Z^4+40*_Z^3-79*_Z^2+76*_Z-27,index=1) - R(_Z^5+_Z^2+1,index=3) = 2

Definition 3. Let p(x) ∈ P, then the set R(p(x), x) ⊂ C is defined by

Maple’s simplify, evala, evalb commands cannot verify this. However, finding the limitations of particular Maple commands is not the point (a sufficiently expert user will be able to guide Maple to this simplification). The point is that working with algebraic numbers is more convenient if they are expressed in a canonical form. Notice that two things must be recognized in the above statement: the relation between the polynomials and the indexing of the roots.

2.

x ∈ R(p(x), x) ⇐⇒ p(x) = 0 .

This corresponds to the non-indexed Maple RootOf. The number of elements in R is deg p. An affine shift does not change the degree of the polynomial, so we must establish that an affine relation between algebraic numbers can only exist between numbers of the same degree. We note that an affine shift of the defining polynomial corresponds to an affine transformation of every root in the set of roots, and the set remains the same size.

AFFINE TRANSFORMATIONS OF ALGEBRAIC NUMBERS

Lemma 1. Let p ∈ P have roots ri ∈ / Q. If q ∈ P has a root s ∈ / Q such that s = αrj + β for some j and α ∈ Q\{0}, β ∈ Q, then the roots of q are precisely αri + β for all i.

Let P ⊂ Q[x] be the set of monic irreducible polynomials over Q. Further, let Pn be the set of monic irreducible polynomials of degree n. We consider the algebraic numbers defined by the roots of the elements of P. From the point of view of Maple, this corresponds to using the output of the factors command, rather than the solve command.

/ Q and s ∈ / Q are roots of irreProof. Assume r ∈ ducible polynomials p and q such that s = αr + β, α ∈ Q, and β ∈ Q. Then s ∈ / Q implies α 6= 0 and r is a root of α− deg(q) q(αx + β) ∈ Q[x]. Since q(x) is irreducible, α− deg(q) q(αx + β) must also be irreducible by theorem 1. But r must be a root of a unique irreducible polynomial. Hence, p(x) = α− deg(q) q(αx + β) which shows that if the roots of p are ri , then the roots of q are αri + β.

Definition 1. For x ∈ C and α, β ∈ Q, an affine transformation T (α, β) of x is defined by T (α, β)x = αx + β . Definition 2. For p(x) ∈ P, α ∈ Q\{0}, β ∈ Q, a monic affine shift M (α, β) of p(x) is defined by M (α, β)p(x) = α

− deg p

p(T (α, β)x) = α

− deg p

Our aim is to relate algebraic numbers through their defining polynomials.

p(αx + β) .

Definition 4. If r and s be algebraic numbers, then r and s are affinely related over Q, written r ≡Q s, if there exist α ∈ Q\{0}, and β ∈ Q such that s = T (α, β)r.

The inverse transformations are T (α, β)−1 x −1

M (α, β)

=

p(x) =

α−1 x − α−1 β ,


αdeg p p T (α, β)−1 x .

(4)

(2) (3)

Lemma 2. Affinely related over Q is an equivalence relation.

We must show that the transformations M (α, β) form a group. This is obvious if M is applied to Q[x], but we apply it only to P. Therefore we first prove that M is closed on P.

Proof. Since the affine transformations define a group, they define a set of orbits of the algebraic numbers. Two algebraic numbers are affinely related if they are members of the same orbit. It is standard theory that orbits define equivalence classes.

Theorem 1. Given p(x) ∈ P, α ∈ Q\{0}, β ∈ Q, then M (α, β)p(x) ∈ P. Proof. First, observe that M (α, β)p(x) = α− deg(p) p(αx + β)

194

Theorem 3. For p(x) ∈ P, α ∈ Q\{0} and β ∈ Q,

The properties of the + operator are summarized in the following theorem.

M (α, β)p(x) = 0 ⇐⇒ T (α, β)−1 R(p(x)) .

Theorem 5. If α ∈ Q\{0}, β ∈ Q, p(x) ∈ P is an irreducible polynomial, r an algebraic number, and n = deg(p), then

Proof. By definition 2 and lemma 1. Theorem 3 shows that algebraic numbers are affinely related if their defining polynomials are related by corresponding affine shifts. Consequently it seems that all affine relations can be deduced by considering the orbits of the defining polynomials. There is a difficulty, however. It is possible for affine relations to exist within a RootOf set. This corresponds to an affine polynomial shift mapping a polynomial nontrivially onto itself. It is therefore important to decide when this can occur.

(a) (M (α, β)p(x))+ = M (α, 0)p+ (x). (b) (T (α, β)r)+ = T (α, 0)r+ . (c) p++ = p+ . (d) r++ = r+ . Proof. For (a), since M (α, β)p(x) =

Theorem 4. Two different roots r ∈ / Q and s ∈ / Q of an irreducible polynomial cannot be linearly related over Q unless r = T (−1, β)s for some β ∈ Q.

+α−n pn−1 (αx + β)n−1 + . . . =

Proof. Assume r and s are different roots of an irreducible polynomial p(x) such that s = T (α, β)r, α ∈ Q, and β ∈ Q. Let rn = α n + β

n−1 X

xn + α−1 (pn−1 + nβ) xn−1 + . . . ,

we have (M (α, β)p(x))+

= =

αi

  pn−1 α−n p αx − −β n α−n p+ (αx) .

For (b), since αr + β has irreducible polynomial
n αn p(α−1 x − α−1 β) = αn α−1 x − α−1 β

i=0

for n ∈ N. Then r0 = r is a root of p. By lemma 1, if rn is a root of p, then rn+1 = α rn + β is a root of p. So, by induction, rn is a root of p for all n ∈ N. Consider the cases α = 1 and α 6= 1 separately. If α = 1, then rn = r + nβ is a root of p for all n ∈ N. The Fundamental Theorem of Algebra requires {rn | n ∈ N } be a finite set. Since α ∈ Q, α 6= 0, and α 6= 1, this implies β = 0 and s = r, a contradiction. If α 6= 1, then   β β n rn = α r + − α−1 α−1

+αn pn−1 α−1 x − α−1 β =


n−1

+... xn + α(pn−1 − nβ)xn−1 + . . .

we get (αr + β)+

α(pn−1 − n β) = (αr + β) + n  pn−1  = α r+ = αr+ . n

Parts (c) and (d) follow from definition 5 and parts (a) and (b).

is a root of p for all n ∈ N. The Fundamental Theorem of Algebra requires {rn |n ∈ N } be a finite set. This implies α = −1 and s = −r + β.

Definition 6. Suppose ρ ∈ Q\{0}, and n is a positive integer. Let ρ = a/b where a ∈ Z and b ∈ N. Let a bn−1 = kn c where k is a positive integer, c is a nonzero integer that is not divisible by any nontrivial nth power of an integer (i.e. other than ±1). Then [ρ]n is defined by [ρ]n = k/b.

2 Example. √ The irreducible √ polynomial x − 2x − 1 has roots r = 1 + 2 and s = 1 − 2 = −r + 2. To summarize this section, we have shown that affine relations between algebraic numbers can be studied by considering the orbits of the defining polynomials under the group action of T (α, β). We now must define an invariant for each orbit, and this will be the number t referred to in the introduction. The set of numbers t will define a cross section in the sense of [2, 3], or in equivalent terminology a canonical form. The selection of the cross-section element is the subject of the next section.

3.

α−n (αx + β)n

Remark. The a and b do not need to be relatively prime. To check that [ρ]n is well defined, observe that if ρ = a1 /b1 = a2 /b2 , and a1 bn−1 = k1n c1 1

and

a2 bn−1 = k2n c2 2

then n−1 n n b1 k2n c2 = bn = a1 bn−1 bn 1 a2 b 2 2 = b2 k 1 c1 , 1

DEFINITION OF A CANONICAL FORM

but b1 , b2 , k1 , k2 are positive integers; c1 and c2 are nonzero integers; and neither c1 nor c2 is divisible by a nontrivial nth power of an integer. Therefore, by the Fundamental Theorem of Arithmetic, we must have c1 = c2 and k1 /b1 = k2 /b2 .

Definition 5. Let p(x) = xn + pn−1 xn−1 + . . . + p0 , where p(x) ∈ P, be an irreducible polynomial. Define p+ (x) ∈ P p+ (x) = M (1, −pn−1 /n)p(x) . If r = R(p(x), x), then r+ ≡Q r is defined by r+ = R(p+ (x), x) = T (1, pn−1 /n)r .

195

(c) By construction, p× = [a0 ]−n n p ([a0 ]n x − pn−1 /n). By theorem 5 and definition 7,   pn−1  p×+ = [a0 ]−n n p [a0 ]n x − n + = [a0 ]−n n p+ ([a0 ]n x) = p× (x)

Lemma 3. Suppose ρ1 ∈ Q, ρ1 6= 0, ρ2 ∈ Q, ρ2 6= 0, and n is a positive integer. Then sgn([ρ1 ]n ) = 1 and [ρn 1 ρ2 ]n = |ρ1 | [ρ2 ]n . Proof. First, sgn([ρ1 ]n ) = 1 is easily seen from definition 6. Second, let ρi = ai /bi where ai ∈ Z and bi ∈ N. Let a2 bn−1 = k2n c2 where k2 is a positive integer; c2 is a nonzero 2 integer; and c2 is not divisible by any nontrivial nth power of an integer. Then ρn 1 ρ2 =

Also, by (a) and lemma 3,   pn−1  p×× = [a0 ]−n = p× (x) n p [a0 ]n x − n ×

an 1 a2 bn 1 b2

(d) By construction, r× = [a0 ]−1 r + pn−1 /n. By theorem 5 and definition 7,
r×+ = [a0 ]−1 r + pn−1 /n + = [a0 ]−1 r+ = r×

and n−1 n an 1 a2 (b1 b2 )

= =

n(n−1) n k 2 c2
n n−1 a1 b 1 k 2 c 2

an 1 b1

Also, by (b) and lemma 3

implying [ρn 1

ρ 2 ]n =

| a1 | bn−1 1 bn 1 b2

k2

r×× = [a0 ]−1 r + pn−1 /n

| a1 | k 2 = = | ρ1 | [ρ2 ]n . b1 b2


×

= r×

Definition 8. If p(x) ∈ Q[x] and p(−x) = p(x), then p(x) is an even polynomial.

Definition 7. If p(x) ∈ Pn , n > 1 is a nonlinear irreducible polynomial and p+ (x) = xn + an−2 xn−2 + . . . + a0 , define irreducible polynomial p× (x) ∈ P by
p× (x) = [a0 ]−n n p+ [a0 ]n x

Definition 9. The complex signum of a complex number z is   0 , z = 0, csgn(z) = 1 , −π/2 < arg z ≤ π/2,  −1 otherwise.

Suppose r ∈ / Q is a root of p(x). Define root r× ≡Q r of p× (x) by r× = [a0 ]−1 n r+ . Theorem 6. If α ∈ Q\{0}, β ∈ Q, p(x) ∈ Pn , n > 1 is a nonlinear irreducible polynomial, r ∈ / Q is an algebraic number, and n = deg(p), then, with sα = sgn α,

Definition 10. Let p(x) ∈ P be a nonlinear irreducible polynomial, p× (x) = xn + an−2 xn−2 + . . . + a0 . If p× is even, define σ(p) to be 1. If p× is not even, define σ(p) to be the sign of the first nonzero coefficient in the sequence an−3 , an−5 , . . .. Suppose r ∈ / Q is a root of p(x). If p× is even, define σ(r) to be csgn(r× ). If p× is not even, define σ(r) to be σ(p).

(a) (M (α, β)p(x))× = (sα )n p× (sα x). (b) (T (α, β)r)× = sα r× (c) p×+ = p×× = p×

Lemma 4. If α ∈ Q, α 6= 0, β ∈ Q, and r ∈ / Q is an algebraic number, then σ(αr + β) = sα σ(r), and sα = sgn α as before.

(d) r×+ = r×× = r× Proof. (a) Let p+ (x) = xn + an−2 xn−2 + . . . + a0 . By theorem 5,
α−n p(αx + β) + = α−n p+ (αx) = xn + . . . + α−n a0 . Therefore,   p(αx + β) αn ×

=

h a i−n 0 αn

n

 
α−n p+ α α−n a0 n x

=


−n |α| [a0 ]−n p+ α|α|−1 [a0 ]n x n α
−n sn α [a0 ]n p+ [a0 ]n sα x

=

sn α p× (sα x)

=

Proof. Let r have irreducible polynomial p(x) ∈ P and p× = xn + an−2 xn−2 + . . . + a0 . Then αr + β has irn −1 −1 reducible polynomial
α p(αn x − α β). By theoremn 6, n −1 −1 α p(α x − α β) × = sα p× (sα x) and this equals x + sα an−1 xn−1 + . . . + sn α a0 .
Therefore, αn p(α−1 x − α−1 β) × is an even polynomial if and only if p× (x) is an even polynomial. By theorem 6, we
know (αr + β)× = sα r× . Hence, if αn p(α−1 x − α−1 β) × is even, we get

n

σ(αr + β)

sα csgn(r× ) = sα σ(r)
If α p(α x − α β) × is not an even polynomial, we get   σ(αr + β) = sgn sgn(α)2l+1 an−2l−1 n

and (αr + β)+ = αr+ . Therefore by definition 7, (αr + β)×

n

a0 ]−1 n αr+

= [α = sα r× .

= |α|

−1

= csgn ((αr + β)× ) = csgn (sα r× ) =

(b) Let r have irreducible polynomial p(x) and let p+ (x) = xn + an−2 xn−2 + . . . + a0 . Then αr + β has irreducible polynomial αn p(α−1 x − α−1 β). By theorem 5,
αn p(α−1 x − α−1 β) + = αn p+ (α−1 x)

−1

−1

= sgn(α) sgn(an−2l−1 ) = sα σ(r)

[a0 ]−1 n αr+

where an−2l−1 is the first nonzero coefficient in the sequence an−1 , an−3 , an−5 , . . ..

196

Definition 11. Suppose p(x) ∈ Pn , n > 1 is a nonlinear irreducible polynomial. Define irreducible polynomial pσ (x) ∈ P by


By (a), pσσ = σ(p)n [a0 ]−n n p (σ(p)[a0 ]n x − pn−1 /n) σ , and this equals pσ . (d) By construction,

pσ (x) = M (σ(p), 0)p× (x)

−1 rσ = σ(r)[a0 ]−1 n r + σ(r)[a0 ]n pn−1 /n

Suppose r ∈ / Q is a root of p(x). Define root rσ ≡Q r of pσ (x) by rσ = T (σ(r), 0)r× .

By theorem 5, definition 7, and definition 11,

Theorem 7. If α ∈ Q, α 6= 0, β ∈ Q, p(x) ∈ P is a nonlinear irreducible polynomial, and r ∈ / Q is an algebraic number, then

σ(r)[a0 ]−1 n r+

rσ+

=

rσ×

(d) rσ+ = rσ× = rσσ = rσ

pσ

(e) σ(pσ ) = 1

Proof. (a) Let p× (x)
= xn +an−1 xn−1 +. . .+a0 . By theorem 6, α−n p(αx + β) × = sgn(α)n p× (sgn(α)x) and this n−1 equals xn + sgn(α)a + . . . + sgn(α)n a0 . Therefore, n−1 x
−n α p(αx + β) × is an even polynomial if and only if p× (x)
is an even polynomial, and if α−n p(αx + β) × is an even polynomial, then
σ α−n p(αx + β) = σ(p) = 1

σ1n sgn(α)n p× (sgn(α)σ1 x)

=

σ(p)n p× (σ(p)x) = pσ (x)

=

σ(αr + β)(αr + β)×

=

sgn(α)σ(r) sgn(α)r× = σ(r)r× = rσ

= =

=

σ(p)n p× (σ(p)x)

=

xn + σ(p)an−1 xn−1 + . . . + σ(p)n a0

σ(p) sgn(an−2l−1 ) = σ(p)2 = 1

Definition 12. If p(x) ∈ P is an irreducible polynomial, then pσ (x) is called a canonical polynomial. If r is an algebraic number, rσ is a canonical root. Lemma 5. Suppose p(x) ∈ Q[x] is an irreducible polynomial and r is a root of pσ (x). If pσ is even, then rσ = csgn(r)r. If pσ is not even, then rσ = r. Proof. By construction


− pn−1 /n) +

pσ (x) = σ(p)n [a0 ]−n n p (σ(p)[a0 ]n x − pn−1 /n)

σ(p)n [a0 ]−n n p+ (σ(p)[a0 ]n x)

and this is xn + 0xn−1 + . . ., which makes r+ = r. Next, pσ (x) = σ(p)n [a0 ]−n n p+ (σ(p)[a0 ]n x), which equals

n

σ(p) p× (σ(p)x) = pσ (x)

n−1 xn + σ(p)[a0 ]−1 + . . . + σ(p)n [a0 ]−n n an−1 x n a0

By theorem 6, lemma 3, and definition 11, pσ×

= rσ (e)

Proof. If r ≡Q s, then there exist α ∈ Q\{0}, β ∈ Q such that r = αs + β. So r ∈ Q if and only if s ∈ Q. If r ∈ Q and s ∈ Q, then rσ = sσ = 0. If r ∈ / Q and s ∈ / Q, then rσ = (αs + β)σ = sσ by theorem 7. If rσ = sσ , then r ≡Q rσ = sσ ≡Q s implies r ≡Q s.

By theorem 5, definition 7, and definition 11,

=

σ

Theorem 8. Let r and s be algebraic numbers. Then r ≡Q s if and only if rσ = sσ .

pσ = σ(p)n [a0 ]−n n p (σ(p)[a0 ]n x − pn−1 /n)

=




If pσ is not even, then σ(rσ ) = σ(pσ ) = 1 by (e).

(c) By construction,

[a0 ]−n n p (σ(p)[a0 ]n x

×

σ(rσ ) = csgn(rσ× ) = csgn(rσ ) = csgn(σ(r)r× ) = σ(r) csgn(r× ) = σ(r)2 = 1

(b) By lemma 4, σ(αr + β) = sgn(α)σ(r) and by theorem 6, (αr + β)× = sgn(α)r× . Therefore,

σ(p)

σ(r)r× = rσ




where an−2l−1 is the first nonzero coefficient in the sequence an−1 , an−3 , an−5 , . . .. (f) Suppose r is a root of p. Then rσ is a root of pσ . If pσ is even, then by (d) and 11,

= p× (x) = pσ (x)
If α−n p(αx + β) × is not even, then by lemma 4, it must
be that σ α−n p(αx + β) = sgn(α)σ(p). For convenient
notation, let σ1 = σ α−n p(αx + β) = sgn(α)σ(p). Then

α−n p(αx + β) σ = σ1n α−n p(ασ1 x + β) ×

=

−1 σ(r)[a0 ]−1 n r + σ(r)[a0 ]n pn−1 /n

=


α−n p(αx + β) ×

=

+

We see that pσ is even if and only if p× is even. If pσ is even, then σ(pσ ) = 1 by definition 10. If pσ is not even, then pσ× = pσ by (c), so   σ(pσ ) = sgn σ(p)2l+1 an−2l−1

(f ) σ(rσ ) = 1

and n is even, which makes
α−n p(αx + β) σ =




= σ(r)r× = rσ

−1 By (b), rσσ = σ(r)[a0 ]−1 n r + σ(r)[a0 ]n pn−1 /n n n−1 Suppose p× = x + an−1 x + . . . + a0 . Then

(c) pσ+ = pσ× = pσσ = pσ

pσ+

= =

(b) (T (α, β)r)σ = rσ

n

−1 σ(r)[a0 ]−1 n r + σ(r)[a0 ]n pn−1 /n

By theorem 6, lemma 3, and 11,

(a) (M (α, β)p(x))σ = pσ (x)

(αr + β)σ

=


σ(p)n [a0 ]−n n p (σ(p)[a0 ]n x − pn−1 /n) ×

Therefore, by lemma 3,  −1 −1 r× = σ(p)n [a0 ]−n n a0 n r+ = [a0 ]n [a0 ]n r+ = r

σ(p)n p× (σ(p)x) = pσ (x)

197

5.

If pσ is even, then pσ× = pσ is even and the relation σ(r) = csgn(r× ) = csgn(r) gives us

EXAMPLES

Example 1 can now be solved as follows. The roots become

rσ = σ(r)r× = csgn(r)r . If pσ is not even, then pσ× = pσ and σ(r) = σ(pσ ) = 1 by theorem 7, implying rσ = σ(r)r× = 1 · r = r. Theorem 9. Suppose p(x) ∈ P is an irreducible polynomial. If pσ is even, then exactly half of the roots of pσ are canonical, and the other half are the negatives of the first half. The roots of pσ expressed in canonical form are √ √ ± s1 , . . . , ± sn/2 √ where the s1 , . . . , sn/2 are the roots of pσ ( x) and the roots √ √ of p(x) corresponding to s1 , . . . , sn/2 are canonical and distinct. If pσ is not even, then all of the roots of pσ are canonical and distinct. √ Proof. If pσ is even, then pσ√ ( x) is a polynomial. If s1 , . . . , sn/2 are the roots of pσ ( x) then the roots of pσ must be √ √ ± s1 , . . . , ± sn/2 √ Since csgn( si ) = 1 for the principal branch of the square √ √ root, we see s1 , . . . , sn/2 are canonical by lemma 5. They are distinct because of irreducibility. If pσ is not even, then all of the roots of pσ are canonical by lemma 5, and again they are distinct.

4.

x1 x2

= =

R(z 5 + z 2 + 1, z, index = 1), R(z 5 + z 2 + 1, z, index = 2),

x3 x4

= =

R(z 5 + z 2 + 1, z, index = 3), R(z 5 + z 2 + 1, z, index = 4),

x5

=

R(z 5 + z 2 + 1, z, index = 5),

x6

= 2 + R(z 5 + z 2 + 1, z, index = 1),

x7

= 2 + R(z 5 + z 2 + 1, z, index = 2),

x8

= 2 + R(z 5 + z 2 + 1, z, index = 3),

x9

= 2 + R(z 5 + z 2 + 1, z, index = 4),

x10

= 2 + R(z 5 + z 2 + 1, z, index = 5).

The statement x8 − x3 = 2, now becomes obvious, as does the statement x1 − xj 6∈ Z, j > 6. Example 2. Let p(x) be the polynomial p(x) = 20736x18 + 179712x17 + 457920x16 − 94656x15 −2769344x14 − 3990464x13 + 4663468x12 + 13336348x11 +1419041x10 − 11999454x9 − 12064413x8 − 11691713x7 +2228047x6 + 20917493x5 + 14885343x4 − 1856968x3 −8277728x2 − 4558616x − 804752. The factored form of p(x) is (having been obtained, for example, from Maple factors)

AN ALGORITHM

The input for the algorithm is p(x) ∈ Q[x], a univariate polynomial over the rationals. The output is a list of the roots of p(x) expressed in the canonical form just described.

20736(x − 1)2 a(x)b(x)c(x)d(x) where

1. Factor p(x) over Q. (The Maple factors command can do this.) Y p(x) = c pi (x)ei i

2. Solve the linear pi (x) to obtain the rational roots of p(x).

a(x) =

x4 − 43 x3 + 23 x2 −

b(x) = c(x) =

x4 + 8x3 + 24x2 + 40x + 16 x4 + 6x3 + 53 x2 + 51 x + 73 4 4 16

d(x) =

x4 − 2x3 + 12 x2 + 12 x +

31 x 27

−

53 81

13 16

Solving the linear factor of p(x) produces the rational root 1 with multiplicity 2. Computing a+ , b+ , c+ , and d+ gives
a+ (x) = a x + 13 = x4 − x − 1

3. For each nonlinear pi (x), obtain the canonical polynomial. If n = deg(pi ), the steps are

b+ (x) =

pi+ (x) = M (1, pn−1 /n)pi (x) pi× = M ([a0 ]n , 0)pi+ (x)

c+ (x) = d+ (x) =

piσ = M (σ(p), 0)pi× (x) Each pi (x) can therefore be expressed canonically as the polynomial M (αi , βi )piσ (x) for computed αi , βi .

b(x − 2) = x4 + 8x − 16
1 c x − 32 = x4 − 41 x2 + 16
d x + 12 = x4 − x2 + 1

Computing a× , b× , c× , and d× gives

Let B be the set of distinct piσ that appear. 4. Each piσ ∈ B that is not an even polynomial has deg(pi ) distinct roots, which in Maple RootOf notation are written R(pi (x)).

a× (x) =

a+ (x) = x4 − x − 1

b× (x) =

1 b (2x) = x4 + x − 1 16 +
16c+ 12 x = x4 − x2 + d+ (x) = x4 − x2 + 1

c× (x) = d× (x) =

5. Each piσ ∈ B that is an even polynomial has deg(pi ) distinct p roots, √which can be written in RootOf notation as ± R(pi ( x).

Computing aσ , bσ , cσ , and dσ gives

6. Collect the R canonical forms from steps (2), (4) and (5) according to multiplicities ei to form the final answer.

198

aσ (x) =

a× (−x) = x4 + x − 1

bσ (x) =

b× (x) = x4 + x − 1

cσ (x) = dσ (x) =

c× (x) = x4 − x2 + 1 d× (x) = x4 − x2 + 1

1

7.

The canonical forms for a, b, c, and d are given by
a(x) = aσ −x + 13
b(x) = 16aσ 12 x + 1 c(x) = d(x) =

1 c (2x 16 σ

dσ x −

[1] J. Della Dora and E. Tournier. Formal solutions of differential equations in the neighborhood of singular points (regular and irregular). In Proceedings of the 1981 ACM Symposium on Symbolic and Algebraic Computation, pages 25–29. ACM Press, 1981. [2] M. Fels and P. Olver. Moving coframes. II. Regularization and theoretical foundations. Acta Appl. Math., 55:127–208, 1999. [3] P. J. Olver. Classical Invariant Theory. Cambridge University Press, 1999. [4] J.-P. Tignol. Galois theory of algebraic equations. Longman, 1988.

+ 3)


1 2

and the basis B is

 B = {aσ (x), cσ (x)} = x4 + x − 1, x4 − x2 + 1

The polynomial aσ (x) is not even and therefore has 4 distinct algebraic roots, represented in Maple by R(aσ (x)). They will contribute 8 of the roots of the p(x), specifically, 13 − R(x4 + x + 1) from the reduction of a(x) and −2+2R(x4 +x+1) from the reduction of b(x). The polynomial cσ p is even and contributes 4 roots in two pairs, namely p − 23 ± 12 R(x2 − x + 1) and 12 ± R(x2 − x + 1). All possible linear relations over Q between roots of p(x) are now explicitly obvious. This concludes the example.

6.

CONCLUDING REMARKS

There remain a number of implementation questions. It should be recalled that Maple allows any polynomial to be an argument of RootOf. For an indexed RootOf, the index must be re-computed for the canonical polynomial. For a non-indexed RootOf, there is a question of what to do with a set that separates into several subsets. Even for the RootOf an irreducible polynomial, there are other properties that might be considered. If pσ is a canonical polynomial with deg pσ = n, then we have n X

R(pσ (x), x, index = k) = 0 .

REFERENCES

(5)

k=1

One way to ensure that this simplification is known to the system would be to express the nth RootOf as the negative of the sum of the first (n − 1) RootOfs. For large n, however, this would be cumbersome, and would not apply to a nonindexed RootOf.

199

Architecture-Aware Classical Taylor Shift by 1 Jeremy R. Johnson

Werner Krandick

Anatole D. Ruslanov

Department of Computer Science Drexel University Philadelphia, PA 19104

Department of Computer Science Drexel University Philadelphia, PA 19104

Department of Computer Science Drexel University Philadelphia, PA 19104

[email protected]

[email protected]

[email protected]

ABSTRACT

cients of the polynomial B(x) = A(x + 1) from the coefficients of the polynomial A(x). Taylor shift by 1 is the most time-consuming subalgorithm of the monomial Descartes method [7] for polynomial real root isolation. Taylor shift by 1 can also be used to shift a polynomial by an arbitrary integer a. Indeed, if B(x) = A(ax) and C(x) = B(x + 1) and D(x) = C(x/a), then D(x) = A(x + a). According to Borowczyk [3], Budan proved this fact in 1811. More recently, von zur Gathen and Gerhard [22, 9] compared six different methods to perform Taylor shifts. The authors distinguish between classical methods and asymptotically fast methods. When the shift amount is 1, the classical methods collapse into a single method which computes n(n + 1)/2 integer sums where n is the degree of the input polynomial. In fact, von zur Gathen’s and Gerhard’s implementation of classical Taylor shift by 1 simply makes calls to an integer addition routine. We will refer to such implementations as straightforward implementations. We present algorithms that outperform straightforward implementations of classical Taylor shift by 1. For input polynomials of low degrees a method of the SACLIB library [8] is faster than straightforward implementations by a factor of at least 2 on our experimental platform (UltraSPARC III); for higher degrees we develop a method that is faster than straightforward implementations by a factor of up to 7 (Figure 7). Our Taylor shift algorithm requires more word additions than straightforward implementations but it reduces the number of cycles per word addition (Figure 10) by reducing memory traffic (Figure 11) and the number of carry computations. The introduction of signed digits, suspended normalization, radix reduction, and delayed carry propagation enables our algorithm to take advantage of the technique of register tiling which is commonly used by optimizing compilers [2, 14]. While our algorithm is written in a highlevel language, it depends on several parameters that can be tuned to the underlying architecture. It is widely believed that computer algebra systems can obtain high performance by building on top of basic arithmetic routines that exploit features of the hardware. It is also believed that only assembly language programs can exploit features of the hardware. This paper suggests that both tenets are wrong. In Section 2 we introduce some notation for the classical Taylor shift by 1. We also characterize two classes of test inputs for the algorithm. In Section 3 we define our notion of straightforward implementations; we single out the GNU-MP-based implementation as our point of reference.

We present algorithms that outperform straightforward implementations of classical Taylor shift by 1. For input polynomials of low degrees a method of the SACLIB library is faster than straightforward implementations by a factor of at least 2; for higher degrees we develop a method that is faster than straightforward implementations by a factor of up to 7. Our Taylor shift algorithm requires more word additions than straightforward methods but it reduces the number of cycles per word addition by reducing memory traffic and the number of carry computations. The introduction of signed digits, suspended normalization, radix reduction, and delayed carry propagation enables our algorithm to take advantage of the technique of register tiling which is commonly used by optimizing compilers. While our algorithm is written in a high-level language, it depends on several parameters that can be tuned to the underlying architecture.

Categories and Subject Descriptors G.4 [Mathematical software]: Algorithm design and analysis; D.3.4 [Software]: Programming Languages—code generation, compilers; I.1.2 [Computing Methodologies]: Symbolic and Algebraic Manipulation—algebraic algorithms

General Terms Algorithms, Design, Performance

Keywords High-performance computing, performance tuning, code generation, register tiling, multiprecision arithmetic, polynomials, Taylor shift, memory hierarchy, ILP scheduling, delayed carry propagation, loop unrolling

1.

INTRODUCTION

Let A be a univariate polynomial with integer coefficients. Taylor shift by 1 is the operation that computes the coeffi-

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200

In Section 4 we describe the specialized algorithm used in SACLIB. In Section 5 we describe our new algorithm; Section 6 documents its performance.

2.

1. ai,j =

2. L(ai,j ) ≤ L(d) + i + j. Proof. Assertion (1) follows from Definition 1 by induction on i + j. Due to assertion (1), şş ť ş ť ş ťť i+j i+j−1 j |ai,j | ≤ + + ... + d j j j ť ş i+j+1 d = j+1

ANALYSIS

We will call a method that computes Taylor shift by 1 classical if the method uses only additions and computes the intermediate results given in Definition 1. Definition 1. For any non-negative integer n let In = {(i, j) | i, j ≥ 0 ∧ i + j ≤ n}. If n is a non-negative integer and

≤

A(x) = an xn + . . . + a1 x + a0 ,

=

0,

ak,−1 ai,j

= =

an−k , ai,j−1 + ai−1,j ,

Remark 1. Theorem 2 implies that, for degree n and max-norm d, the binary length of all intermediate results is at most L(d) + n. The algorithm in Section 4 can be slightly improved for small-degree polynomials by tightening that bound for n ∈ {8, . . . 39} to L(d) + n − 1, for n ∈ {40, . . . , 161} to L(d)+n−2, and for n ∈ {162, . . . , 649} to L(d) + n − 3.

as shown in Figure 1.

an an−1 an−2 an−3

→ → → →

0 ↓

0 ↓

0 ↓

0 ↓

a0,0 a1,0 a2,0 a3,0

a0,1 a1,1 a2,1

a0,2 a1,2

a0,3

We will use Theorem 3 to prove lower bounds for the computing time of two classes of input polynomials. Theorem 3. Let n be a non-negative integer. Then at ş ť n , 0 ≤ k ≤ n, k have binary length ≥ n/2.

least n/2 of the binomial coefficients

Figure 1: By Theorem 1, the pattern of integer additions in Pascal’s triangle, ai,j = ai,j−1 + ai−1,j , can be used to perform Taylor shift by 1.

Proof. The assertion is clearly true for all n ∈ {0, . . . , 19}, so we may assume n ≥ 20. We then have n − bn/4c + 1 ≥ 42 .

Theorem 1. Let n be a non-negative integer, and let A(x) = an xn + . . . + a1 x + a0 be an integer polynomial. Then, in the notation of Definition 1, A(x + 1) =

n X

Also, for 0 < i < bn/4c, n n n−i > ≥ = 4, bn/4c − i bn/4c n/4

an−h,h xh .

h=0

so that ş ť n bn/4c

Proof. The assertion clearly holds for n = 0; so we may assume n > 0. For every k ∈ {0, . . . , n} let Ak (x) = Pk h h=0 ak−h,h x . The coefficients of the polynomial Ak reside on the k-th diagonal of the matrix of Figure 1. Then, for all k ∈ {0, . . . , n − 1}, we have Ak+1 (x) = (x + 1)Ak (x) + an−(k+1) . Now an easy induction on k shows that Ak (x) = P k (x + 1)h for all k ∈ {0, . . . , n}. In particular, h=0 an−k+h Pn An (x) = h=0 ah (x + 1)h = A(x + 1).

L(a) =

=

n − bn/4c + 1 n n−1 · · ··· · bn/4c bn/4c − 1 1

>

4bn/4c+1 = 22bn/4c+2 > 2n/2 .

Hence, the binary length of each binomial coefficient ş ť ş ť ş ť n n n , , ..., bn/4c bn/4c + 1 n − bn/4c is > n/2. But the number of those coefficients is > n/2.

Definition 2. Let a be an integer. The binary-length of a is defined as ¡

2i+j d

which proves assertion (2).

is an integer polynomial we let, for k ∈ {0, . . . , n} and (i, j) ∈ In , a−1,k

ąi+j ć ą ć ąć an + i+j−1 an−1 + . . . + jj an−i , and j j

Definition 4. For any positive integers n, d we define the polynomials

1 if a = 0, blog2 |a|c + 1 otherwise.

Bn,d (x) = dxn + dxn−1 + · · · + dx + d,

Definition 3. The max-norm of an integer polynomial A = an xn + · · · + a1 x + a0 is |A|∞ = max(|an |, . . . , |a0 |).

Cn,d (x) = xn + d. Theorem 5 and the proof of Theorem 4 characterize the computing time functions of classical Taylor shift on the sets of polynomials Bn,d and Cn,d . We use the concept of dominance defined by Collins [6] since it hides fewer constants than the more widely used big-Oh notation; Collins also defines the maximum computing time function.

The algorithm in Section 4 requires a bound on the binary lengths of the intermediate results ai,j . Theorem 2. Let n be a non-negative integer, and let A(x) = an xn +. . .+a1 x+a0 be an integer polynomial of max-norm d. Then, for all (i, j) ∈ In ,

201

Theorem 4. Let t+ (n, d) be the maximum computing time function for classical Taylor shift by 1 where n ≥ 1 is the degree and d is the max-norm. Then t+ (n, d) is co-dominant with n3 + n2 L(d).

2. Horner’s scheme—ascending order of output coefficients (a0,0 , a1,0 , a0,1 , a2,0 , a1,1 , a0,2 , . . . , an,0 , . . . , a0,n ). 3. Synthetic division—ascending order of output coefficients

Proof. The recursion formula in Definition 1 is invoked |In | = n(n + 1)/2 times. Hence the number of integer additions is dominated by n2 . By Theorem 2, the binary length of any summand is at most L(d) + n. Thus the computing time is dominated by n2 · (L(d) + n). We now show that, for the input polynomials Bn,d , the computing time dominates n3 + n2 L(d). Since, for any fixed n ≥ 1, the computing time clearly dominates L(d) we may assume n ≥ 2. By Theorem 2 (1), ş ť i+j+1 ai,j = d j+1

(a0,0 , a1,0 , . . . , an,0 , a0,1 , . . . , an−1,1 , . . . , a0,n ). 4. Descending order of output coefficients (a0,0 , a0,1 , . . . , a0,n , a1,0 , . . . , a1,n−1 , . . . , an,0 ). Von zur Gathen and Gerhard use method (1) [10]. The computer algebra system Maple [15, 17, 16], version 9.01, uses method (3) in its function PolynomialTools[Translate]. In methods (3) and (4) the output coefficients appear earlier in the sequence than in the other methods. The computing times of the four methods are very similar; they differ typically by less than 10%. In our experiments we will use method (3) to represent the straightforward methods; Figure 2 gives pseudocode.

for all (i, j) ∈ In . For k = i + j ≥ 2, Theorem 3 yields that at least (k + 1)/2 of the binomial coefficients ş ť ş ť ş ť k+1 k+1 k+1 , , ..., 1 2 k+1

for i = 0, . . . , n bi ← ai assertion: bi = an−i,−1 for j = 0, . . . , n − 1 for i = n − 1, . . . , j bi ← bi + bi+1 assertion: bi = an−i,j

have binary length ≥ (k + 1)/2. So, for all k ∈ {2, . . . , n} there are least (k + 1)/2 integers ai,j with i + j = k and L(ai,j ) ≥ L(d) − 1 +

k+1 2

Now the assertion follows by summing all the lengths.

Figure 2: The straightforward method we consider uses integer additions to compute the elements of the matrix in Figure 1 from top to bottom and from left to right.

Our proof of Theorem 5 assumes that the time to add two non-zero integers a, b is co-dominant with L(a)+L(b); Collins [6] makes the same assumption in his analyses.

The efficiency of straightforward methods depends entirely on the efficiency of the underlying integer addition routine. Von zur Gathen and Gerhard use the integer addition routine of NTL [19, 18] in their experiments. But Tables 1 and 2 imply that the integer addition routine of GNU-MP [11] is faster. We use the GNU-MP routine in our implementation of a straightforward method. The GNU-MP package represents integers in sign-lengthmagnitude representation. On the UltraSPARC III we have the package use the radix β = 264 . Let n be a non-negative integer, and let u = u0 + u1 β + · · · + un β n , where 0 ≤ ui < β for all i ∈ {0, . . . , n} and un 6= 0. The magnitude u is represented as an array u of unsigned 64-bit integers such that u[i]=ui for all i ∈ {0, . . . , n}. Let v = v0 + v1 β + · · · + vn β n be a magnitude of the same length. The routine mpn add n is designed to add u and v in n + 1 phases of 4 cycles each. Phase i computes the carry-in ci−1 and the result digit ri = (ui + vi + ci−1 ) mod β. Figure 3 gives a high-level description of the routine; all logical operators in the figure are bit-wise operators. In each set of four successive phases the operation address computes new offset addresses for ui+1 , vi+1 , and ri+1 , respectively, during the first three phases; in the fourth phase, the operation address is replaced by a loop control operation. The routine consists of 178 lines of assembly code. In-place addition can be performed. Whenever the sum does not fit into the allocated result array, GNU-MP allocates a new array that is just large enough to hold the sum. The NTL library represents integers using a sign-lengthmagnitude representation similar to the one GNU-MP uses. But while GNU-MP allows the digits to have word-length,

Theorem 5. The computing time function of classical Taylor shift by 1 on the set of polynomials Cn,d of Definition 4 is co-dominant with n3 + L(d). Proof. By Theorem 2, an,0 = d + 1 and, for (i, j) ∈ In − {(n, 0)}, ş ť i+j ai,j = . j Hence, by Theorem 3, for any k ∈ {0, . . . , n}, at least half of the integers ak,0 , ak−1,1 , . . . , a0,k have binary length ≥ k/2. Since all of them—except possibly an,0 —have binary length ≤ k, we have that −L(an,0 ) +

n X k X

L(ak−j,j ) ∼ n3 .

k=0 j=0

But the time to compute an,0 is co-dominant with L(d), and so the total computing time is co-dominant with n3 + L(d).

3.

STRAIGHTFORWARD METHODS

We call an implementation of classical Taylor shift by 1 straightforward if it uses a generic integer addition routine to compute one of the following sequences of the intermediate results ai,j . 1. Horner’s scheme—descending order of output coefficients (a0,0 , a0,1 , a1,0 , a0,2 , a1,1 , a2,0 , . . . , a0,n , . . . , an,0 ).

202

IEU1 IEU2 MEM

cycle 1 a ← (ui−1 ∨ vi−1 ) ∧ ¬ri−1 b ← ui−1 ∧ vi−1 load ui+3

cycle 2 a←a∨b address load vi+3

cycle ě 3 ę ci−1 ← a/263 b ← ui + vi mod β store ri−1

cycle 4 ri ← b + ci−1 mod β ui ∨ vi —

Figure 3: The UltraSPARC III has two integer execution units (IEU1, IEU2) and one load/store unit (MEM). The GNU-MP addition routine adds each pair of 64-bit words in a phase that consists of 4 machine cycles. Digit additions are performed modulo β = 264 ; carries are reconstructed from the leading bits of the operands and the result. Pk−1 (j) j (r) P [i + 1] = an−q where j=0 an−q β is the coefficient of n−q x in the input polynomial. After these preparations the Taylor shift can be executed using just the two nested loops of Figure 4. The principal disadvantage of the method is the cost of adding many zero words due to padding. This makes the method impractical for large inputs. Also, the carry computation generates branch mispredictions.

NTL-digits have 2 bits less than a word. As opposed to GNU-MP, NTL needs 1 bit of the word to absorb the carry when it adds two digits. This explains why NTL-digits are 1 bit shorter than GNU-MP-digits. Another bit is lost for the following reason. While GNU-MP represents an integer as a C-language struct, NTL represents it as an array and uses the first array element to represent the signed length of the integer. Since all array elements are of the same type, NTLdigits are signed as well—even though their sign is never used. Finally, due to its way of performing multiplications, NTL cannot take full advantage of a 64-bit word-length. In our experiments on the UltraSPARC III the NTL radix was 230 . The NTL addition routine ntl zadd consists of 113 lines of C++ code.

4.

Step4: /* Apply synthetic division. */ m = k * (n + 1); for (h = n; h >= 1; h--) { c = 0; m = m - k; for (i = 1; i <= m; i++) { s = P[i] + P[i + k] + c; c = 0; if (s >= BETA) { s = s - BETA; c = 1; } else if (s <= -BETA) { s = s + BETA; c = -1; } P[i + k] = s; } }

THE SACLIB METHOD

The SACLIB library of computer algebra programs [8] performs classical Taylor shift by 1 using the routine IUPTR1. The routine, consisting of 144 lines of C-code, is due to G. E. Collins and was originally written for the SAC-2 computer algebra system [5]. Figure 7 shows that the method is faster than straightforward methods for polynomials of small degrees. SACLIB represents integers with respect to a radix β that is a positive power of 2; in our experiments, β = 262 . Integers a such that −β < a < β are called β-digits and are represented as variables of type int or long long. Integers a such that a ≤ −β or β ≤ aPare represented as lists h i (d0 , . . . , dh ) of β-digits with a = i=0 di β where dh 6= 0 and, for i ∈ {0, . . . , h}, di ≤ 0 if a < 0 and di ≥ 0 if a > 0. SACLIB adds integers of opposite signs by adding their digits. None of these digit additions produces a carry. The result is a list (d0 , . . . , dk ) of β-digits that may be 0 and that may have different signs. If not all digits are 0, the non-zero digit of highest order has the sign s of the result. The digits whose sign is different from s are adjusted in a step called normalization. The normalization step processes the digits in ascending order. Digits are adjusted by adding s · β and propagating the carry −s. The routine IUPTR1 performs Taylor shift by 1 of a polynomial of degree n and max-norm d by performing the n(n+ 1)/2 coefficient additions without normalizing after each addition. A secondary idea is to eliminate the loop control for each coefficient addition. To do this the program first computes the bound n + L(d) of Remark 1 for the binary length of the result coefficients. The program determines the number k of words required to store n + L(d) bits. The program then copies the polynomial coefficients in ascending order, and in ascending order of digits, into an array that provides k words for each coefficient; the unneeded highorder words of each coefficient are filled with the value 0. This results in an array P of k(n + 1) entries such that, for i ∈ {0, . . . , k(n + 1) − 1} and i = qk + r with 0 ≤ r < k,

Figure 4: Taylor shift by 1 in SACLIB. The key ideas are the use of signed digits, suspended normalization, and the elimination of loop overhead for each integer addition.

5.

THE TILE METHOD

Our new method outperforms the existing Taylor shift methods by reducing the number of cycles per word addition. The GNU-MP addition routine of Figure 3 requires 4 cycles per word addition. To improve on this we reduce the number of carry computations by using a smaller radix and allowing carries to accumulate inside a computer word. Further, we reduce the number of read and write operations by performing more than one word addition once a set of digits has been loaded into registers. This requires changing the order of operations; only certain digits of the intermediate integer results ai,j in Definition 1 will be computed in one step. We perform only additions; signed digits will implicitly distinguish between addition and subtraction. The technique we use is an instance of register tiling—a computation method that groups the operands, loads them into machine registers, and operates on the operands without referencing the memory [2, 14]. We call our method tile method. The routine consists of 275 hand-written lines of Ccode. In addition, we use a code generator to automatically unroll and schedule some parts of the code, which further improves performance but results in a total number of 848 lines of C-code.

5.1

Description of the algorithm

We partition the set of indices In of Definition 1 as shown in Figure 5 (a).

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Definition 5. Let n, b be positive integers. For non-negative integers i, j let Ti,j = {(h, k) ∈ In | bh/bc = i ∧ bk/bc = j}, and let T be the set of non-empty sets Ti,j .

(r)

(r)

into Si,j and Ei,j for all levels r. Referring to Figure 5 (b) we call the collection of register tile computations for all levels r a register tile stack. The maximum value of r for each stack of index (i, j) depends on the precision of the stack which we define as follows. Definition 8. The precision L∗i,j of the register tile stack with index (i, j) is defined recursively as follows.

Remark 2. The set T is a partition of the set of indices In ; some elements of T can be interpreted as squares of sidelength b, others as triangles and pentagons.

L∗−1,j = max({L(ah,k ) | (h, k) ∈ N0,j }),

Definition 6. Let Ti,j ∈ T . The sets of input indices to Ti,j are

L∗i,−1 = max({L(ah,k ) | (h, k) ∈ Wi,0 }),

Ni,j = {(h, k) ∈ In | h = ib − 1 ∧ bk/bc = j},

L∗i,j = max({L∗i−1,j , L∗i,j−1 }∪{L(ah,k ) | (h, k) ∈ Si,j ∪Ei,j }).

Wi,j = {(h, k) ∈ In | bh/bc = i ∧ k = jb − 1}.

To facilitate block prefetching we place the input digits to a register tile next to each other in memory. We thus have the following interlaced polynomial representation of the polynomial A(x) in Definition 1 by the array P. If i is a non-negative integer such that i = g(n + 1) + f and 0 ≤ f < n + 1 then (g) P[i] contains the value af,−1 defined in Definition 7.

The sets of output indices for Ti,j are Si,j = {(h, k) ∈ In | h = ib + b − 1 ∧ bk/bc = j}, Ei,j = {(h, k) ∈ In | bh/bc = i ∧ k = jb + b − 1}. Remark 3. Clearly, Ni,j = Si−1,j , Wi,j = Ei,j−1 whenever these sets are defined.

Theorem 6. The computation of a register tile requires at most L(β − 1) + 2b − 2 bits for the magnitude of each intermediate result. Proof. Let n = 2b − 1, and let Bn,β−1 (x) be the polynomial defined in Definition 4. For all (h, k) ∈ T0,0 we have 0 ≤ h, k ≤ b − 1. Then, by Theorem 2, L(ah,k ) ≤ L(β − 1) + h + k ≤ L(β − 1) + 2b − 2.

T1,2

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Theorem 7. If L(B) ≥ L(β − 1) + 2b − 2 and 1 bit is available for the sign then the tile method is correct. Remark 4. The UltraSPARC III has a 64-bit word. We use 16 registers, and we let b = 8, β = 249 , and B = 263 .

a. b. Figure 5: a. Tiled Pascal triangle. b. Register tile stack. A register tile is computed for each order of significance. Carries are propagated only along lower and right borders.

5.2

Theorem 8. The tile method has the following properties. 1. Assuming the straightforward method must read all operands from memory and write all results to memory, the tile method will reduce memory reads by a factor of b/2, and memory writes by a factor of b/4.

Definition 7. Let ah,k be one of the intermediate integer results in Definition 1, and let β be an integer > 1. We write X (r) r ah,k = ah,k β , r

2. Since the UltraSPARC processor architecture is capable of concurrent execution of 2 integer instructions and 1 memory reference instruction with a memory reference latency of at least 2 cycles, a b × b register tile compu2 tation takes at least b2 + 7 processor cycles.

ŕ ŕ ŕ (r) ŕ where ŕah,k ŕ < β, and we define, for all i, j, (r)

(r)

Ni,j = {ah,k | (h, k) ∈ Ni,j } (r)

(r)

Properties of the algorithm

(r)

and, analogously, Wi,j , Si,j , and Ei,j .

Proof. (1) Obvious. (2) In the register tile, the addition at the SE-corner must follow the other b2 − 1 additions, and the addition at the NW-corner must precede all other additions. The first addition requires two summands in registers, which takes at least 3 cycles for the first summand and 1 more cycle for the second summand. The last sum needs to be written to two locations; the first write requires 3 cycles and the second 1 more cycle. Since we can per2 form the other b2 − 2 additions in (b 2−2) cycles, the register 2 tile will take at least 3 + 1 + (b − 2)/2 + 1 + 3 = b2 /2 + 7 cycles.

Let I = max{i | Ti,j ∈ T } and J = max{j | Ti,j ∈ T }. The tile method computes, for i = 0, . . . , I and j = 0, . . . , J − i, the intermediate integer results with indices in Si,j ∪ Ei,j from the intermediate integer results with indices in Ni,j ∪ Wi,j . The computation is performed as follows. A register (r) (r) tile computation at level r takes Ni,j and Wi,j as inputs and performs the additions described in Figure 6 (a); the additions are performed without carry but using a radix B > β. Once the register tile computations have been performed for all levels r, a carry propagation transforms the results

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The 8 × 8 register tile computation should take at least 82 /2 + 7 = 39 processor cycles, see Figure 6 (b). By unrolling and manually scheduling the code for the register tile, the code compiled with the Sun Studio 9 C compiler and the optimization options -fast -xchip=ultra3 -xarch=v9b required 53 cycles. When the compiler was used to schedule the unrolled code the computation required 63 cycles.

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1 (L1) on-die caches that use 32-byte blocks. The external Level-2 (L2) cache features a 1- to 8-megabyte unified 2-way set-associative design with a variable block size of 64 bytes to 512 bytes with 64-byte sub-blocks. The Sun Blade 2000 workstation has an 8-megabyte L2 cache. Compilers We wrote all our code in C and compiled it using the Sun Studio 9 [20] compiler with the optimization options -xO3 -xarch=v9b. The optimization options -fast xchip=ultra3 -xarch=v9b yielded slightly slower run times. The Sun compiler generated executable code that was 10%70% faster than code generated by versions 2.95.2 and 3.3.2 of the GNU C/C++ compiler with either of the optimization options -O3 and -O3 -mcpu=ultrasparc3 -m64. We compiled version 4.1.2 of the GNU-MP library using the Sun Studio 7 compiler; we installed the library using the standard installation but substituting CFLAGS by -fast. Performance counter measurements We accessed the performance counters on our processor through the CPClibrary that was provided with the operating system. We monitored processor cycles, instructions, and branch mispredictions as well as cache misses for the L1 instruction cache, the L1 data cache, and the L2 external cache. Execution times were calculated from the number of processor cycles; on our machine, 1 cycle corresponds precisely to 1/900 µs. Before each measurement, we flushed the L1 and L2 data caches by declaring a large integer array and writing and reading it once. We did not flush the L1 instruction cache; our measurements show that its impact on performance is insignificant. We obtained each data point as the average of at least 3 measurements. The fluctuation within these measurements was usually well under 1%. We did not remove any outliers.

31 Cycles

load load add

add store store

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Figure 6: a. A scheduled 8 × 8 register tile. Arrows represent memory references, “+”-signs represent additions. Numbers represent processor cycles. The 2 integer execution units (IEU) perform 2 additions per cycle. b. Register tile: a sketch of the proof of Theorem 8.

5.3

Automatic tuning

The algorithm has several parameters such as tile size b, radix β, and the amount of unrolling and scheduling of various parts of the code. These parameters can be tuned to a particular architecture. On the UltraSPARC III, b = 8 turned out to be the largest register tile size for which the Sun Studio 9 C compiler stored all summands in registers.

Performance - high degrees

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In the RAM-model of computation [1] the tile method is more expensive—with respect to the logarithmic cost function—than straightforward methods. Indeed, by reducing the radix the tile method increases the number of machine words needed to represent integers and hence requires more word additions than straightforward implementations. However, modern computer architectures [12, 4] are quite different from the RAM-model. We show that, on the UltraSPARC architecture, the tile method outperforms straightforward methods by a significant factor—essentially by reducing the number of cycles per word addition. We compare our computing times with those published by von zur Gathen and Gerhard [22, 9]. Hardware and operating system We use a Sun Blade 2000 workstation. The machine has two 900-MHz Sun UltraSPARC III processors, two gigabytes of random-access memory, and it operates under the Solaris 9 operating system. The UltraSPARC III has 32 user-available 64-bit integer registers [13, 21]. Its superscalar architecture provides six 14stage pipelines, four of which can be independently engaged. Two of the pipelines perform integer operations, two floating point operations, one memory access, and one pipeline performs branch instructions. The processor is capable of speculative execution of branch instructions and memory loads. The data cache has 64 kilobytes, the instruction cache 32 kilobytes. Both caches are 4-way set-associative Level-

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Memory traffic reduction Figure 11 shows that the tile method reduces the number of memory reads with respect to the straightforward method by a factor of up to 7. The polynomials Bn,d were used as inputs. The number of memory reads in the GNU-MP-based straightforward method is independent of the compiler since the implementation relies to a large extent on an assembly language routine. However, the number of memory reads in the tile method depends on how well the compiler is able to take advantage of our Ccode for the computation of register tiles. The figure shows that the Sun Studio 9 C-compiler with the option -xO3 xarch=v9b works best for the tile method.

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results grow. Indeed, pre-allocating memory accelerates the straightforward method by a factor of only 1.25 for degree 50. As the degree increases that factor approaches 1. In Figure 8 the polynomials Cn,d reveal a weakness of the tile method. The tile method does not keep track of the individual precisions of the intermediate results ai,j but instead uses the same precision for all the integers in a tile. The tile stack containing the constant term d of C22,d and C25,d consists of 28 and 3 integers ai,j , respectively. Thus, when the degree stays fixed and d tends to infinity, the tile method becomes slower than the straightforward method by a constant factor. The figure shows that—even when the degree is small—the constant term d must become extremely large in order to degrade the performance.

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Figure 10 shows the number of cycles per word addition for the GNU-MP-based straightforward Taylor shift described in Section 3 and for the tile method described in Section 5; the polynomials Bn,d were used as inputs. For large degrees the methods require about 5.7 and 1.4 cycles per word addition, respectively. Since the tile method uses the radix 249 and the straightforward method uses the radix 264 the tile method executes about 64/49 ≈ 1.3 times more word additions than the straightforward method. As a result the tile method should be faster than the straightforward method by a factor of 5.7/(1.4 · 1.3) ≈ 3.1. The measurements shown in Figure 7 agree well with this expectation.

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Branch mispredictions Both the straightforward and tile methods produce at most one branch misprediction every 200 cycles which is not enough to significantly affect performance. However, the branch misprediction rate of the SACLIB method is 60 times greater than that of the straightforward method when the degree is high. Computing times in the literature Von zur Gathen and Gerhard [22, 9] published computing times for the NTLbased implementation of the straightforward method described in Section 3. Tables 1 and 2 quote those computing times and compare the NTL-based straightforward method with the GNU-MP-based straightforward method and the tile method. The computing times we quote were obtained on an UltraSPARC workstation rated at 167 MHz [22] and on a Pentium III 800 MHz Linux PC; the latter experiments were performed using the default installation of version 5.0c of NTL [9]. We installed NTL in the same way on our experimental platform but while the default installation uses the gcc compiler with the -O2 option we used the Sun compiler with the options -fast -xchip=ultra3. This change of compilers sped-up the NTL-based straightforward method by factors ranging from 1.06 to 1.63. Von zur Gathen and Gerhard ran their program letting k = 7, . . . , 13 and n = 2k − 1 for input polynomials of degree n and max-norm ≤ n for Table 1, and max-norm < 2n+1 for Table 2; the integer coefficients were pseudo-randomly generated. We used the same input polynomials in our experiments. The NTL-based straightforward method runs faster on the UltraSPARC III than on the Pentium III, but the speedup ratios vary. This is likely due to differences between the processors in cache size and pipeline organization. The computing time ratios between the NTL- and GNU-MP-based straightforward methods on the UltraSPARC III are more uniform and range between 0.9 and 1.7. If these computing time ratios can be explained by the difference in radix size—230 for NTL and 264 for GNU-MP—then there is no justification for the use of assembly language in the GNUMP-addition routine. Again, the tile method outperforms the straightforward methods.

kindly made his Taylor shift code [22, 9] available. Our research was supported in part by NSF grants ECS-0424475 (W.K.) and ITR/NGS-0325687 (J.R.J.).

8.

Table 1: Computing times (s.) for Taylor shift by 1 —“small” coefficients. degree 127 255 511 1023 2047 4095 8191

straightforward NTL-addition UltraSPARC [22] Pentium III [9] 0.004 0.001 0.019 0.005 0.102 0.030 0.637 0.190 4.700 2.447 39.243 22.126 — 176.840

0.001 0.004 0.016 0.101 0.710 4.958 44.200

tile GMP-add UltraSPARC III 0.00076 0.00010 0.00327 0.00046 0.01475 0.00286 0.08261 0.03183 0.56577 0.27114 3.73049 1.97799 29.91298 18.48445

Table 2: Computing times (s.) for Taylor shift by 1 —“large” coefficients. degree 127 255 511 1023 2047 4095 8191

7.

straightforward tile NTL-addition GMP-add UltraSPARC [22] Pentium III [9] UltraSPARC III 0.006 0.002 0.001 0.00096 0.00016 0.036 0.010 0.005 0.00434 0.00099 0.244 0.068 0.029 0.02154 0.00838 1.788 0.608 0.231 0.17607 0.09183 13.897 8.068 1.773 1.27955 0.83963 111.503 65.758 13.878 9.97772 6.27948 — 576.539 140.630 151.27732 61.04515

ACKNOWLEDGEMENTS

Thomas Decker invented the interlaced polynomial representation to improve the SACLIB method. J¨ urgen Gerhard

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REFERENCES

[1] Alfred V. Aho, John E. Hopcroft, and Jeffrey D. Ullman. The Design and Analysis of Computer Algorithms. Addison-Wesley Publishing Company, 1974. [2] Randy Allen and Ken Kennedy. Optimizing Compilers for Modern Architecture: A Dependence-Based Approach. Morgan Kaufmann Publishers, New York, 2002. [3] Jacques Borowczyk. Sur la vie et l’œuvre de Fran¸cois Budan (1761–1840). Historia Mathematica, 18:129–157, 1991. [4] Randal E. Bryant and David R. O’Hallaron. Computer Systems: A Programmer’s Perspective. Prentice Hall, 2003. [5] G. E. Collins and R. G. K. Loos. Specifications and index of SAC-2 algorithms. Technical Report WSI-90-4, Wilhelm-Schickard-Institut f¨ ur Informatik, Universit¨ at T¨ ubingen, 1990. [6] George E. Collins. The computing time of the Euclidean algorithm. SIAM Journal on Computing, 3(1):1–10, 1974. [7] George E. Collins and Alkiviadis G. Akritas. Polynomial real root isolation using Descartes’ rule of signs. In R. D. Jenks, editor, Proceedings of the 1976 ACM Symposium on Symbolic and Algebraic Computation, pages 272–275. ACM Press, 1976. [8] George E. Collins et al. SACLIB User’s Guide. Technical Report 93-19, Research Institute for Symbolic Computation, RISC-Linz, Johannes Kepler University, A-4040 Linz, Austria, 1993. [9] J¨ urgen Gerhard. Modular Algorithms in Symbolic Summation and Symbolic Integration, volume 3218 of Lecture Notes in Computer Science. Springer-Verlag, 2004. [10] J¨ urgen Gerhard. Personal communication, 2005. [11] Torbj¨ orn Granlund. GNU MP: The GNU Multiple Precision Arithmetic Library. Swox AB, September 2004. Edition 4.1.4. [12] John L. Hennessy, David A. Patterson, and David Goldberg. Computer Architecture: A Quantitative Approach. Morgan Kaufmann, 3rd edition, 2002. [13] Tim Horel and Gary Lauterbach. UltraSPARC-III: Designing third-generation 64-bit performance. IEEE MICRO, 19(3):73–85, 1999. [14] Marta Jim´ enez, Jos´ e M. Llaber´ıa, and Agust´ın Fern´ andez. Register tiling in nonrectangular iteration spaces. ACM Transactions on Programming Languages and Systems, 24(4):409–453, 2002. [15] Maplesoft. Maple 9: Learning Guide, 2003. [16] M. B. Monagan, K. O. Geddes, K. M. Heal, G. Labahn, S. M. Vorkoetter, J. McCarron, and P. DeMarco. Maple 9: Advanced Programming Guide. Maplesoft, 2003. [17] M. B. Monagan, K. O. Geddes, K. M. Heal, G. Labahn, S. M. Vorkoetter, J. McCarron, and P. DeMarco. Maple 9: Introductory Programming Guide. Maplesoft, 2003. [18] Victor Shoup. NTL: A Library for Doing Number Theory. http://www.shoup.net/ntl. [19] Victor Shoup. A new polynomial factorization algorithm and its implementation. Journal of Symbolic Computation, 20(4):363–397, 1995. [20] Sun Microsystems. Sun Studio Collection. http://www.sun.com/software/products/studio/. [21] Sun Microsystems. UltraSPARC III Cu: User’s Manual, 2004. [22] Joachim von zur Gathen and J¨ urgen Gerhard. Fast algorithms for Taylor shifts and certain difference equations. In W. W. K¨ uchlin, editor, International Symposium on Symbolic and Algebraic Computation, pages 40–47. ACM Press, 1997.

On the Complexity of Factoring Bivariate Supersparse (Lacunary) Polynomials* Erich Kaltofen

Pascal Koiran

Dept. of Mathematics North Carolina State University Raleigh, North Carolina 27695-8205, USA

Laboratoire LIP ´ Ecole Normale Superieure de Lyon ´ 46, Allee ´ d’Italie 69364 Lyon Cedex 07, France

[email protected]

[email protected] lacunary† polynomials. A supersparse polynomial

ABSTRACT We present algorithms that compute the linear and quadratic factors of supersparse (lacunary) bivariate polynomials over the rational numbers in polynomial-time in the input size. In supersparse polynomials, the term degrees can have hundreds of digits as binary numbers. Our algorithms are Monte Carlo randomized for quadratic factors and deterministic for linear factors. Our approach relies on the results by H. W. Lenstra, Jr., on computing factors of univariate supersparse polynomials over the rational numbers. Furthermore, we show that the problem of determining the irreducibility of a supersparse bivariate polynomial over a large finite field of any characteristic is co-NP-hard via randomized reductions.

f (X1 , . . . , Xn ) =

t X

α

α

ai X1 i,1 · · · Xn i,n

i=1

is input by a list of its coefficients and corresponding term degree vectors. One defines the size of f as size(f ) =

t “ X

” size(ai ) + ⌈log2 (αi,1 · · · αi,n + 2)⌉ ,

(1)

i=1

where size(ai ) is the bit-size of the scalar coefficients. One thus allows very high degrees, say with hundreds of digits as binary numbers, in distinction to the usual sparse representation [28, 16]. If the coefficients are integers, one cannot evaluate a supersparse polynomial at integer values in polynomial-time in its size, because the value of the polynomial can have exponential size, say 2100 digits. Important exceptions are evaluating at 0 or ±1. A supersparse polynomial can be represented by a straight-line program [13] of size O(sizef ) via evaluating its terms with repeated squaring. It is NP-hard to test if two integral univariate supersparse polynomials have a non-trivial greatest common divisor [22]. A breakthrough polynomial-time result is in [3]. Any integral root of a univariate supersparse polynomial with integral coefficients can be found in (sizef )O(1) bit operations. H. W. Lenstra, Jr., [19, 20] has generalized the result to computing factors of fixed degree in an algebraic extension of fixed degree, in particular to computing rational roots in polynomial-time. Using interpolation and divisibility testing ` a la [1] in connection with Lenstra’s algorithm, in section 3 we present an algorithm for computing linear and quadratic rational factors of integral bivariate (n = 2) supersparse polynomials in (sizef )O(1) bit operations. Our algorithm is randomized of the Monte Carlo kind, and in section 4 we show how the linear bivariate factors can be found deterministically. Several hardness results for supersparse polynomials over finite fields have been derived from Plaisted’s approach [7, 17]. For example, Plaisted’s hardness of GCD 6= 1 extends to polynomials over Zp [7] and can be used to show NPhardness (via randomized reduction) of the irreducibility of supersparse bivariate polynomials for sufficiently large p (cf.

Categories and Subject Descriptors I.1.2 [Computing Methodologies]: Symbolic and Algebraic Manipulation—Algorithms; F.1.2 [Theory of Computation]: Analysis of Algorithms and Problem Complexity—Nonnumerical Algorithms and Problems

General Terms algorithms, theory

Keywords sparse polynomials, lacunary polynomials, multivariate polynomials, polynomial factorizations, polynomial-time complexity, NP-hardness

1. INTRODUCTION The algorithms in this paper take as inputs “super”sparse polynomials, which A. Schinzel and H. W. Lenstra, Jr., call ∗

This material is based on work supported in part by the National Science Foundation under Grant No. CCR-0305314 (Kaltofen).

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. ISSAC’05, July 24–27, 2005, Beijing, China. Copyright 2005 ACM 1-59593-095-7/05/0007 ...$5.00.

†

A lacuna is a hole as in the word ‘lake;’ the polynomials have, so to speak, “lagoons of zero coefficients.”

208

[17, Proof of Theorem 1]). In section 5 we summarize those results and generalize them to finite fields of any characteristic. For all problems that we consider there are deterministic and/or probabilistic algorithms whose bit complexity is of order (size(f ) + deg(f ))O(1) [14, 15]. We remark that our representation of the coefficients of f and the modulus p is by dense vectors of digits, not by supersparse lists of nonzero digits and their positions in the integers (cf. [25]). We note that Barvinok’s representation by short rational generating functions [2] is related to our supersparse representation, and short rational functions have been successfully employed to solve combinatorial counting problems [4].

coefficient bound techniques [6], all factors have coefficients ` ´O(D) of size t + log kf k + log kϕk , which is independent of deg(f ). For example, for ϕ = ζ−1, that is, K = Q, and d = 1 = D, Lenstra’s algorithm finds all rational roots of a supersparse integral polynomial f in polynomial-time in size(f ).

3.

2. THE RESULTS BY CUCKER ET AL. AND LENSTRA

Algorithm Supersparse Factorization P Input: a supersparse f (X, Y ) = ti=1 ai X αi Y βi ∈ Z[X, Y ] that is monic in X and an error probability ǫ = 1/2l . Output: a list of polynomials gj (X, Y ) with degX (gj ) ≤ 2 and degY (gj ) ≤ 2 and corresponding multiplicities, which with probability no less than 1−ǫ are all linear and quadratic irreducible factors of f over Q together with their true multiplicities.

In [3] it is shown how to compute an integer root of a supersparse polynomial f (X) = a1 + a2 X α2 + · · · + at X αt ∈ Z[X] in polynomial time in the size of the polynomial. The result has a short proof based on finding gaps: suppose that f (X) = g(X) + X u h(X) with g 6= 0, h 6= 0, deg(g) ≤ k and let u − k ≥ δ = log2 kf k1 = log2 (|a1 | + · · · + |at |). For an integer a 6= ±1, we have f (a) = 0 =⇒ g(a) = h(a) = 0. Assume the contrary, namely that a 6= 0, ±1 and h(a) 6= 0. Then |g(a)| < kf k1 · |a|k ≤ 2u−k · |a|k ≤ |a|u ≤ |au h(a)|,

Step 0. Factor out the maximum powers of X and Y that divide f . The non-zero coefficients of f do not change. Compute all linear and quadratic irreducible factors of f that are in Q[Y ] by applying Lenstra’s method to the coefficients of X αi . The multiplicities are also provided by Lenstra’s algorithm.

(2)

thus |f (a)| ≥ |au h(a)| − |g(a)| > 0. Note the similarity of (2) with the proof of Cauchy’s root bound. The estimate for δ can be sharpened [3, Proposition 2]. The polynomial time algorithm can now proceed by computing the integer roots of those polynomial segments ai X αi +· · ·+aj X αj in f whose terms have degree differences αl − αl−1 < δ, for all i < l ≤ j. After dividing out X αi , we have polynomials of degree ≤ (t − 1)(δ − 1), whose common integer roots are found by p-adic lifting [21]. In section 4 we give a variant of the gap technique for high degree sums of linear forms. H. W. Lenstra, Jr. has used the gap method to computing rational roots and low degree factors of supersparse rational polynomials via the height of an algebraic number (see section 4). The algorithm presented in P[19] receives as input a supersparse polynomial f (X) = ti=1 ai X αi ∈ K[X], where the algebraic number field K is represented as K = Q[ζ]/(ϕ(ζ)) with a monic irreducible minimum polynomial ϕ(ζ) ∈ Z[ζ]. Furthermore, a factor degree bound d is input. The algorithm produces a list of all irreducible factors of f over K of degree ≤ d and their multiplicities. Let D = d · deg(ϕ). There are at most 2

D

O(t · 2 · D · log(2D t))

LINEAR AND QUADRATIC FACTORS

We now present our randomized algorithm for computing linear and quadratic factors (and their multiplicities) of bivariate supersparse polynomials. For simplicity, we shall consider polynomials with rational coefficients only, although our method would allow coefficients in an algebraic number field. Our algorithm calls the univariate algorithm by H. W. Lenstra, Jr. [19]. For simplicity (see Remark 1 below), we assume that the input polynomial is monic in X.

Step 1. Compute all linear and quadratic irreducible factors in Q[X] of f (X, 0), f (X, 1) and f (X, −1) by Lenstra’s method. The algorithm will also provide the multiplicities of those factors. Step 2. Interpolate all factor combinations. Test if a factor candidate g(X, Y )µ of candidate multiplicities µ divides f (X, Y ) by testing if 0 ≡ f (X, a) mod (g(X, a)µ , p) where a ∈ S ⊂ Z, and p ≤ B a prime integer are randomly selected. The cardinality |S| of S and the bound B are chosen in dependence of f and the input error probability ǫ (see below). The algorithm may fail to sample a prime p ≤ B and return “failure,” which is interpreted as an incorrect answer in the output specification of the probability of correctness. 2 We now show that our algorithm Supersparse Factorization can be implemented as to run in ` ´O(1) t + log(deg f ) + log kf k + log 1/ǫ (5)

(3)

irreducible factors of degree ≤ d [20, Theorem 1], each of which, with the exception of the possible factor X, has multiplicity at most t [19, Proposition 3.2]. The algorithm finishes in ` ´O(D) t + log(deg f ) + log kf k + log kϕk (4)

bit operations. Note that the measure (5) is polynomial in size(f ) and l = − log ǫ. By (3) in section 2 and our restriction to D ≤ 2, the polynomials f (X, 0), f (X, −1) and f (X, 1) each have no more than O(t2 log t) linear or irreducible quadratic factors. In Step 2, one interpolates factors that are monic in X and whose coefficients have size (t + log kf k)O(1) . There are O(t4 (log t)2 ) combinations of linear factors and O(t12 (log t)6 ) combinations of quadratic factors, the latter because we must also consider products of univariate linear factors as

bit operations. Here kϕk is the (infinity) norm of the coefficient vector of ϕ and kf k is the norm of the vector of norms of the coefficients ai (ζ). We assume that a common denominator has been multiplied through and all coefficients of the ai (ζ) are integers. We note that by standard factor

209

images of bivariate quadratic factors. In practice, of course, the number of combinations can be much smaller. At least one of the univariate factors in each combination is 6= X in the linear case and 6= X 2 in the quadratic case, because the interpolated bivariate factor cannot be X or X 2 . Therefore the multiplicity m of one of the univariate factors satisfies m ≤ t, and we need to check all µ ≤ m. For each candidate factor G(X, Y ) = g(X, Y )µ we consider the division with remainder in X, f (X, Y ) − q(X, Y )G(X, Y ) = h(X, Y ), where degX (h) < degX (G).

The bit complexity measure (5) follows from the bounds (9) and (10) together with the repeated squaring algorithm and a polynomial primality test used in Step 2. Remark 1. Our algorithm can be extended to compute in polynomial time all irreducible factors gj with degX (gj ) = O(1), i.e., of constant degree in X, and simultaneously of degY (gj ) ≤ 2. The input condition of monicity of f can be relaxed to accept polynomials with a leading coefficient (or trailing coefficient) in X that does not vanish for Y = 0, Y = −1 or Y = 1. One imposes a factor of the leading coefficient on the interpolated polynomials, which is a technique from sparse Hensel lifting [11]. One may also switch the roles of X and Y . However, at this time we do not know at all how to interpolate the factors of polynomials such as X 2d (11) (X i − 1)(Y 2ei − 1)fi (X, Y )

(6)

By considering (6) as a (unimodular) linear system over Q(Y ) with degX (f ) + 1 equations and variables, we obtain bounds for degY (h) and khk [8]: degY (h) ≤ degY (f ) + degY (G)× (degX (f ) + 1 − degX (G)) = O(t deg(f ))

i

(7)

where the fi are supersparse.

and However, in the next section, we can show how to compute in deterministic polynomial time all factors of total degree 1 of any supersparse bivariate rational polynomial, including those of the form (11).

khk2∞ ≤ t · kf k21 × ((degX (G) + 1) · kGk21 )degX (f )+1−degX (G) .

(8)

From (7) and ǫ we derive a bound for |S| in Step 2, and from (8) and ǫ a bound for B in Step 2. Suppose G does not divide f , that is there is a coefficient hi (Y ) 6= 0 of X i in h. First, we wish to have 0 6= hi (a) with probability ≥ 1 − η/3, where η = ǫ/A with A = O(t13 (log t)6 ) being the number of factor combinations and multiplicities that have to be tested. The probability to pick a root of hi (Y ) among the elements in S ⊂ Z is no more than degY (h)/|S|. By (7), for a set S of cardinality |S| = (t + deg f + 1/ǫ)O(1)

4.

(9)

we can succeed with probability ≥ 1 − η/3. Let H = hi (a) for a ∈ S. We get by (9) and again by (7) and (8) that H = (t + deg f + kf k + 1/ǫ)O(t deg f ) . Second, we choose B such that 0 6≡ H (mod p) with probability ≥ 1 − η/3. By facts on the prime number distribution (see [23] for explicit estimates), there is a constant γ1 such that H has at most γ1 log H/ loglog H distinct prime factors. Since there are no fewer than γ2 B/ log B primes ≤ B, the probability that 0 ≡ H (mod p) is no more than γ3 (log H/ loglog H)/(B/ log B) for some constants γ2 and γ3 . Because one has γ3

t X

aj X αj (a + bX)βj

(12)

j=0

is identically equal to zero. Here a and b and the aj are rational numbers; the αj and βj are non-negative integers. This algorithm can be used to check with certainty whether a “candidate factor” Y − bX − a (for instance generated by an interpolation technique asPin section 3) really is a factor of the bivariate polynomial j aj X αj Y βj . In general, deciding deterministically whether a straight-line program computes the identically zero polynomial is a notorious open problem. It turns out, however, that for polynomials of the form (12) this problem has an efficient solution. We will then see that this verification algorithm can be easily converted into an algorithm that actually finds all linear factors. Even though our input polynomials have rational coefficients as in the remainder of the paper, the results of this section rely heavily on algebraic number theory.‡ We review the necessary material in section 4.1. A suitable gap theorem is established in section 4.2. Here, some crucial ideas are borrowed from Lenstra’s [19] paper. In particular, Proposition 1 closely follows Proposition 2.3 of [19]. Finally, our deterministic algorithms are presented in section 4.3.

log H/ loglog H 3A log H ǫ B ≤ ⇐⇒ γ3 ≤ , B/ log B 3A ǫ loglog H log B

one may choose B = (A · log H · 1/ǫ)O(1)

DETERMINISTIC LINEAR FACTORS

In this section we give a deterministic polynomial time algorithm that finds the linear factors of a supersparse polynomial. In contrast to the randomized algorithm of section 3, this deterministic algorithm can handle all (bivariate) supersparse polynomials. Our approach is based on the observation that a polynomial g(X, Y ) is divisible by Y − bX − a iff g(X, a + bX) = 0. We will first give an algorithm that decides whether a polynomial of the form

(10)

and achieve failure probability ≤ η/3. Note that the number of digits in p is of order (t+log(deg f )+log kf k+log 1/ǫ)O(1) . The algorithm must succeed to pick a prime p ≤ B. By iterating the prime selection process O(log(A/ǫ) · log B) times we can assume that to happen with probability ≥ 1 − η/3. Thus a single false factor combination is eliminated with probability ≥ (1 − η/3)3 ≥ 1 − 1/η. Therefore no false factor combination or multiplicity is accepted with probability ≥ (1 − η)A ≥ 1 − Aη ≥ 1 − ǫ.

‡

It is an interesting open problem whether they have more elementary proofs such as the one given in section 2.

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4.1 Heights of algebraic numbers

A nice feature of H(f ) is its invariance by scalar multiplication: if λ ∈ Q \ {0}, H(λf ) = H(f ). Indeed, if we multiply a polynomial by pα where p is prime, the archimedean absolute value is multiplied by pα and the p-adic absolute value is divided by pα . The other absolute values are unchanged. Note also that H(f ) = maxj |aj | if the aj are relatively prime integers. Computing H(f ) in the general case aj ∈ Q is therefore quite easy: reduce to the same denominator to obtain integer coefficients, divide by their gcd and take the maximum of the absolute values of the resulting integers (so in particular H(f ) ∈ Z>0 for any f ). Finally, a word of caution: our notion of height is not intrinsic to the given polynomial in X, since it is not invariant of the particular representation (12). Given a bivariate polynomial g(X, Y ) one could, however, define an intrinsic height H(G) as done above (i.e., as the projective height of its tuple of coefficients), and we would have H(f (X, a + bX)) = H(G).

In this section we quickly recall some number theoretic background. For any prime number p, the p-adic absolute value on Q is characterized by the following properties: |p|p = 1/p, and |q|p = 1 if q is a prime number different from p. For any x ∈ Q \ {0}, |x|p can be computed as follows: write x = pα y where p is relatively prime to the numerator and denominator of y, and α ∈ Z. Then |x|p = 1/pα (and of course |0|p = 0). We denote by MQ the union of the set of p-adic absolute values and of the usual (archimedean) absolute value on Q. Let d, e ∈ Z be two non-zero relatively prime integers. By definition, the height of the rational number d/e is max(|d|, |e|). There is an equivalent definition in terms of absolute Q values: for x ∈ Q, H(x) = ν∈MQ max(1, |x|ν ). Note in particular that H(0) = 1. More generally, let K be a number field (an extension of Q of finite degree). The set MK of normalized absolute values is the set of absolute values on K which extend an absolute value of MQ . For ν ∈ MK , we write ν|∞ if ν extends the usual absolute value, and ν|p if ν extends the p-adic absolute value. One defines a “relative height” HK on K by the formula Y HK (x) = (13) max(1, |x|ν )dν .

Theorem 1. Let f (X) be a polynomial of the form (12) where (a, b) is a pair of rational numbers different from the five pairs (0, 0), (±1, 0), (0, ±1). Assume without loss of generality that the sequence (βj ) is nondecreasing, and assume also that there exists l such that βl+1 − βl > log(t H(f ))/ log κ, where κ > 1 is an absolute constant definedPin Lemma 2. If f is identically zero, the polynomials g = lj=0 aj X αj (a + P bX)βj and h = tj=l+1 aj X αj (a+bX)βj are both identically zero.

ν∈MK

Here dν is the so-called P “local degree”. For every p (either : Q]. Sometimes, instead prime or infinite), ν|p dν = [KQ of (13) one just writes HK (x) = ν max(1, |x|ν ) if it is understood that each absolute value may occur several times (in fact, dν times) in the product. The absolute height H(x) of x is HK (x)1/n , where n = [K : Q]. It is independent of the choice of K. There is a nice connection between the height of algebraic numbers and the Mahler measure of polynomials. Recall that the Mahler measure M (f ) of a polynomial of degree d: f (X) = ad X d + · · · + a1 X + a0 = ad

d Y

Proof. Let U (a, b) be the set of roots of unity defined in Lemma 2 below. By hypothesis, f (θ) = 0 for each θ ∈ U (a, b). By Proposition 1 below, g and h are both identically zero on U (a, b). The result follows since U (a, b) is an infinite set. We denote by U the set of complex roots of unity of prime order, and by U≥5 the set of complex roots of unity of prime order ≥ 5. Lemma 1. There is an absolute constant κ1 > 1.045 such that the following holds. For any θ ∈ U≥5 , if a ∈ Z \ {0} and b ∈ Z \ {0} then H(a + bθ) ≥ κ1 .

(X − αi )

i=1

Q is by definition equal to |ad | di=1 max(1, |αi |). It turns out that if α is an algebraic number of degree d and f ∈ Z[X] its minimal polynomial, M (f ) = H(α)d ([27], section 3.3). This connection is often helpful when one has to estimate heights, but here we will use directly the definition of height in terms of absolute values. In Proposition 1 we will also use the product formula: Y (14) |x|dνν = 1

Remark 2. The hypothesis that θ is of order at least 5 is necessary. Indeed, if θ is of order 3 then H(1 + θ) = 1 since 1 + θ = −θ2 . Moreover, the restriction to roots of prime order can probably be removed with some additional work. Proof of Lemma 1. Note that |a + bθ|ν ≤ 1 for any ultrametric absolute value. Indeed, |a + bθ|ν ≤ max(|a|ν , |bθ|ν ) = max(|a|ν , |b|ν ).

ν∈MK

for any x ∈ K \ {0}. More details on absolute values and height functions can be found for instance in [18] or [27].

Hence we only need to take the archimedean absolute values into account to estimate the height. Recall that if θ is of order d, its conjugates are the other roots of unity of order d. Hence

4.2 A gap theorem We define a notion of height for an expression of the form (12) by the formula Y H(f ) = |f |ν ,

H(a + bθ)d−1 =

d−1 Y

max(1, |a + be2ikπ/d |).

k=1

Assume first that a and b are of the same sign, and for instance positive. Then |a + be2ikπ/d | ≥ a + b cos(2ikπ/d) ≥ 1 + cos(2π/5) if k ≤ d/5. Hence

ν∈MQ

where |f |ν = max0≤j≤t |aj |ν . There is a classical notion of height for a point in projective space ([10], section B.2) and in fact H(f ) is simply the height of the point (a0 , a1 , . . . , at ).

H(a + bθ) ≥ (1 + cos(2π/5))⌊d/5⌋/(d−1) .

211

This lower bound is always > 1.045 since d ≥ 5, and its limit as d → +∞, which is equal to (1 + cos(2π/5))1/5 , is > 1.055. To complete the proof, we now consider the case where a and b have opposite signs. Assume for instance that a ≥ 1 and b ≤ −1. Then |a + be2ikπ/d | ≥ a + b cos(2ikπ/d) ≥ 3/2 if d/3 ≤ k ≤ 2d/3. Hence H(a + bθ) ≥ (3/2)⌊d/3⌋/(d−1) . This lower bound is again always > 1.10 and its limit as d → +∞, which is equal to (3/2)1/3 , is > 1.14.

Let n ∈ U (a, b). Using again the binomial formula, it follows from (15) that ` ´ y n + ncy n−1 + n2 c2 y n−2 + · · · + ncn−1 y = dn − cn .

Since |dn − cn |ν = 1, we must have |y|ν ≥ 1 (so that in fact |y|ν = 1). We conclude that H(x) ≥ p ≥ 2 if θ ∈ U (a, b). (iii) The last subcase occurs when c 6= d and p | d − c. We can write y = dθ −c = c(θ −1)+(c−d). By hypothesis |c − d|ν ≤ 1/p, and by subcase (i) |c(θ − 1)|ν = 1 if θ belongs to the set U (a, b) defined in that subcase. We may therefore take the same U (a, b), and we conclude again that H(x) ≥ 2 if θ ∈ U (a, b).

We now deal with the case where a and b are rational numbers. Lemma 2. There is an absolute constant κ > 1.045 such that the following holds: for any pair (a, b) of rational numbers, different from the 5 excluded pairs of Theorem 1, there exists an infinite set U (a, b) of roots of unity such that H(a+ bθ) ≥ κ for any θ ∈ U (a, b).

We have shown that H(x) ≥ 2 whenever θ ∈ U (a, b) and a 6∈ Z \ {0}, b 6∈ Z \ {0}, a = 0 or b = 0. One may therefore take κ = min(2, κ1 ) (so in fact κ = κ1 ).

Proof. Let (a, b) be a pair of rational numbers different from the 5 excluded pairs. If b = 0, H(a + bθ) = H(a) ≥ 2 since a 6∈ {−1, 0, 1}. If a = 0, H(a + bθ) = H(bθ) = H(b) ≥ 2 since b 6∈ {−1, 0, 1} (indeed, for any ν we have |bθ|ν = |b|ν |θ|ν = |b|ν ). One may therefore take for U (a, b) the set of all roots of unity if a = 0 or b = 0. Also, we have shown in Lemma 1 that one may take U (a, b) = U≥5 if a ∈ Z \ {0} and b ∈ Z \ {0}. We therefore assume for the remainder of the proof that a and b are both nonzero, and that they are not both integers. By reduction to the same denominator one finds integers c, d, e ∈ Z\{0} such that e ≥ 2, gcd(c, d, e) = 1, and a+bθ = (dθ − c)/e for any root of unity θ. Let x = a + bθ, let p be a prime factor of e, and fix any ν such that ν|p. Since |x|ν ≥ p|y|ν where y = dθ − c, it remains to lower bound |y|ν (note that we have the upper bound |y|ν ≤ 1). If θ is a n-th root of unity, we have (y + c)n = dn .

Proposition 1. Let (a, b) be a pair of rational numbers different from the five excluded pairs of Theorem 1. Let f be a polynomial of the form (12), and let k ≥ 1 be an integer. Write f = g + h where g collects all the terms of f with βj ≤ k and h collects all the terms of f with βj > k. Let u = min{βj ; βj > k}. Assume that θ is a zero of f , and that θ belongs to the set U (a, b) of Lemma 2. If u − k > log(t H(f ))/ log κ,

(16)

where κ is as in Lemma 2, then θ is a common zero of g and h. Proof. We may assume that each of the two polynomials g and h collects at most t of the t + 1 terms of f (otherwise, the result is clear). Assume by contradiction that g(θ) 6= 0. Let K = Q[θ] and ν ∈ MK . If |a + bθ|ν ≥ 1, each term of g(θ) satisfies |aj θαj (a + bθ)βj | ≤ |f |ν |a + bθ|kν , therefore

(15)

|g(θ)|ν ≤ max(1, |t|ν )|f |ν |a + bθ|kν if |a + bθ|ν ≥ 1.

We first assume that p divides c. In this case p cannot divide d since gcd(c, d, e) = 1. Hence (15) implies that |y|ν = 1, so that |x|ν ≥ p. Since this is true for any ν such that ν|p, we have H(x) ≥ p ≥ 2. If p divides c, one may therefore take U (a, b) equal to the set of all roots of unity. We now examine the case p ∤ c. We assume that θ 6= 1 is a n-th root of unity, and distinguish 3 subcases.

A similar argument shows that |h(θ)|ν ≤ max(1, |t|ν )|f |ν |a + bθ|uν if |a + bθ|ν ≤ 1. We have |g(θ)|ν = |h(θ)|ν , so we can combine these two statements in max(1, |a + bθ|ν |)u−k · |g(θ)|ν ≤ max(1, |t|ν ) · |f |ν · |a + bθ|uν .

(i) If c = d, we shall see that |y|ν = 1 whenever p ∤ n. Indeed, |y|ν = |c|ν |θ − 1|ν = |θ − 1|ν . Set z = θ − 1. Since (z + 1)n = 1 and z 6= 0, it follows from the binomial formula that ` ´ ` ´ z n−1 + nz n−2 + n2 z n−3 + · · · + n2 z = −n.

Raise this to the power dν /[K : Q] and take the product over ν ∈ MK . Using the fact that H(t) = t, and applying (14) to g(θ) and a+bθ (which are both supposed to be nonzero) one finds that H(a + bθ)u−k ≤ t · H(f ). However, H(a + bθ) ≥ κ by Lemma 2. This is in contradiction with (16).

4.3

Deterministic algorithms

Hence |z|ν = 1 since |n|ν = 1. We conclude that H(x) ≥ p ≥ 2, and one may take U (a, b) equal to the union for all integers n such that p ∤ n of the set of n-th roots of unity different from 1.

Theorem 2. We have a polynomial-time deterministic algorithm for deciding whether a polynomial of the form (12) is identically zero.

(ii) The second subcase (c 6= d and p ∤ d − c) is similar, but slightly more involved. Let U (a, b) be the set of positive integers n such that p ∤ (dn − cn ). Note that U (a, b) is infinite since n ∈ U (a, b) or n + 1 ∈ U (a, b) for any n ≥ 1 , as is shown as follows: assume the contrary, namely that p | dn+1 − cn+1 and p | dn − cn . It follows that p | (dn+1 −cn+1 )−d(dn −cn ) = cn (d−c). This is impossible since p ∤ c.

Note that there is a trivial algorithm which deals with the case where (a, b) is one of the five excluded pairs of Theorem 1. In the following we therefore assume that (a, b) is not one of these five excluded pairs, and we fix a rational number ǫ > 0 such that one may take κ = 2ǫ in Lemma 2. Set δ = ⌈n/ǫ⌉, where n is the unique integer such that 2n−1 ≤ t H(f ) < 2n . Assume that the βj ’s are sorted by nondecreasing order as in Theorem 1. There is a unique

212

integer s ≥ 1 and a unique partition of the set {0, 1, . . . , t} in subsets U1 , . . . , Us of consecutive integers with the following property: if an integer j belongs to Ul then j + 1 also belongs to Ul if βj+1 < βj + δ, otherwise j + 1 belongs to Ul+1 (to obtain this partition, just sweep the list of the βj ’s from left to right and create a new subset whenever P an element βj such that βj+1 − βj ≥ δ is found). P Let fl = j∈Ul aj X αj (a + bX)βj . By construction f = sj=1 fl and by Theorem 1, f is identically zero iff all the fl are identically zero. Indeed, we have δ > log(t H(f ))/ log κ, where κ = 2ǫ . Furthermore, we can write fl = (a + bX)γl gl where X gl = aj X αj (a + bX)δj,l , (17)

bX − a. There are five special cases for the pair (a, b), which correspond to the five excluded pairs of Theorem 1. As pointed out in the proof of Theorem 2, one can check easily for each of these five pairs whether g(X, a + bX) = 0. In the following we therefore look for factors Y −bX −a where (a, b) is different from the five excluded pairs. As in Theorems 1 and 2, we assume that the βj are sorted by nondecreasing order. The idea is to use Theorem 1 to reduce this problem to several factoring problems about dense polynomials. Let U1 , . . . , Us be the partition of the set of indices {0, 1, . . . , t} which is constructed when the algorithm of Theorem 2 is run on the polynomial f (X) = g(X, a + bX). Crucially, this partition is in fact independent of the pair P(a, b). As in the proof of Theorem 2, one can write g = sj=1 Y γl gl , P αj δj,l where gl = Y , γl = min{βj ; j ∈ Ul }, and j∈Ul aj X δj,l = βj − γl . By Theorem 1, the linear factors of g are (excluding excluded pairs!) the common linear factors of the gl . We have therefore reduced our initial problem to the computation of the linear factors of each gl . This progress is significant since, as shown in the proof of Theorem 2, in every gl the exponents δj,l of variable Y are “small” (polynomially bounded in the size of the input polynomial g). The exponents of X may still be large, however. To deal with this problem we run the same factoring algorithm on input gl instead of g, with the roles of variables X and Y interchanged. This reduces the problem to the computation of the linear factors of polynomials where the exponents of X and Y are all “small”. One can then use any deterministic polynomial time algorithm that finds the linear factors of a dense polynomial.

j∈Ul

γl = min{βj ; j ∈ Ul }, and δj,l = βj − γl . Each exponent δj,l satisfies 0 ≤ δj,l < δ. The gl are all identically zero iff f is identically zero. We can now describe our main algorithm. 1. Compute H(f ) as explained before Theorem 1 and the integer δ defined above. 2. Construct the list (g1 , . . . , gs ) defined by (17). 3. Express each polynomial (a + bX)δj,l as a sum of powers of X. 4. Substitute in (17) to express each gl as a sum of powers of X, and decide whether the gl are all identically zero. If so, output “f = 0”. Otherwise, output “f 6= 0”. The correctness of this algorithm follows from the discussion after Theorem 2, and it is clear that steps 1 and 2 run in polynomial time. Step 3 also runs in polynomial time since δj,l < δ and δ is bounded by a polynomial in the input size (so we can simply expand (a + bX)δj,l by brute force). Finally, in step 4 we express gl as a sum of at most δ|Ul | ≤ δ(t + 1) terms. This completes the proof of the running time estimate, and of Theorem 2.

5.

In [22] and the earlier papers cited there, NP-hardness results are derived for supersparse polynomials over the integers. In [7, 17] several hard problems are extended for supersparse polynomials over finite fields. We give similar NP-hard problems over finite fields, but now for finite fields of arbitrary characteristic. Figure 1 shows D. Plaisted’s model for 3-SAT in n Boolean variables z1Q , . . . , zn . Clauses correspond to factors of X N −1 with N = n j=1 pj , where pj distinct primes. We note that all Poly(Ci ) are supersparse polynomials for any clause Ci with one, two or three literals. An immediate consequence of the construction, justified by the rootsets in Figure 1 that are associated with the polynomials, is that the conjunctive normal form C1 ∧ · · · ∧ Cs is satisfiable if and only if GCD(Poly(C1 ), . . ., Poly(Cs )) 6= 1. We first generalize that reduction to coefficients from an arbitrary field. Let p ∤ N be a fresh prime and let Y ΨN (y) = (X − e2bπi/N ) ∈ Z[y]

Remark 3. One can deal at no additional expense with polynomials of the slightly more general form: f (X) =

t X

NP-HARDNESS OF SUPERSPARSE BIVARIATE IRREDUCIBILITY

aj (c + dX)αj (a + bX)βj .

j=0

Indeed, the change of variable Y = c + dX yields a polynomial g(Y ) of form (12). The only case which cannot be handled in this way is the seemingly trival one b = d = 0. Here one has to decide whether the rational number P t αj βj a is equal to zero. It is not clear whether this j=0 aj c can be done in deterministic polynomial time, even if a, c and the aj are all integers. Theorem 2 yields a deterministic algorithm for Step 2 with µ = 1 in algorithm Supersparse Factorization in Section 3. However, that variant still fails to compute linear factors of polynomials of the form (11), which we remedy next.

1≤b
be the cyclotomic equation of order N . Since ζ = (y mod ΨN (y)) is a representation for a primitive N -th root of unity, we have over the integers

Theorem 3. We have a polynomial-time deterministic algorithm that finds Pall linear factors of a supersparse polynomial g(X, Y ) = tj=0 aj X αj Y βj .

X N −1 ≡ (X −y 1 )(X −y 2 ) · · · (X −y N ) mod (ΨN (y)). (18)

Proof. We first find all linear factors of f that are in Q[X] by applying Lenstra’s method to the coefficients of Y βj (this is similar to step 0 of the algorithm of section 3). After that, it remains to find all factors of the form Y −

Let Fq ⊇ Zp be the splitting field of ΨN (y) mod p (one has q = pλ where λ is the multiplicative order of p modulo N ) and let ζ ∈ Fq with ΨN (ζ) = 0 in Fq . Taking (18) modulo p

213

Formula

Polynomial

Rootset

zj

X N/pj − 1 N X − 1 = Ppk −1 X iN /pk i=0 X N/pk − 1

{(e2πi/N )a | a ≡ 0 (mod pj )}

¬zk L1 ∨ L2 ∨ L3 zj ∨ zk ∨ zl zj ∨ ¬zk ∨ zl ¬zj ∨ ¬zk ∨ zl ¬zj ∨ ¬zk ∨ ¬zl C1 ∧ · · · ∧ Cs

LCM(Poly(L1 ), Poly(L2 ), Poly(L3 )) N/pj (X − 1)(X N/pk − 1)(X N/pl − 1)(X N/(pj pk pl ) − 1) N/(pk pj ) (X − 1)(X N/(pk pl ) − 1)(X N/(pj pl ) − 1) N/(pj pk ) (X − 1)(X N − 1)(X N/(pk pl ) − 1) N/pk (X − 1)(X N/(pj pk pl ) − 1) N (X − 1)(X N/(pj pk pl ) − 1) X N/(pj pk ) − 1 XN − 1 N/(pj pk pl ) X −1 GCD(Poly(C1 ), . . . , Poly(Cs ))

{(e2πi/N )b | b 6≡ 0 (mod pk )} S3

Roots(Lj )

Ts

Roots(Ci )

j=1

i=1

Figure 1: D. Plaisted’s polynomials for literals, clauses and CNFs (N =

j=1

pj ).

polynomials in K[X], and the set of irreducible supersparse polynomials in K[X, Y ] are co-NP-hard under randomized reduction for K = Q and K = Fq with arbitrary p and sufficiently large q = pm . Co-NP-hardness of irreducibility remains valid if we assume that the supersparse bivariate polynomials are monic in X.

and evaluating the resulting polynomial identity at y = ζ, that is, taking it modulo y − ζ | ΨN (y), one obtains X N − 1 = (X − ζ 1 )(X − ζ 2 ) · · · (X − ζ N ) in Fq [X].§ (19) Since p ∤ N , the polynomial X N − 1 mod p has no multiple roots, and we can replace e2πi/N by ζ in D. Plaisted’s construction. We thus have established the following.

Proof. Reduction to two polynomials follows from Proposition 2 and Lemma 3 (cf. [7]). NP-hardness of P squarefreeness follows by considering the product f1 (f2 + si=3 ci fi ). Since D. Plaisted’s polynomials fi = Poly(Ci ) are divisors of X N − 1 and therefore are squarefree, with high probability both factors will be squarefree. Therefore their product is not squarefee if and only if the two factors have a common GCD, which is NP-hard under randomized reduction. NP-hardness of irreducibility follows by considering the polP ynomial F (X, Y ) = X u f1 (X) + Y (f2 (X) + X si=3 ci fi (X)) for sufficiently large u to make F (X, Y ) monic in X (cf. [17, Proof of Theorem 1]). Note that D. Plaisted’s polynomials fi = Poly(Ci ) are not divisible by X. Shifting f3 , . . . , fs by a factor of X ensures that f2 + c3 Xf3 + · · · + cs Xfs is relatively prime to X u . Clearly, F (X, Y ) is irreducible if and P only if GCD(f1 , f2 + X si=3 ci fi ) = 1, that if and only if GCD(f1 , . . . , fs ) = 1 with high probability.

Proposition 2. The set of tuples of relatively prime supersparse polynomials in Fq [X] is co-NP-hard for arbitrary q = pµ . In [22], co-NP-hardness is shown for pairs of supersparse relatively prime polynomials over Z. However, D. Plaisted’s pairs do not remain relatively prime modulo all primes p. We overcome that deficiency via randomized reductions. Lemma 3. Let fi (X) ∈ K[X] be nonzero polynomials for i = 1, . . . , s, s ≥ 2, K a field, d = deg(f1 ) and S ⊂ K. Then for randomly chosen ci ∈ S, 3 ≤ i ≤ s, we have the probability estimate s “ ” X ‹ Prob GCD(fi ) = GCD(f1 , f2 + ci fi ) ≥ 1 − d |S|. 1≤i≤s

Qn

i=3

(20) Furthermore, if f2 is squarefree and e = deg(f2 ) ≥ deg(fi ) for i ≥ 3, then with probability no less than 1 − (2e − 1)/|S| P the polynomial f2 + si=3 ci fi will remain squarefree.

For example, co-NP-hardness of supersparse bivariate irreducibility yields via [1] the following reduction to integer factoring.

Proof. The estimate (20) is Lemma 2 in [5]. Squarefreeness follows by the same techniques as P given there by considering the discriminant of F = f2 + si=3 yi fi as a Sylvester resultant of F and ∂F/∂X with symbolic yi .

Corollary 1. Suppose we have a Monte Carlo polynomial-time irreducibility test for supersparse polynomials in F2m [X, Y ] for sufficiently large m. Then large integers can be factored in Las Vegas polynomial-time.

We obtain the following NP-hardness problems under randomized reduction, which generalize the results in [17] to arbitrary characteristic.

Already in [17], Hilbert irreducibility is mentioned as a means to establish NP-hardness of irreducibility of supersparse polynomials in Z[X]. Note that one can evaluate the linear variable Y at a polynomial-sized integer. However, no proven effective versions seem to be known that would yield a randomized polynomial reduction (cf. [26, 24]). Nonetheless, if a fast irreducibility test of supersparse polynomials in Z[X] were discovered, we believe that Hilbert irreducibility would yield fast algorithms for NP-complete problems,

Theorem 4. The set of pairs of relatively prime supersparse polynomials in K[X], the set of squarefree supersparse §

With N = pµ − 1, 1 ≤ µ, we have proven that the multiplicative group of a finite field Fpµ is cyclic. For that proof the only property of ΨN needed is that it is the monic integral minimum polynomial of a primitive N -th root of unity in C.

214

thus resulting in what we call a “good heuristic” for NPcompleteness [15]. Of course, the Hilbert irreducibility theorem is not valid over Fq and the hardness of supersparse irreducibility in Fq [X] remains open. In addition, the complexity of root finding of supersparse polynomials over finite fields is open. In [15], we have posed two open problems: Given a prime number p and integers b, c ∈ Zp and α, β with with p − 1 > α > β > 0, compute a ∈ Zp such that aα + baβ + c ≡ 0 (mod p) in (log p)O(1) bit operations. Alternatively, prove that computing a root in Zp of a polynomial given by straight-line program over Zp is NP-hard. Acknowledgments: Pascal Koiran would like to thank Nicolas Brisebarre for useful number-theoretic discussions on Lemma 1. We thank Joachim von zur Gathen and Igor Shparlinksi for pointing out their work on NP-hardness to us. The authors would like to thank the referees whose comments where very useful for improving this paper.

[11] Kaltofen, E. Sparse Hensel lifting. In EUROCAL 85 European Conf. Comput. Algebra Proc. Vol. 2 (Heidelberg, Germany, 1985), B. F. Caviness, Ed., Lect. Notes Comput. Sci., Springer Verlag, pp. 4–17. Proofs in [12]. [12] Kaltofen, E. Sparse Hensel lifting. Tech. Rep. 85-12, Rensselaer Polytechnic Instit., Dept. Comput. Sci., Troy, N. Y., 1985. [13] Kaltofen, E. Greatest common divisors of polynomials given by straight-line programs. J. ACM 35, 1 (1988), 231–264. [14] Kaltofen, E. Polynomial factorization 1987-1991. In Proc. LATIN ’92 (Heidelberg, Germany, 1992), I. Simon, Ed., vol. 583 of Lect. Notes Comput. Sci., Springer Verlag, pp. 294–313. [15] Kaltofen, E. Polynomial factorization: a success story. In ISSAC 2003 Proc. 2003 Internat. Symp. Symbolic Algebraic Comput. (New York, N. Y., 2003), J. R. Sendra, Ed., ACM Press, pp. 3–4. Abstract for invited talk. Slides: http://www.math.ncsu.edu/ ~kaltofen/bibliography/03/issac.pdf. [16] Kaltofen, E., and Lee, W. Early termination in sparse interpolation algorithms. J. Symbolic Comput. 36, 3–4 (2003), 365–400. Special issue Internat. Symp. Symbolic Algebraic Comput. (ISSAC 2002). Guest editors: M. Giusti & L. M. Pardo. [17] Karpinski, M., and Shparlinski, I. On the computational hardness of testing square-freeness of sparse polynomials. In Proc. AAECC-13 (Heidelberg, Germany, 1999), vol. 1719 of Lect. Notes Comput. Sci., Springer Verlag, pp. 492–497. [18] Lang, S. Algebra. Addison-Wesley, 1993. [19] Lenstra, Jr., H. W. Finding small degree factors of lacunary polynomials. In Gy˝ ory et al. [9], pp. 267–276. [20] Lenstra, Jr., H. W. On the factorization of lacunary polynomials. In Gy˝ ory et al. [9], pp. 277–291. [21] Loos, R. Computing rational zeros of integral polynomials by p-adic expansion. SIAM J. Comput. 12, 2 (1983), 286–293. [22] Plaisted, D. A. New NP-hard and NP-complete polynomial and integer divisibility problems. Theoretical Comput. Sci. 13 (1984), 125–138. [23] Rosser, J. B., and Schoenfeld, L. Approximate formulas of some functions of prime numbers. Illinois J. Math. 6 (1962), 64–94. [24] Schinzel, A., and Zannier, U. The least admissible value of the parameter in Hilbert’s Irreducibility Theorem. Acta Arithm. 65 (1995), 371–391. [25] Shparlinski, I. E. Computing Jacobi symbols modulo sparse integers and polynomials and some applications. J. Algorithms 36 (2000), 241–252. [26] Sprindˇ zuk, V. G. Arithmetic specializations in polynomials. J. reine angew. Math. 340 (1983), 26–52. [27] Waldschmidt, M. Diophantine approximation on linear algebraic groups. Springer Verlag, Heidelberg, Germany, 2000. [28] Zippel, R. E. Probabilistic algorithms for sparse polynomials. PhD thesis, Massachusetts Inst. of Technology, Cambridge, USA, Sept. 1979.

6. REFERENCES [1] Agrawal, M., Kayal, N., and Saxena, N. PRIMES is in P. Manuscript, 2002. Available from http://www.cse.iitk.ac.in/news/primality.pdf. [2] Barvinok, A. I., and Woods, K. Short rational generating functions for lattice point problems. J. Amer. Math. Soc. 16 (2003), 957–979. [3] Cucker, F., Koiran, P., and Smale, S. A polynomial time algorithm for diophantine equations in one variable. J. Symbolic Comput. 27, 1 (1999), 21–29. [4] De Loera, J. A., Hemmecke, R., Huggins, P., Sturmfels, B., and Yoshida, R. Short rational functions for toric algebras and applications. J. Symbolic Comput. 38, 2 (2004), 959–973. [5] D´ıaz, A., and Kaltofen, E. On computing greatest common divisors with polynomials given by black boxes for their evaluation. In Proc. 1995 Internat. Symp. Symbolic Algebraic Comput. (ISSAC’95) (New York, N. Y., 1995), A. H. M. Levelt, Ed., ACM Press, pp. 232–239. [6] von zur Gathen, J., and Gerhard, J. Modern Computer Algebra. Cambridge University Press, Cambridge, New York, Melbourne, 1999. Second edition 2003. [7] von zur Gathen, J., Karpinski, M., and Shparlinski, I. Counting curves and their projections. Computational Complexity 6, 1 (1996/1997), 64–99. [8] Goldstein, A. J., and Graham, R. L. A Hadamard-type bound on the coefficients of a determinant of polynomials. SIAM Rev. 16 (1974), 394–395. ˝ ry, K., Iwaniec, H., and Urbanowicz, J., Eds. [9] Gyo Number Theory in Progress (1999), vol. 1 Diophantine Problems and Polynomials, Stefan Banach Internat. Center, Walter de Gruyter Berlin/New York. Proc. Internat. Conf. Number Theory in Honor of the 60th Birthday of Andrzej Schinzel, Zakopane, Poland June 30–July 9, 1997. [10] Hindry, M., and Silverman, J. H. Diophantine Geometry: an Introduction, vol. 201 of Graduate Texts in Mathematics. Springer, 2000.

215

Generic Matrix Multiplication and Memory Management in LinBox* Erich Kaltofen

Dmitriy Morozov

George Yuhasz

Dept. of Mathematics North Carolina State University Raleigh, North Carolina 27695-8205, USA

Department of Computer Science Duke University Durham, North Carolina 27708-0129, USA

Dept. of Mathematics North Carolina State University Raleigh, North Carolina 27695-8205, USA

[email protected]

[email protected]

[email protected]

ABSTRACT

Canada, France and the USA. See http://www.linalg.org for the list of participating researchers and for the open source code. Our goal is to supply “efficient black box solutions for a variety of problems including linear equations and matrix normal forms with the guiding design principle of reusability” [5]. The LinBox library utilizes two different abstraction devices. The first is algorithmic, and it introduces the notion of a black box matrix [10]† , which is a matrix representation by a procedure that efficiently computes the product of the matrix times an arbitrary vector. The second abstraction is the programming methodology of generic, reusable software design. We use the C++ template instantiation mechanism to compile code for the most efficient ways of performing the arithmetic in the various entry fields [5, 6, 13]. This paper describes two components that exhibit the generic programming techniques LinBox provides. The first is the implementation of blackbox matrix multiplication and the second is the incorporation of external libraries that utilize garbage collection. We have implemented matrix multiplication by a lazy matrix-times-matrix product for the black box matrix type of the LinBox library. Let A, B be black box matrices that have matrix-times-vector functions y = Ax and y = Bx, where x and y are vector objects. In LinBox, the matrixtimes-vector functions are named “apply”, and are member template functions with parametric types for both the input vector x and the output vector y, which can be sparse or dense vector types or columns of matrices. The apply function for the matrix product y = (A · B)x is implemented as the function composition z = Ax; y = Bz. The issue is the choice for the vector data type of the intermediate vector z. Each black box matrix class defines a preferred input and a preferred output vector type. Our composed black box class now has a template parameter switch that lets the user choose at compile time different vector types for z: either the preferred output type of B or preferred input type of A. Or one may force a conversion from the preferred output vector type of B to the preferred input vector of A, or select a default intermediate vector type. Of course, neither must be done when the output/input types are the same. We use C++’s partial template specialization rules for building the proper instantiations of the template class for composition (see sections 2 and 3).

We describe the design and implementation of two components in the LinBox library. The first is an implementation of black box matrix multiplication as a lazy matrixtimes-matrix product. The implementation uses template meta-programming to set the intermediate vector type used during application of the matrix product. We also describe an interface mechanism that allows incorporation of external components with native memory management such as garbage collection into LinBox. An implementation of the interface based on SACLIB’s field arithmetic procedures is presented.

Categories and Subject Descriptors I.1.3 [Computing Methodologies]: Symbolic and Algebraic Manipulation—Languages and Systems; D.1.5 [Software]: Programming Techniques—Object-oriented programming

General Terms design, languages

Keywords black box matrix, C++ templates, C++ allocator, system integration, garbage collection, memory management, exact linear algebra

1. INTRODUCTION LinBox is a C++ template library that provides generic implementations of black box linear algebra algorithms [5]. The library was developed by a consortium of universities in ∗

This material is based on work supported in part by the National Science Foundation (USA) under Grants CCR-0113121 and CCR0305314.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. ISSAC’05, July 24–27, 2005, Beijing, China. Copyright 2005 ACM 1-59593-095-7/05/0007 ...$5.00.

†

The term “black box” matrix seems to have first been coined in our paper.

216

template class Blackbox{ public: typedef Field Field; typedef Vector In PreferredInputVector; typedef Vector Out PreferredOutputVector; // Constructors and destructor ... template OutVector& apply(OutVector& y, const InVector&); template OutVector& applyTranspose(OutVector& y, const InVector& x); const Field& field(); size t rowdim(); size t coldim(); private: // Internal storage and methods . . . };

In sections 4–6, we describe an interface mechanism by which one can plug a library of functions whose objects are garbage collected, such as Maple procedures or Java methods, into our LinBox algorithms. Our benchmark test is with SACLIB’s [15, 2] modular integer arithmetic. The problem is that LinBox algorithms need to allocate temporary intermediate values which are arrays of SACLIB’s modular digits that must be registered with SACLIB’s garbage collector. Watt [16] gives a solution in the Aldor-Maple setting. Our solution is based on C++’s STL allocator template class, so that genericity in our algorithms is maintained and minimal reprogramming is needed.

2. DESIGN ISSUES OF THE COMPOSITION CLASS The design of the composition class described in this paper is predicated on a change made in the black box archetype in LinBox. In the initial design, all black box matrices were templated by the vector type they expected as input and output vectors [5]. The current version of the black box archetype moves the vector type template parameter from the archetype itself to its member functions. The methods apply and applyTranspose are member template functions [9, 14.5.2], with two template parameters, an input vector type and an output vector type. A member template apply allows the design of a generic matrix times vector function that can be instantiated with several vector types that adhere to the vector object interface used. Different vector types may arise in the future, which then can be directly plugged into the matrix code. After the decision was made to have member template functions apply and applyTranspose in the black box archetype, the idea of preferred input and output vector types was introduced. Black box matrices may have efficient implementations of their apply methods when working with a particular vector type. If this is true, then a user or an algorithm working with the black box matrix in question could choose this input and output vector type accordingly to speed up computation. Placing typedefed members PreferredInputVector and PreferredOutputVector in the definition of a black box matrix gives users access to the preferred input and output types of the matrix. Figure 1 shows the black box archetype with preferred input and output vectors. The composition class is based on the lazy evaluation scheme for black box matrix multiplication, meaning the class will use an intermediate vector in its calculations. Since the black box matrices themselves are not templated by the vector type, there are many ways to choose the vector type of the intermediate vector. Suppose A is user defined blackbox matrix with preferred input vector type dense and preferred output vector type sparse. Assume P is a preconditioner that can be used as a left or right multiplier and let P have preferred input and output vector types sparse. When P is used as a left preconditioner, the composed matrix P A should use a sparse vector as the intermediate type, since the types are the same. If P is used as a right preconditioner and code efficiency is the highest priority, then the input type of A, a dense vector, should be used as the intermediate type of AP . However, if memory is limited and a top priority, then the output type of P , a sparse vector, would be the best choice for the intermediate vector. Fi-

Figure 1: Black box Archetype with preferred vectors nally, if space is not a problem and if the cost of copying one vector type into another is made up for by the gain in efficiency of the apply methods, then a conversion between types would be the best choice for handling the intermediate vectors. The design of the new composition class will provide the user with four methods for selecting the type of the intermediate vector. Assume you are composing two black box matrices A, B into the black box matrix AB. The default method will compare the preferred input of A and the preferred output of B, and if they are the same use that type, else use a dense vector as the default type. In addition, the preferred input type of A or the preferred output type of B can be used as the intermediate vector type. Finally, the user can use both types and do a conversion between the two types during the computation in apply methods. A selection method that allows users to control how the intermediate vector type is chosen will be provided. The intermediate vector type will be chosen upon instantiation of the composition class and each composition object will use one instance of the intermediate vectors to be used for all applications of the composed matrix. There are two design ideas not incorporated into the composition class described within this paper that warrant explanation. A user may wish to provide an intermediate vector type to be used regardless of the preferred input and output types. This is not allowed in the implementation and design of the composition class currently, but the option may be added later. Second, the apply methods (not the entire class) could be templated to choose the intermediate vector type. Allowing the apply methods themselves to choose the intermediate vector type is not possible under the black box archetype in LinBox. The composition class must follow the black box archetype, since it is itself a black box matrix, and so the apply methods can only be templated by the input and output vector type. Further, having the apply methods choose the intermediate vector type would mean the creation of many temporary vectors, which could slow down running times. Having the class choose the type and construct one intermediate vector that will be reused many times is more efficient and less prone to memory leaks.

217

3. IMPLEMENTATION OF THE COMPOSITION CLASS

template class TypeChooser { public: typedef D TYPE; TypeChooser() { } }; template class TypeChooser { public: typedef T TYPE; TypeChooser() { } };

The composition class will use partial template specialization to implement all the features listed in section 2. Since partially specialized template classes are instantiated instead of primary template classes when the template parameters match the partial definition, it becomes possible to program a conditional “if-then-else” during compile time expansion (template “meta” programming, see [14] and [9, 14.5.4.2]). Partial template specialization is used several ways in the composition class. First it is the driving mechanism behind the selection method presented to a user. The user can select how the intermediate vector is chosen by passing a flag as a template parameter. The flag is an enumerated type defined in the composition header file with the declaration in figure 2.

Figure 4: Type choosing class typedef TypeChooser< typename Blackbox2::PreferredOutputType, typename Blackbox1::PreferredInputType, std::vector<Element> > VectorType; Figure 5: Instantiation of TypeChooser in DEFAULT

enum IntermediateVector { DEFAULT, INPUT, OUTPUT, CONVERSION };

ing an apply call, while a conversion between vector types must be performed if the types are not equal. As recommended by a referee, the comparison of types and choice of apply methods is made at compile time. The CONVERSION specialization contains a nested class that is templated by the preferred intermediate vector types and implements the apply methods. The encapsulated class has a partial specialization that eliminates the conversion of types when the two vector types are equal.

Figure 2: Enumerated type IntermediateVector This flag defines all the choices of how the intermediate vector type can be selected. The composition class is passed the user’s choice as a template parameter as you can see in the declaration of the class Compose, shown in figure 3.

4.

template class Compose;

EXTERNAL LIBRARIES AND MEMORY MANAGEMENT IN LINBOX

The memory management problems created when incorporating an external library into LinBox arise because LinBox uses C++ pointers and references as well as operators new and delete to address, manipulate, allocate, and deallocate memory. This fact ties the library to the details and representation of the physical memory and does not allow for the use of such abstractions as garbage collected memory or memory coming from a pre-allocated pool. It is important to introduce an interface that will provide a link between the external libraries and the LinBox objects that interact with them. The following paragraphs describe the LinBox objects that need such an interface. LinBox uses several external libraries to perform field operations. The different implementations of the field arithmetic are contained in the objects that adhere to the field archetype, a common object interface defined by LinBox. It is these implementations that encapsulate the exact representation of the field elements and allow their manipulation by providing methods such as add, mul, etc. All these methods operate directly on the elements, and, therefore, need to be aware not only of how the elements are represented in the memory, but also how the memory is represented in the system. Black box matrices may need to allocate field elements and then use field objects to operate on those elements. In such cases, a black box acts as a container and needs to be aware of the way in which elements are allocated and deallocated (the operations that replace operators new and delete, respectively) as well as how elements are addressed. Vectors in LinBox are usually containers of field elements, and in such case need to be provided with the same information as black boxes that contain elements: the ways in which ele-

Figure 3: Declaration of Compose The Compose class is specialized by the third template parameter, with each specialization making the appropriate choice for the type of the intermediate vector. Further the flag is defaulted to DEFAULT so a user does not need to make a selection. The DEFAULT specialization requires that a comparison of the preferred input and output vector types be done and if they are the same, then we use that type, else we will use a dense vector for the intermediate type. This check and choice of types is done at compile time to avoid any unnecessary computations during run time. To perform this selection of types at compile time, a type choosing class is used. The DEFAULT specialization of Compose will instantiate a type choosing class, passing the preferred vector types as template parameters, and the template expansion mechanism will make the appropriate choice for the intermediate vector type. The code for the type choosing class TypeChooser is given in figure 4 and the instantiation of the class in DEFAULT is shown in figure 5. The type choosing class in figure 4 compares the types T and S. If they are different then the top implementation of the class TypeChooser is instantiated by the compiler and the default type D is chosen. If the types T and S are the same then the second definition is instantiated and the type T=S is chosen. The CONVERSION specialization compares the preferred intermediate vector types to specialize the apply methods. If the types are equal then no conversion needs to be done dur-

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template class std::allocator { public: typedef T value type; typedef size t size type; typedef ptrdiff t difference type; typedef T* pointer; typedef const T* const pointer; typedef T& reference; typedef const T& const reference; pointer address(reference r) const { return &r; } const pointer address(const reference r) const { return &r; } allocator() throw(); template allocator(const allocator&) throw(); ˜allocator() throw(); // space for n Ts pointer allocate(size type n, allocator::const pointer hint = 0); // deallocate n Ts, don’t destroy void deallocate(pointer p, size type n); // initialize *p by val void construct(pointer p, const T& val) { new(p) T(val); } // destroy *p but don’t deallocate void destroy(pointer p) { p−>˜T(); } size type max size() const throw(); // in effect: typedef allocator other template struct rebind { typedef allocator other; } }; template bool operator==( const allocator&, const allocator&) throw(); template bool operator!=( const allocator&, const allocator&) throw();

ments can be allocated, deallocated, and addressed. Vectors may also be used to encapsulate the functionality of external vectors (for example, those native to the computer algebra systems that are using LinBox), and in such cases usually are accompanied by the field objects that describe how the elements stored in such external vectors should be manipulated. The last important part of the library that needs to be aware of how memory is represented and managed is the algorithm implementations. A concrete example of the problems that may arise if an external library is to be used with LinBox is presented with SACLIB [2, 15]. SACLIB provides a number of facilities that can be of use to LinBox; from the perspective of memory management it is not important which specific facilities we employ, so we will concentrate on SACLIB’s functions that perform field arithmetic over Zp fields. What is important, however, is the fact that SACLIB uses a garbage collector to manage its memory, and applying SACLIB functions to the memory allocated using operator new or its underlying malloc will cause the program to crash since the elements will not be registered with the garbage collector. In fact, any memory that is used by SACLIB needs to be allocated by the library’s own functions. For instance, to allocate a continuous array of elements, SACLIB provides a function GCAMALLOC(n) that takes the size of the array n as its argument. Since the memory is automatically garbage collected one may simply remove all the references to such an array in order for it to be returned into the pool of available memory.

5. STL ALLOCATORS The problem of providing generic algorithms and generic abstractions for memory management is not unique to LinBox. It has been addressed and solved before by the C++ Standard Template Library [11]. STL employs the technique of allocators, objects “used to insulate implementers of algorithms and containers that must allocate memory from the details of physical memory” [14, page 567]. Allocators accomplish this function by providing a common interface that encapsulates the memory management functionality by providing standard names for types and functions that are involved in memory management such as pointers, references, functions to allocate, deallocate memory, and construct C++ objects in that memory. To understand better what functionality an allocator is supposed to provide, we examine the way in which the standard allocator may be declared. The standard allocator is provided by the STL in the header <memory> and is used by default by all the STL standard containers. The example in figure 6 is from [14, page 567]: The functionality of each element of the allocator’s design is implied by its name. For more information on the allocators and their use in the STL, see [11, Chapter 24] and [14, §19.4], here we will examine some of the important aspects of their design. A notable element of the allocator design are the typedef declarations at the beginning of the std::allocator declaration. While in the implementation of the standard allocator the basic types (pointers and references) are defined to be called pointer, reference, and so on, one can easily imagine how, for example, a smart pointer class could be typedefed to be called pointer. The fact that the allocators provide not only the functions that are used to allocate and deallocate memory, but also the data types used to repre-

Figure 6: STL Allocator sent the memory is very important: one can imagine a class that captures not only the information about the address of an object in the main memory, but also an address of the machine on which that object is stored, thus, providing a representation for distributed memory. One has to note that the C++ Standard [9, 20.1.5 4] relaxes the requirements for the STL container implementations: Implementations of containers described in this International Standard are permitted to assume that their Allocator template parameter meets the following two additional requirements beyond those in Table 32: — All instances of a given allocator type are required to be interchangeable and always compare equal to each other. — The typedef members pointer, const pointer, size type, and difference type are required to be T*, T const*, size t, and ptrdiff t, respectively. However, at the same time it encourages the implementors of the libraries to not make such assumptions [9, 20.1.5 5]: Implementors are encouraged to supply libraries that can accept allocators that encapsulate more

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general memory models and that support nonequal instances. In such implementations, any requirements imposed on allocators by containers beyond those requirements that appear in Table 32, and the semantics of containers and algorithms when allocator instances compare nonequal, are implementation-defined.

We should also note that some fields may need to store field elements as part of the field’s description. Such elements can no longer be just members of the field since fields may be allocated on the stack: rather, the references (or pointers) to such elements should be stored in the field itself, and the elements should be allocated using the field’s allocator. See subsection 6.3 for the details of how the solution to the same problem with temporary elements used in the algorithm implementations should be implemented. It is also interesting to note that the only modification necessary to the fields that have already been implemented to allow them to retain their current functionality is typedefing std::allocator<Element> to be their ElementAllocator member, and defining getElementAllocator() member function to return ElementAllocator(). While such modifications are sufficient to retain their functionality, for many fields it is acceptable to operate on top of many different memory models† , so many common field implementations may become template classes themselves, and allow a user to supply an allocator type which they would in turn pass to other library facilities as well as use to address the field elements. In such cases, LinBox shall follow the C++ Standard’s encouragement ([9, 20.1.5 5] — cited in section 5), and use ElementAllocator::reference (pointer, etc.) in its fields’ method declarations to accept more general memory models.

Unfortunately, common implementations of the STL used today, including the one supplied by the C++ compiler from the GNU Compiler Collection [7] (one of the key compilers targeted by LinBox), do make assumptions about the requirements on the typedef members of the allocators, which strictly limits the kinds of models that can be described. One element of allocators’ design that deserves closer attention is the member struct rebind. As the comment in the code in figure 6 states, rebind effectively typedefs its member other to be of type allocator. This manipulation is provided so that storage for objects of a type other than the container element type can be managed. See [12, Chapter 4, p. 101]. For example, STL list nodes can thus be allocated via a user defined allocator.

6. ALLOCATORS IN LINBOX 6.1 Fields

6.2

As we have established earlier (in section 4) the part of the library whose addressing, allocation, deallocation, etc. requires special attention is the field elements: they most often occupy the most storage in any library operation, and they are the end target of black box manipulations. While all parts of the library need to be adjusted to be ready to accept different memory models, we should note that LinBox has already been designed in a way where all the details of the representation of field elements are hidden both from other parts of the library and the user by the field objects. In fact, field elements themselves do not even need to be classes (for example, in an implementation of Zp field where p fits in a word, the elements themselves may be of type unsigned int), so it is only natural that we add another piece of information about elements — namely, how the memory in which they are stored is represented — to the field objects. As a result, the field archetype and all the compliant field implementations get two new components:

Vectors and black boxes

All the vectors used in LinBox internally adhere to the interfaces of various STL containers (most notably, std::vector<Element> for storing dense vectors of field elements, std::vector< std::pair<size t, Element> > for storing sparse sequence vectors of elements, and std::map< size t, Element > for storing sparse associative vectors). As it was mentioned above, STL defines its containers to be parametrized by an additional allocator type specifically for providing descriptions of alternative memory models. Since we have defined Field::ElementAllocator to adhere to the STL allocator object interface, and the declarations of containers are aware of which field objects are used (technically, they only need to be aware of the element type, but in reality such types are always obtained from the field object which is known in the context of the particular declaration), we simply require the library and the user to provide the allocator type to the vector declarations. As a result, the typical declarations of the vector objects are now of the form shown in figure 7. We also note that an allocator may contain auxil-

• a typedef member ElementAllocator • a method ElementAllocator getElementAllocator() const.

std::vector // dense vectors, std::vector<std::pair<size t, Field::Element>, Field::ElementAllocator>// sparse sequence vectors std::map<size t, Field::Element, Field::ElementAllocator> // sparse associative vectors

The first is the actual allocator type — the class that must adhere to the STL allocator requirements (see [9, 20.1.5]). The second member is the function that returns an instance of the allocator that the containers, algorithms and the user should use after possibly copying to a rebound allocator of appropriate type. This instance contains all the information necessary for memory management. A typical example of an allocator that needs additional information is a pool allocator.∗

Figure 7: Vector declarations with allocators With the pointer to the segment and its size one can construct a pool allocator that would abstract the memory to the library’s facilities. A sample implementation of a pool allocator can be found in Boost memory library [1]. † This is not the case, however, when the underlying library that implements field arithmetic is tied to some specific memory model. For an example, see section 7.2.

∗

A pool allocator is an excellent example of an allocator for different memory models because of its conceptual simplicity and usefulness: several POSIX facilities (such as shared memory and memory mapped files) describe the memory that they provide by supplying a pointer to the appropriate segment of memory, the size of which is known.

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iary information that is necessary to describe the underlying memory model, so an instance of the allocator object has to be passed to a container. Internally, vectors are used primarily by various algorithms, so we consider an example of such use in subsection 6.3. LinBox may also use “external vectors” — vectors that are native to and adhere to some internal representation of the computer algebra systems that are employing LinBox’s functionality. In such cases, those vectors have to be adapted to conform to some interface that LinBox understands (e.g., the STL vector interface), so a wrapper class has to be provided. Then all the memory management information that is related to the representation of the vector depends on individual implementations and should be encapsulated inside that class. While some of the black boxes provided by LinBox do not need to store field elements (for example, Transpose black box) the majority perform operations on field elements as part of the matrix times vector product. Such black boxes always have a field object passed to them, which now also contains information about how the memory used by the field elements needs to be managed. Black boxes in turn can simply pass allocators from such field objects to the underlying containers in the same way that is described above for vectors. If a black box uses temporary field elements in the implementation of its apply method, special attention needs to be paid to such elements, namely, to the fact that they cannot be allocated on the run-time stack as before. The same problem is present for algorithm implementations, so we discuss the details in the following section.

tion” technique [14, page 366] to avoid potential mistakes. See figure 9 for an example. The same technique should be some algorithm() { Field f; ... std::vector tmp vec(2, Field::Element(), f.getElementAllocator()); Field::ElementAllocator::reference one = tmp vec[0]; Field::ElementAllocator::reference two = tmp vec[1]; . . .} Figure 9: Creating temporary elements using allocators used in the implementations of black boxes, fields, and any other places that use temporary elements or where elements could be placed on the stack (for example, as members of field objects that could themselves be placed on the stack).

7.

SAMPLE CODE

This section provides some sample code that illustrates the design and implementation ideas described previously.

7.1

Composition

The following shows how to use the Compose class in LinBox. First, figure 10, shows a blackbox matrix that extends the current implementation of diagonal matrices, and provides preferred input and output vector types. Figure 11

6.3 Algorithms

template class MyDiagonal{ public: //Diagonal matrix data structures typedef Input PreferredInputType; typedef Output PreferredOutputType; // Implementation of required functions . . .};

Just like any other part of the library, when a LinBox algorithm needs to manipulate field elements it is passed a field object that encapsulates information about the field including the representation of the field elements. When an algorithm needs to create a vector, it needs to allocate such a vector by passing the allocator type and object to it, so a typical declaration of a vector now may be of the form shown in figure 8. Here f is the field object (of the type Field) and n is the size of the vector that is being declared.

Figure 10: Diagonal matrix implementing preferred vectors is an example of a program that illustrates how to use the Compose class and all of the options for choosing the intermediate vector type. The code in figure 11 has an example of each option for selecting the intermediate vector type. The black box matrix AA will use the preferred input type of A, which is a dense vector. The matrix BC will use the output type of C which is an STL list of pairs. The matrix CB uses the DEFAULT option, and since the input type of C and the output type of B are the same, then the intermediate vector will be a sparse sequence vector. Finally the matrix BB uses the CONVERSION option. Because the input and output types of B are different, then the composed matrix will have to do a conversion between the two types.

std::vector v(n, Field::Element(), f.getElementAllocator()); Figure 8: Declaration of LinBox vectors An issue that requires special attention are temporary elements that an algorithm may allocate. Such elements may no longer be allocated on the stack due to the fact that a memory model that the given field uses may not allow it: for example, SACLIB (discussed in section 7.2) implements specific routines to scan the stack for its Words; however, the authors of that library could make a requirement (for example, for the sake of system-independence) that the memory used by the library can only be allocated using the library’s own functions, i.e., placing data on the stack would not be allowed. When an algorithm needs temporary elements for its implementation, it has to utilize the field’s allocator for their allocation, construction, etc. While one may use the allocator’s methods allocate, construct, etc. directly, it is advisable to employ the “resource acquisition is initializa-

7.2

Allocators

Next is an illustration of allocators in LinBox based on the SACLIB library. “SACLIB is a library of C programs derived from the SAC2 system” [2, 15]. It includes facilities for list processing, integer, modular number, and rational number arithmetic, polynomial arithmetic, linear algebra,

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Word allocated GCA list = NIL; int num of allocators = 0; template class SacLibAllocator { public: ... SacLibAllocator() throw() { if (num of allocators == 0) { GCGLOBAL(&allocated GCA list); // register allocated GCA list with // the garbage collector } ++num of allocators; } pointer allocate(size type n, const void* hint = 0) {int size to alloc = n * sizeof (T) / sizeof (Word) + (n * sizeof (T) % sizeof (Word) == 0 ? 0 : 1); Word h = GCAMALLOC(size to alloc, GC CHECK); allocated GCA list = COMP(h, allocated GCA list); return (pointer) GCA2PTR(h); } void deallocate(pointer p, size type n) { std::pair<Word, Word> res = remove from list(p, allocated GCA list); allocated GCA list = res.second; } . . . };

int main(int argc, char **argv){ //Declarations typedef MyDiagonal::Dense, Vector::Dense > Blackbox1; typedef MyDiagonal::Dense, Vector::SparseSeq > Blackbox2; typedef MyDiagonal::SparseSeq, std::list<std::pair<size t, Field::Element> > > Blackbox3; //Initialization Blackbox1 A(F, d1); Blackbox2 B(F, d2); Blackbox3 C(F, d3); //Compose the matrices and apply them to vectors Compose AA(&A, &A); Compose BC(&B, &C); Compose CB(&C, &B); Compose BB(&B, &B); y1=AA.apply(y1, x1); y2=BC.apply(y2, x2); y3=CB.apply(y3, x3); y4=BB.apply(y4, x4); return 0; } //End main Figure 11: Composition test program computing polynomial GCD and resultants, polynomial factorization. All of SACLIB’s objects are presented to the user via their handles each of which occupies one word in memory (which SACLIB conveniently calls Words). While most of the SACLIB’s functions manipulate lists, the library also provides a facility to allocate garbage collected arrays: arrays that can both be referred to by the SACLIB structures (and garbage collected when they become inaccessible) and contain handles to other SACLIB structures that will be taken care of by the garbage collector. The specific functions that are of interest are GCAMALLOC(n) that allocates a garbage collected array of size n and GCA2PTR(A) that returns a pointer to the actual elements of the array, thus, allowing the user to refer to the elements directly without using the supplementary accessor functions. One cannot place a reference to a SACLIB structure in the dynamically allocated memory since the garbage collector will not be aware of such references, will collect the structures that it will consider inaccessible, and further behavior of the program will be undefined. In order to make SACLIB’s facilities available to LinBox one has to define not only a field object (see SacLibModularField in figure 13) that uses SACLIB’s functions to implement arithmetic over Zp , but also an allocator (SacLibAllocator) that communicates to the algorithms and containers how the memory needs to be allocated for SACLIB. It is interesting to note that SacLibAllocator has very few differences from the standard allocator described in section 5, so here we present only those key differences in figure 12. The main differences appear in methods allocate and deallocate. In method allocate, after computing the size of the garbage collected array that needs to be allocated, the allocator calls function GCAMALLOC to allocate the actual array, then it adds it to a globally registered list

Figure 12: SacLib Allocator allocated GCA list to make sure the array will be accessible and will not be removed by the garbage collector, and returns the pointer to the actual elements stored in the array. Method deallocate, in turn, removes a previously allocated array from the allocated GCA list, the actual collection of the memory occupied by the array occurs during the next invocation of the garbage collector. SacLibModularField provides the allocator and uses the appropriate SACLIB functions to implement various field operations, see figure 13. SACLIB has to be initialized using its function BEGINSACLIB before it can be used in a program, and after SACLIB is uninitialized using ENDSACLIB, all of its structures will be unavailable, therefore, all the SacLibModularFields should be destroyed by the time ENDSACLIB is called. To achieve this goal one should again use “resource acquisition is initialization” technique by putting all of the operations that utilize SACLIB in an unnamed scope as in figure 14.

8.

CONCLUDING REMARKS

First, we address the issue of code efficiency for our generic framework. All decisions in matrix composition are resolved at compile time and there is no loss of efficiency due to genericity. Similarly, if a standard STL std::allocator is used, the templates are compiled out, and the generated code incurs no additional run-time overhead. Our customization of an allocator to handle SACLIB objects can introduce an inefficiency: each time a SACLIB Word is allocated, it is prepended to a list that is registered with the SACLIB garbage collector, which can be both time and space inefficient. However, as an STL allocator, SacLibAllocator can

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class SacLibModularField { public: typedef Word Element; typedef SacLibAllocator<Element> Allocator; typedef ElementAllocator::reference ElementReference; typedef ElementAllocator::const reference ElementConstReference; typedef ElementAllocator::pointer ElementPointer; ... SacLibModularField (const integer& m, const ElementAllocator& alloc = ElementAllocator()) : alloc(alloc) { modulus = alloc.allocate(1); . . . } ElementReference add (ElementReference x, ElementConstReference y, ElementConstReference z) const { return x = MISUM(* modulus, y, z); } ... ElementAllocator getElementAllocator() const { return alloc; } private: ElementAllocator alloc; ElementPointer modulus; // cannot be just Element because the field could // be allocated in the dynamic memory, see section 6.1 . . . };

two anonymous referees for their informative and constructive suggestions.

9.

REFERENCES

[1] Boost C++ libraries, 2003. URL: http://www.boost.org. [2] Brown, C. Saclib2.1 on Linux. http://www.cis. udel.edu/~saclib/linux/SaclibLinux.html, Mar. 2000. [3] Chen, L., Eberly, W., Kaltofen, E., Saunders, B. D., Turner, W. J., and Villard, G. Efficient matrix preconditioners for black box linear algebra. Linear Algebra and Applications 343–344 (2002), 119–146. [4] Cohen, A. M., Gao, X.-S., and Takayama, N., Eds. Proc. First Internat. Congress Math. Software ICMS 2002, Beijing, China (Singapore, 2002), World Scientific. [5] Dumas, J.-G., Gautier, T., Giesbrecht, M., Giorgi, P., Hovinen, B., Kaltofen, E., Saunders, B. D., Turner, W. J., and Villard, G. LinBox: A generic library for exact linear algebra. In Cohen et al. [4], pp. 40–50. [6] Dumas, J.-G., Giorgi, P., and Pernet, C. FFPACK: Finite field linear algebra package. In Gutierrez [8], pp. 119–126. [7] GNU compiler collection, 2003. FSF - Free Software Foundation Gnu Project: URL: http://gcc.gnu.org. [8] Gutierrez, J., Ed. ISSAC 2004 Proc. 2004 Internat. Symp. Symbolic Algebraic Comput. (New York, N. Y., 2004), ACM Press. [9] ISO/IEC. International standard, programming languages—C++. No. 14882:1998(E). American National Standards Institute, New York, 1998. [10] Kaltofen, E., and Trager, B. Computing with polynomials given by black boxes for their evaluations: Greatest common divisors, factorization, separation of numerators and denominators. J. Symbolic Comput. 9, 3 (1990), 301–320. [11] Musser, D. R., Derge, G. J., and Saini, A. STL Reference Guide, second ed. Addison-Wesley, Reading, MA, 2001. [12] Plauger, P. J., Stepanov, A. A., Lee, M., and Musser, D. R. The C++ Standard Template Library. Prentice Hall PTR, Upper Saddle River, New Jersey, 2001. [13] Saunders, D., and Wan, Z. Smith normal form of dense integer matrices, fast algorithms into practice. In Gutierrez [8], pp. 274–281. [14] Stroustrup, B. The C++ Programming Language, third ed. Addison–Wesley, Reading, Massachusetts, 1997. [15] Vielhaber, H., Buchberger, B., Collins, G. E., ´ n, M. J., Hong, H., Johnson, J. R., Encarnacio Krandick, W., Loos, R., and Mandache, A. N. A. M. SACLIB 1.1 user’s guide. Tech. Rep. 93-19, RISC Linz, Linz, Austria, 1993. [16] Watt, S. M. A study in the integration of computer algebra systems: Memory management in the Maple-Aldor environment. In Cohen et al. [4], pp. 405–410.

Figure 13: SacLib field implementation #include ... int main(int argc, char* argv[ ]) { BEGINSACLIB(&argc); { // “SACLIB safety scope” integer m; . . .// initialize m to some large prime number SacLibModularField F(m); // perform field operations, create black boxes, //invoke algorithms, etc ... } // end “safety scope” ENDSACLIB(SAC FREEMEM); }//End main Figure 14: SacLib test program also allocate an array of SACLIB Words by a single call, thus allowing the application program to “pool” native memory chunks. One can also provide automatic memory blocking via the allocator mechanism, but we have not done so. In its current state, the LinBox library contains numerous algorithms for sparse, structured and black box matrices. We have described a framework that permits black box matrix multiplication, which can be employed, for example, in the pre-conditioners needed in some of the algorithms [3]. We have also given a means to incorporate external memory managers, which allows the use of external garbage collected libraries in LinBox and which can implement a memory model where allocation is distributed over several computers. Acknowledgments: We thank members of the LinBox project for their helpful comments on our design. We would also like to thank Matt Austern, Gillmer Derge, and David Musser for the useful information and pointers to the design intricacies of C++ that they have provided. We thank the

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¨ Exact Analytical Solutions to the Nonlinear Schrodinger Equation Model Biao Li

Yong Chen

Qi Wang

1. Nonlinear Science Center and Department of Mathematics Ningbo University Ningbo 315211, China 2. MM Key Lab Chinese Academy of Sciences Beijing 100080, China

1. Nonlinear Science Center and Department of Mathematics Ningbo University Ningbo 315211, China 2. MM Key Lab Chinese Academy of Sciences Beijing 100080, China

1. Dept. Appl. Math. Dalian University of Technology Dalian 116024, China 2. MM Key Lab Chinese Academy of Sciences Beijing 100080, China

[email protected]

[email protected]

ABSTRACT A method is developed for constructing a series of exact analytical solutions of the nonlinear Schr¨ odinger equation model (NLSE) with varying dispersion, nonlinearity, and gain or absorption. With the help of symbolic computation, a broad class of analytical solutions of NLSE are obtained. From our results, many previous known results of NLSE obtained by some authors can be recovered by means of some suitable selections of the arbitrary functions and arbitrary constants. Further, the formation, interaction and stability of solitons have been investigated.

Categories and Subject Descriptors J.2 [Computer Applications]: Physical Sciences and Engineering

General Terms Algorithms

Keywords Nonlinear Schr¨ odinger Equation, Symbolic Computation, Soliton, Soliton Propagation and Interaction.

bulence and phase transitions, biophysics, and star formation and so on [5]. Since solitary waves, or solitons, which are the best known solutions of NLSE, were introduced and developed in 1971 by Zakharov and Shabat [37], there have been many significant contributions, inclusive of theoretical and numerical aspects, to the development of the NLSE solitons theory [1, 4, 12, 13, 14, 15, 16, 17, 18, 19, 20, 27, 29, 30, 31, 32, 33, 34]. After predictions of the possibility of the existence [14] and experimental discovery by Mollenauer, stolen, and Gordon [27], today, NLSE optical solitons are regarded as the natural data bits and as an important alternative for the next generation of ultrahigh speed optical telecommunication systems [12, 13, 17, 27, 32, 33]. Unfortunately, due to the pulse exponential attenuation along the fiber in real optical fibers, the balance between nonlinear and dispersion is broken and it affects the transmission of optical soliton. Usually, by slowly changing the fiber parameter slow changing along the longitudinal direction or using the heat-insulating amplify, we maintain the transmission of soliton in the actual communication system. Therefore, in the real communication system of optical solitons, the transmission of soliton is described by the NLSE model with varying coefficients i

1.

INTRODUCTION

wangqi [email protected]

∂u 1 ∂2u + β(z) 2 + δ(z)|u|2 u = iα(z)u, ∂z 2 ∂t

(1)

where β(z) and δ(z) are slowly increasing dispersion coefficient and nonlinear coefficient, respectively; α(z) represents the heat-insulating amplification or lose. Serkin and Hasegawa [17, 32] developed an effective mathematical algorithm to discover and investigate infinite number of novel soliton solutions for NLSE (1) and discussed the problem of soliton management [1, 13, 17, 32] described by NLSE (1). Ruan et al [29] studied NLSE (1) by the symmetry approach and reported some exact solutions. Hong and Liu [15] simulated NLSE (1) by a novel numerical approach based on the discovery of a new and intrinsic conservation law for NLSE (1). The finding of some mathematical algorithms to discover and investigate exact analytical solutions of nonlinear dispersive systems is important to the field, and might have significant impact on future research. The motivation for the present study lies in the physical

The nonlinear Schr¨ odinger equation model (NLSE) is one of the most important and ”universal” nonlinear models of modern science. NLSE appears in many branches of physics and applied mathematics, including nonlinear quantum field theory, condensed matter and plasma physics, nonlinear optics and quantum electronics, fluid mechanics, theory of tur-

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importance of the NLSE (1) and the need to have some exact solutions. To have some explicit analytical solutions of NLSE (1) may enable one to better understand the physical phenomena which it describes. The exact solutions, which are accurate and explicit, may help physicists and engineers to discuss and examine the sensitivity of the model to several physical parameters. In this work, we will develop a method to construct a series exact solutions for NLSE (1). With the help of symbolic computation, six families of solutions of NLSE (1) are derived. Then based on these solutions, the formation, interaction and stability of solitons are discussed. This rest of this paper is organized as follows. In section 2, a methodology for constructing exact analytical solutions of nonlinear evolution equations (NEEs) is proposed. In section 3, we apply the method to the NLSE (1) and obtain a broad class of exact solutions for these models, and further investigate the main features of various analytical solutions obtained by using direct computer simulations. Section 4 is a short summary and discussion.

2.

Now we outline our proposed method as follows: Given a nonlinear NEEs with, say, two variables: {z, t} p(ut , uz , uzt , utt , uzz , · · · ) = 0.

Step 1. We assume that the solutions of Eq. (14) are as follows: m

(2)

tanhq x =

sinhq x 1 , sechq x = , x∈C coshq x coshq x

(3)

where a0 = a0 (z, t), ai = ai (z, t), bi = bi (z, t), (i = 1, · · · , m), ξ = ξ(z, t) are all differentiable functions of {z, t}, σ(ξ) and τ (ξ) satisfy Eqs. (11)-(12). The parameter m can be determined by balancing the highest order derivative term and the nonlinear terms in (14). m is usually a positive integer, if not, some proper transformation u(z, t) → um (z, t) may be in order to satisfy this requirement. Step 2. Substituting (15) along with (11) and (12) into (14), we obtain a set of algebraic polynomials for σ i (ξ)τ j (ξ) (i = 0, 1, · · · ; j = 0, 1). Setting the coefficients of these terms σ i (ξ)τ j (ξ) to zero, we get a system of over-determined PDEs (or ODEs) with respect to unknown functions {a0 , ai , bi (i = 1, 2, · · · , m), ξ}. Step 3. Solving the over-determined PDEs (or ODEs) system by use of a symbolic computation system-Maple, we would end up with the explicit expressions for {a0 , ai , bi (i = 1, 2, · · · , m), ξ} or the constraints among them. Step 4. Thus according to {(13), (15)} and the conclusions in Step 3, we can obtain many families of exact solutions for Eq. (14). Remark 1. The method proposed is more general than the methods [8, 36, 25, 26, 28]. Firstly, compared with various tanh method [8, 36, 9, 24, 25, 26, 28], projective Riccati equations method [6, 7, 35], the restriction on ξ(z; t) as merely a linear function {z, t} and the restriction on the coefficients a0 , ai , bi (i = 1, 2, · · · , m; j = 0, 1) as constants are removed. Secondly, generalized hyperbolic-function method [11], generalized Riccati equation method [21, 23] and generalized projective Riccati equations methods [22] can be recovered by selecting different parameters: a0 , ai , bi (i = 1, 2,· · · , m; j = 0, 1), ξ(z, t), µ and q.

where q > 0 is a deformation parameter. Note that, if q 6= 1, then sinhq x is not odd and coshq x is not even: sinhq (−x) = −q sinh1/q x, coshq (−x) = q cosh1/q x, x ∈ C (4) The following formulae can be easily proven: (sinhq x)0 = coshq x,

(5)

(coshq x)0 = sinhq x,

(6)

cosh2q x − sinh2q x = q,

(7)

(tanhq x)0 = qsech2q x,

(8)

0

(sechq x) = −(tanhq x)(sechq x),

(9)

tanh2q = 1 − qsech2q x.

(10)

3. ANALYTICAL SOLUTIONS OF NLSE We now investigate NLSE (1) with our algorithm. In order to obtain some exact solutions of NLSE (1), firstly we make the transformation

Now using the above q-deformed hyperbolic functions and the ideas in [7, 8, 9, 15, 16, 24, 25, 26, 28, 29, 34, 35], we introduce a system of ODEs as follows: dσ(ξ) dτ (ξ) = −σ(ξ)τ (ξ), = 1 − µσ(ξ) − τ 2 (ξ), dξ dξ

u(z, t) = V (z, t) exp[i(θ(z, t))],

(11)

−V θz + (12) Vz +

We know that Eqs. (12)-(13) have the following solutions: σ(ξ) =

1 sinhq ξ , τ (ξ) = . µ + coshq ξ µ + coshq ξ

(16)

Then substituting (16) into (1) and setting the real and imaginary parts of the resulting equation equal to zero, we obtain the following sets of PDEs

where µ, q > 0(q 6= µ2 ) are constants. τ 2 (ξ) = 1 − 2µσ(ξ) + (µ2 − q)σ 2 (ξ).

(15)

i=1

METHOD ex − qe−x ex + qe−x , coshq x = , 2 2

σ i−1 (ξ)[ai σ(ξ) + bi τ (ξ)],

u(z, t) = a0 +

In [2, 3], Arai introduced q-deformed hyperbolic functions as follows: sinhq x =

(14)

1 β(z)(Vtt − V θt2 ) + δ(z)V 3 , 2

1 β(z)(2Vt θt + V θtt ) − α(z)V = 0, 2

(17) (18)

By balancing Vtt and V 3 in (17), we obtain m = 1. Therefore we assume the solutions of Eqs. (17)-(18) in the following special forms

(13)

where the variable ξ is complex, so that the two functions can be trigonometric or hyperbolic. For simplicity, in the paper, we only consider the case of hyperbolic functions

V (t, z) = a0 (z) + a1 (z)σ(ξ) + b1 (z)τ (ξ), ξ = tλ(z) + η(z), (19)

225

Θ(t, z) = t2 ∆(z) + tΓ(z) + Ω(z),

(20)

Case 1.

where a0 (z), a1 (z), b1 (z), λ(z), η(z), ∆(z), Γ(z) and Ω(z) are functions of z to be determined, τ (ξ) and σ(ξ) satisfy Eqs. (11)-(12). Substituting (11), (12), (19) and (20) into (17)-(18), collecting coefficients of monomials of τ (ξ), σ(ξ) and t of the resulting system (Notice that β(z), δ(z), α(z), a0 (z), a1 (z), b1 (z), λ(z), η(z), ∆(z), Γ(z) and Ω(z) are independent of t), then setting each coefficients to zero, we obtain an overdetermined ODEs system with respect to differentiable functions β(z), δ(z), α(z), a0 (z), a1 (z), b1 (z), λ(z), η(z), ∆(z), Γ(z) and Ω(z) as follows: a00 − αa0 + β∆a0 = 0,

(21)

b01 + β∆b1 − αb1 = 0,

(22)

1 − b1
4 δµb21 + βλ2 µ − 12 δa0 a1  = 0, 2

(23)

−a0 Γ0 + 2 βΓ∆ = 0,

(24)

−a0 (∆0 + 2 β∆2 ) = 0,

(25)

µ b1
2 βλ∆ + λ0  = 0,

(26)

0

−a1 (Γ + 2 βΓ∆) = 0,

(27)

−a1 (∆0 + 2 β∆2 ) = 0,

(28)

−b1 (∆0 + 2 β∆2 ) = 0,

(29)

0

−a1 (2 βλ∆ + λ ) = 0,

(30)

1 a0 (2a20 δ − βΓ2 + 6 δb21 − 2Ω0 ) = 0, 2

(31)

−b1 (−q + µ2 )(η 0 + βλΓ) = 0,

(32)

−b1 (−q + µ2 )(2 βλ∆ + λ0 ) = 0,

(33)

µ = a0 = a1 = ∆ = 0, β = β, b1 = b1 , δ = −

α= Ω=−

−6 δa1 b21 µ −

3 δa1 b21 + 3 δa20 a1 −

1 2 2  β(z)dzC1 −  βdzC2 + C3 , λ = C2 , 2

µ = a0 = b1 = ∆ = 0, a1 = a1 , β = β, Γ = C1 , δ =

α=

(42)

C2 2 βq , (a1 )2

a01 1 1 , Ω =  βdzC2 2 −  βdzC1 2 + C3 , a1 2 2 η = −C2 C1  β(z)dz + C4 , λ = C2 ,

(43)

where C1 , C2 , C3 , C4 are arbitrary constants, β and a1 are d a1 (z). all arbitrary function of z, a01 = dz Case 3. µ = a0 = a1 = 0, ∆ = ∆, b1 = b1 , λ = C2 ∆, β = −

α=

1 ∆0 , 2 (∆)2

1 2 ∆b01 − b1 ∆0 1 ∆ 0 C2 2 , δ= , Γ = C1 ∆, 2 b1 ∆ 2 (b1 )2

1 1 1 2 C1 ∆ + C2 2 ∆ + C3 , η = C1 C2 ∆ + C4 , (44) 4 2 2 where C1 , C2 , C3 , C4 are arbitrary constants, ∆ and b1 are d ∆(z) . all arbitrary function of z, ∆0 = dz Case 4. Ω=

µ = a0 = b1 = 0, λ = C2 ∆, β = −

(34)

1 ∆0 , Γ = C1 ∆, 2 (∆)2

1 1 2 ∆a01 − a1 ∆0 C1 C2 ∆ + C4 , ∆ = ∆, α = , 2 2 a1 ∆

1 1 1 C2 2 q∆0 Ω = − C2 2 ∆ + C1 2 ∆ + C 3 , a 1 = a 1 , δ = − , 4 4 2 (a1 )2 (45) where C1 , C2 , C4 , C7 are arbitrary constants, q 6= µ2 , a1 (6= d ∆(z) . 0) and ∆(6= 0) are arbitrary function of z, ∆0 = dz Case 5. a1 ∆ = a0 = 0, b1 = ±  , Γ = C2 , λ = C4 , −q + µ2

3 βa1 λ2 µ + 3 δa0 a21 + 3 δa0 b21 µ2 − 3 δa0 b21 q = 0, 2 (35) −b1 (2 βΓ∆ + Γ0 ) = 0,

b01 , Γ = C1 , η = −C2 C1  βdz + C4 , b1

where C1 , C2 , C3 , C4 are arbitrary constants, β and b1 are d arbitrary functions of z, b01 = dz b1 (z). case 2.

η= 1 b1 (−2Ω0 − βΓ2 + 6 δa20 + 2b21 δ) = 0, 2

C22 β , (b1 )2

(36)

1 1 βa1 Γ2 − a1 Ω0 + βa1 λ2 − 6 δa0 b21 µ = 0, 2 2 (37)

1 1 βC4 2 (−q + µ2 ) 1 βdzC4 2 −  βdzC2 2 +C7 , δ = − ,  4 2 4 (a1 )2

βb1 λΓµ + a01 + β∆a1 + b1 η 0 )µ − αa1 = 0,

(38)

Ω=−

−a1 (η 0 + βλΓ) = 0,

(39)

a01 , a1 = a1 , β = β, η = −C4 C2  β(z)dz + C6 , a1 (46) where C2 , C4 , C6 , C7 are arbitrary constants, q 6= µ2 , a1 (6= d 0) and β(6= 0) are all arbitrary function of z, a01 = dz a1 (z) . Case 6.

a1 (δa21 + 3 δb21 µ2 − 3 δb21 q + βλ2 µ2 − βλ2 q) = 0,

(40)

b1 (−δb21 q + βλ2 µ2 − βλ2 q + δb21 µ2 + 3 δa21 ) = 0,

(41)

α=

Note: In the above ODEs system, a0 and a1 denote a0 (z) and a1 (z), respectively, and so on. Solving Eqs. (21)-(41) by means of Maple, we obtained the following results.

a0 = 0, β = −

226

1 ∆0 1 −2 ∆a01 + a1 ∆0 , λ = C4 ∆, α = − , 2 2 (∆) 2 a1 ∆

a1 = a 1 , δ =

1 ∆0 C4 2 (−q + µ2 ) 1 , η = C4 C2 ∆(z) + C6 , 8 (a1 )2 2 1 1 ∆C4 2 + C2 2 ∆ + C7 , 8 4 a1 ∆ = ∆, b1 = ±  , −q + µ2

Γ = C2 ∆, Ω =

(47)

where C2 , C4 , C6 , C7 are arbitrary constants, q 6= µ2 , a1 (6= d ∆(z). 0) and ∆(6= 0) are all arbitrary function of z, ∆0 = dz Therefore from (13), (16), (19), (20), (42)-(47), we obtain six families of exact solutions for NLSE (1) as follows: Family 1-2. From Case 1-2, two solutions of NLSE (1) are as follows: u1

u2

=

where C2 , C4 , C6 , C7 are arbitrary constants and

β tanhq  C2 t − C1 C2  βdz + C4  δ C 2 + 2C22 × exp  i  Ct − 1  βdz + C3  , (48) 2

β=−

C2  −

α=

where C1 , C2 , C3 , C4 are arbitrary constants and β 0 δ − βδ 0 . 2βδ

(50)

Family 3-4. From Case 3-4, two solutions of NLSE (1) are as follows: u3

β 1 tanhq  C2 ∆(t + ) + C4  δ 2 C 2 + 2C22 exp  i  ∆(t2 + C1 t + 1 ) + C3  , (51) 4

= C2 ∆  −

qβ 1 sechq  C2 ∆(t + ) + C4  δ 2 C 2 + 2C22 exp  i  ∆(t2 + C1 t + 1 ) + C3  , (52) 4 where C1 , C2 , C3 , C4 are arbitrary constants and u4

= C2 ∆ 

β=−

1 ∆0 δβ 0 − δ 0 δ − β∆. , α= 2 2∆ 2δβ

(53)



=

µ2 − q C4 sinhq (ξ) β  − δ  µ + cosh (ξ) ± µ + cosh (ξ) 2 q q × exp  i  Ct −

C42 + 2C22  βdz + C7  , (54) 4

where C2 , C4 , C6 , C7 are arbitrary constants and ξ = C4 t − C4 C2  βdz + C6 , α = Family 6. u6

=



δβ 0 − δ 0 β . 2δβ

(57)

Figure 1 shows the snake-solitons and boomerang-like solitons scenario given by Eq. (49). Dispersion managed (DM) soliton u3 with periodic dispersion coefficient is shown in Figure 2. Figure 3 represent the two DM solitons interaction in the case of equal phase and three exponentially decreasing bright quasi-solitons interaction. Figure 4 shows the snakesoliton propagation and snake-solitons interaction scenario given by u5 on the space periodic surface. It is necessary to point out that when β and δ are linearly dependent in u5 , the figures are similar to Figure 1. Figure 5 shows the propagation of W -shaped soliton u6 and two W -shaped solitons interaction with the initial separation equal to 4.

Family 5. From Case 5, a solution of NLSE (1) is derived as follows: u5

δβ 0 − δ 0 δ − β∆. 2δβ

Remark 2. When setting C1 = 0, q = 1, the Theorems 1-2 [32] and Theorems 1-2 [33] can be reproduced by our results (51)-(53); 2) The solutions obtained in [15, 22, 29] are special cases of our results Families 1-4 and can also be recovered by selecting arbitrary functions and arbitrary constants suitably. But to our knowledge, the other solutions have not been reported earlier. In order to understand the significance of these solutions expressed by (48)- (57), the main soliton features of them were investigated by using direct computer simulations with the accuracy as high as 10−9 . We have investigated the interaction dynamics of particle-like solutions obtained and the in influence of high-order effects on the dynamics of dispersion and amplification management. As follows from numerical investigations elastic character of chirped solitons interacting does not depend on a number of interacting solitons and their phases. For simplicity, we only consider some examples for each solution under q = 1 and some special parameters. Here, U = |u(z, t)|2 denotes the intensity of solution, the dispersion coefficient β is taken as periodic function and nonlinear coefficient δ is taken as periodic function or constant and 1 ∆=  . (58) 2βdz + C0

βq sechq  C2 t − C1 C2  βdz + C4  δ C 2 − C12 × exp  i  C1 t + 2  βdz + C3  , (49) 2

= C2 

α=

1 ∆0 C2 , ξ = C4 ∆(t + ) + C6 , 2 ∆2 2

(55)

4. SUMMARY AND DISCUSSION In summary, the method developed provides a systematic way to discover and investigate an infinite number of various solitons for NLSE (1) with varying dispersion, nonlinearity, and gain or absorption. The solutions obtained include some more general solutions (e.g., u3 , u4 under C1 6= 0) and some

µ2

−q C4 ∆ β sinhq (ξ)  − δ  µ + cosh (ξ) ± µ + cosh (ξ) 2 q q exp  i  ∆(t2 + C2 t +

C42 + 2C22 ) + C7  , (56) 8

227

Figure 3: Fig.3(a) depicts two DM bright solitons interaction given by u4 with β = 1 − 0.9 sin(z/2)2 , δ = 100, C0 = 40, C1 = 0, C2 = 30, C3 = C4 = 0; Fig.3(b) depicts mutual interaction of three exponentially decreasing bright quasi-solitons u4 in the case of the equal amplitudes and phase with β =  exp(0.003z)/10000, δ = 1/( 2βdz + 2)2 = 1/∆2 , C0 = 2, C1 = C3 = C4 = 0, C2 = 1.

Figure 1: snake-solitons and boomerang-like solitons interaction scenario given by bright solitons u2 . Input conditions: 1) β = − cos(z/4), δ = −10 cos(z/4), C1 = 6, C2 = 0.6, C4 = 0 in Fig.1(a); 2) β = δ = 1 − 0.14z, C1 = C3 = 1, C2 = 1/2, C4 = 0 in Fig.1(b).

Figure 4: Fig.4(a) depicts snake-soliton propagation given by u5 with µ = 2, β = δ = cos(z/4), C2 = 6, C4 = 1.2, C6 = C7 = 1.2. Fig.4(b) depicts snake-solitons interaction scenario given by u5 on the space periodic surface f (z, t) = 1/16 exp(sin(z/3)). Input conditions: µ = 2, β = cos(z/3) exp(sin(z/3)), δ = cos(z/3), C2 = 8, C4 = 1/2, C6 = 1.

Figure 2: Evolution of the dispersion managed dark solitary waves given by u3 with β = − cos(4z), δ = −10 cos(4z), C0 = 3, C1 = 0, C2 = 2, C3 = 1, C4 = 0 in Fig.2(a); β = 1 − 0.8 sin(z)2 , δ = 0.01, C0 = 60, C2 = −30, C4 = 0 in Fig.2(b).

228

[4] [5]

[6]

Figure 5: Fig.5(a) depicts one W -shaped soliton propagation and Fig.5(b) depicts two W -shaped solitons interaction scenario given by u6 with the initial soliton separation equal to 4. Input conditions: µ = 2, β = 1 − 0.8 sin(z)2 , δ = 0.1, C0 = 20, C2 = 0, C4 − 30, C6 = 1.

[7]

[8]

[9] new types of solutions (e.g., u5 , u6 under µ 6= 0). As shown in Fig.5, the new formal solutions u5 , u6 possess the dark soliton-like properties. The other solutions’ pure solitonlike features are confirmed by accurate direct computer simulations. The results obtained in this letter are of general physics interest and should be readily experimentally verified. We hope the method given here is useful to study the soliton phenomena in soliton theory. At the same time, it is necessary to point out that the problem with optical solitons is not merely the factors mentioned in the paper. The PDE governing the envelope for pulse propagation is the High-Order Nonlinear Schr¨ odinger equation (HONSE) for which the NLSE is only an approximation. The higher-order effects have a destructive effect on the NLSE soliton solutions which are felt for sufficiently small pulses or sufficiently large distances. Mathematical Soliton solutions have been found for the HONSE [10] but unfortunately these do not correspond to physical regimes. The algorithm presented is an unified method and can be also applied to a broad class of PDEs in mathematical physics. In addition, whether we can further improve the algorithm and develop new algorithm is still an interesting and open problem to construct more types of analytical solutions of PDEs.

5.

[10]

[11]

[12]

[13]

[14]

[15]

[16]

ACKNOWLEDGMENTS

The work is partially supported by the National Natural Science Foundations of China (No. 90203001 and No. 10475055) and a National Key Basic Research Project of China (No. 2004CB318000), Postdoctoral Science Foundation of China and Zhejiang Provincial Natural Science Foundation of China (No.Y604056).

6.

[17]

[18]

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230

Half-GCD and Fast Rational Recovery Daniel Lichtblau Wolfram Research, Inc. 100 Trade Center Drive Champaign, IL 61820, USA

[email protected]

ABSTRACT

the corresponding middle values for the original pair of inputs. Various analyses explain why this is asymptotically fast compared to direct computation of the full Euclidean sequence (we refer below to this latter as the “standard” Euclidean algorithm). Since its introduction in the early 1970’s, the asymptotically fast HGCD idea has given rise to several variants and descriptions thereof. This state of affairs has come to pass because of difficulties encountered in proofs of correctness. It turns out that the integer case is particularly troublesome due to the possibility of carries that may cause intermediate values to be too large or too small relative to what the algorithm requires. Several papers [3, 15, 13] redress this with fix-up steps that involve a limited number of Euclidean steps or reversals thereof. These papers tend to have proofs that involve analysis of many detailed cases, thus making them difficult to follow, let alone implement. (To be fair, they strive for greater generality in some respects). The main contribution of this paper is to provide a simple formulation with straightforward proofs. As testimony to its relative simplicity, the GCD method we present is now implemented as of version 5.1 of Mathematica (TM) [17]. It is an improved version of that which appeared in version 5.0. The prior work was coded by David Terr, with assistance from Mark Sofroniou and the author, in early 2001. It could be described as a “Las Vegas” approach insofar as it is always correct but only probabilistically fast; in practice we have never noticed it to falter. The fully deterministic method of this paper was coded by the author and Mark Sofroniou. Some important uses of asymptotically fast GCD to date are in finding greatest common divisors of pairs of large univariate polynomials or integers. An important advantage it enjoys is that, with little loss in efficiency, it finds corresponding cofactors when needed (that is, it computes the extended GCD). This is required, for example, in Hermite normal form computations. Moreover in finding cofactors for steps that take us half the distance to the GCD, the HGCD is ideally suited to fast recovery of rationals from padic images (as we will see, the code involved is trivial). The second contribution of this paper is to show this as applied to linear equation solving. This will give some indication of speed improvement over a standard Euclidean algorithm based recovery method. In another recent paper, [14] take a different direction by operating on the low (rather than high) end of the inputs. This has the advantage that carries are no longer an issue. A possible drawback is that rational recovery becomes slightly

Over the past few decades several variations on a “half GCD” algorithm for obtaining the pair of terms in the middle of a Euclidean sequence have been proposed. In the integer case algorithm design and proof of correctness are complicated by the effect of carries. This paper will demonstrate a variant with a relatively simple proof of correctness. We then apply this to the task of rational recovery for a linear algebra solver.

Categories and Subject Descriptors F.2.1 [Analysis of Algorithms and Problem Complexity]: Numerical Algorithms and Problems—Numbertheoretic computations; I.1.2 [Symbolic and Algebraic Manipulation]: Algorithms—Algebraic algorithms; G.4 [Mathematical Software]: Algorithm Design and Analysis

General Terms Algorithms, Performance

Keywords Integer gcd, subquadratic arithmetic, rational recovery

1.

INTRODUCTION WORK

AND

RELATED

The “half GCD” (HGCD) algorithm, as described in [12, 1], works by taking the high parts of a pair of inputs in a Euclidean domain, first recursively finding the pair of elements in the Euclidean sequence for that pair that straddles the middle of the larger input as well as the 2 × 2 matrix that converts to this middle pair. It then uses these values to convert the original pair to something sufficiently smaller. This is repeated one time, along with Euclidean steps at appropriate points, in such a way that one obtains

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less transparent though they show how it may still be done. They present a timing comparison that clearly demonstrates the efficacy of their code. At present it is not simple to compare directly to ours due, to different installations of the underlying GMP bignum arithmetic facility [10] as well as possible differences in memory management and timing thereof, but they appear to be in the same ballpark. The author thanks two anonymous referees for detailed remarks and suggestions that improved the exposition of this paper, and thanks the second referee as well for bringing several errors in the draft to his attention.

2.

Lemma 2. Assume Rj is a nontrivial product of elementary matrices. Then | sj |≤ n/mj−1 and | tj |≤ m/mj−1 .

Proof. This is done by induction. The base case gives equalities. For the inductive step we will show that | sj+1 |≤ n/mj ; the case for | tj+1 | is handled similarly. By lemma 1(iii) we know b|sj+1 /sj |c = qj . Hence |sj+1 /sj | ≤ qj . Thus | sj+1 |≤ qj sj . By the inductive hypothesis sj ≤ n/mj−1 . Hence sj+1 |≤ qj n/mj−1 = bmj−1 /mj cn/mj−1 ≤ n/mj . This lemma is used to bound various quantities in the lemmas of the next section.

A QUICK REVIEW OF EUCLIDEAN REMAINDER SEQUENCES

Lemma 3. For m > n > 0 „ suppose « we are given a product „ « m u of elementary matrices times such that the result, n v „ « u satisfies u > v > 0. Then are a consecutive pair in the v „ « m remainder sequence for . n

Much of what we discuss in this section applies to general Euclidean domains once one adjusts definitions (e.g. of the floor function) as needed, but we restrict our attention to integers as the„case « of interest. We are given a pair of m (we will use column vector notation positive integers n throughout, as we frequently multiply on the left by a matrix) with m > n. We are interested in elements in the Euclidean remainder sequence m = m0 , n = m1 , . . . , mk . The integer quotients are the floor of the divisions of successive terms in this sequence. bmj−1 /m „ q«j = „ « j c. We define m mj = . For example, the matrix Rj such that Rj n mj+1 « „ 0 1 (matrices of this form are called “elemenR0 = 1 −q1 „ « 0 1 tary”) and Rj = Rj−1 . From this last it is clear 1 −qj that the top row of Rj+1 „ is just the bottom row of Rj . « sj tj . We state a few Hence we may write Rj = sj+1 tj+1 basic facts about these quantities.

This is presented as Fact 1 in [15]. It is important because it tells us that we may take a product of elementary matrices of the form Rj above, computed with respect to a new pair of integers, and still arrive at a consecutive pair in the remainder sequence for the original pair. Finally we remark that „ « there is obviously an index k for √ √ mk which the pair straddle m, i.e. mk ≥ m > mk+1 mk+1 . These together, and in order, are referred to as the “middle” pair in the remainder sequence (regardless of where the index occurs in the sequence of such indices).

3. BASIC THEORY FOR THE HGCD ALGORITHM „

« m with n m > n. Take k to be a positive integer less than the size of m in bits (initially it will be b log22 m c but we do not use that « „ « „ k m 2 f0 + f 1 with until the next section). We write = n 2 k g0 + g 1 {f1 , g1 } < 2k and recursively compute the middle „ pair, « and f0 corresponding multiplier matrix, for the pair . This g „ „ « 0 „ « « ri f0 ri gives a matrix Ri and pair with Ri = ri+1 g0 ri+1 √ want to use R , or a close relaand ri+1 < f0 ≤ ri . We i « „ k „ « 2 f0 + f 1 m = tive, on the original pair . We have Ri k n 2 g0 + g 1 « „ „ « « „ vi f ri . The +Ri 1 . We will call this product 2k vi+1 g1 ri+1 next two lemmas will find bounds, one upper and one lower, for these elements. We first handle vi+1 . We bound the absolute value and, under certain circumstances, we place a tighter bound on how negative it may become. This is important because, in order to invoke lemma 3, we will need a way to correct for the negative case. √ Lemma 4. (i) | v√i+1 |< 2k+1 f0 . √ (ii) Suppose ri > 2 f0 . Then vi+1 > −2k−1 f0 . √ Proof. Note that {si+1 , ti+1 } < f0 /ri < f0 . We have k vi+1 = 2 ri+1 + si+1 f1 + ti+1 g1 . Again we begin with a pair of positive integers

Lemma 1. Assume Rj is a nontrivial product of elementary matrices. (i) sj−1 =„sj+1 « tj−1 = tj+1 + qj tj . « + q„j sj and m mj (ii) If Rj then mj−1 = mj+1 + qj mj . = n mj+1 (iii) The signs in Rj alternate in both rows and columns: sj tj < 0 and sj sj+1 < 0. (iv) The sizes grow top to bottom and right to left: |sj+1 | > |sj |, |tj+1 | > |tj |, and |tj | > |sj |. (v) qj = b|sj+1 /sj |c = b|tj+1 /tj |c. Proof.„(i)-(iv) « Quickly proven by writing out the prod0 1 uct Rj = Rj−1 . 1 −qj (v) As qj sj = sj−1 − sj+1 , parts (iii) and (iv) together imply that |sj+1 | ≥ qj |sj | > (|sj+1 |−|sj |). This suffices to give the first floor equality for qj . The second is done similarly. This lemma shows how to compute „ Rj−1 and « mj−1 given mj Rj and the remainder sequence pair . The signifmj+1 icance is that we can “go back” in the Euclidean sequence should we happen to overshoot (this will be discussed later). Note also that in the special case of R0 , which is an elementary matrix, we can obtain q0 immediately. We easily recognize this case, as it arises if and only if the first matrix element is zero. We also use part (iii) to prove the next lemma.

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« «„ 0 1 1 0 is also an elementary 1 −qj h 1 matrix. In particular this holds for any negative integer h. « „ 0 1 and by defProof. The product is simply 1 h − qj inition this is elementary precisely when h − qj < 0.

(i) Using the upper bound of 2k on {f1 , g1 } and the alternating signs in √ the matrix value is bounded √ Ri , the absolute √ by | vi+1 |< 2k f0 +√2k f0 = 2k+1 f0 . f0 (ii) Since ri > 2 f0 we have vi+1 > −2k 2√ = f0 √ k−1 f0 . −2

h < qj the product

We will use these same notions in subsequent lemmas (particularly the sign alternation, in effect to ignore one of the three terms) without further mention. We now look at vi = 2k ri + si f1 + ti g1 . √ √ Lemma 5. (i) Suppose ri > 2 f0 . Then vi > 2k−1 f0 . √ k−1 . (ii) Suppose ri ≤ 2 f0 . Then vi > 2

„

We will use such to repair deficiencies in sign or « „ products vi . order of a pair vi+1

4. THE HGCD ALGORITHM Input: A pair of nonnegative integers m > n. „ « „ « vi m Output: A pair = Ri of consecutive invi+1 n „ « m tegers in the Euclidean remainder sequence for with n √ vi ≥ m > vi+1 , and a matrix Ri which is the product „ « of m = elementary transformation matrices, such that Ri n „ « vi . vi+1 Step 1: With the same input specification as in the previous sections, we begin by choosing k = b log22 m c. Thus, as „ « „ k « m 2 f0 + f 1 above, we write = with {f1 , g1 } < 2k . n 2 k g0 + g 1 √ k m > 2k−1 and Moreover the √ choice of k gives 2 > g0 ≤ f0 < 2 m. „ « f Step 2: Recursively compute HGCD 0 . With notation g0 as the last section, « is a matrix Ri and pair „ result „ « the „ in « √ ri f ri and 0 ≤ ri+1 < f0 < ri . with Ri 0 = ri+1 g0 ri+1 « „ „ « vi m = Ri Step 3: Compute . Note that we aln vi+1 ready have „ the«“upper part” of the resulting vector comri ; this can be used to reduce the size of the puted as ri+1 multiplications in this step. Step 4: The bounds presented in the lemmas do not rule out the possibility that vi+1 may be negative, or „ « „ that « vi < u vi vi+1 . If vi > vi+1 > 0 then we set = and vi+1 v move to step 5 at this point. Otherwise we must repair the pair, in such a way that the transformation matrix remains a product of elementary matrices. This is necessary so that we may invoke lemma 3 to know the resulting „ vector « is a m consecutive pair in the remainder sequence for . We n split into three cases that together comprise all possibilities. Suppose vi+1 > vi . Take the matrix H = „ Case (i). « 1 0 where h = bvi+1 /vi c ≥ 1. By lemma 7 HRi −h 1 is a product of elementary The new « pair thus ob«matrices. „ „ « „ vi u vi which satisfies = tained is =H vi+1 − hvi vi+1 v the requirement that u > v > 0. For purposes of notation we continue to call the resulting matrix Ri . Note that the value of u is unchanged (hence lemma bounds still apply), while the absolute value of v has diminished. We now move on to step 5.

Proof. Lemmas 1 and 2 sign alteration and size bounds f0 in Ri gives vi = 2k ri + si f1 + ti g1 > 2k ri − 2k ri−1 . (i) The hypothesis √and the fact that ri−1 > ri yield vi > √ f0 2k (ri − 2√ ) > 2k 2f0 = 2k−1 f0 . f 0

k

2 (ri ri−1 − (ii) Now we write the lower bounding value as ri−1 √ f0 ). Since f0 ≤ ri < ri−1 and the latter two are in√ tegers, there is an a > 1 with ri−1 = f0 + a. So we √ k have vi > √f2 +a (ri ( f0 + a) − f0 ). This in turn is larger ( 0 ) √ √ √ k k than √f2 +a ( f0 ( f0 + a) − f0 ) = √f2 +a a f0 which is ( 0 ) ( 0 ) bounded below by 2k−1 . „ « „ « m vi For the pair lemmas 4 and 5 give = Ri n vi+1 an upper bound on one element and a lower bound on the other. There will be situations in which we must backtrack a Euclidean step to use Ri−1 , that is, the multiplier „ « matrix f0 preceding Ri in the remainder sequence for . In this g « „ „ « 0 vi−1 m = Ri−1 case we need to bound . n vi

√ k−1 Lemma 6. (i) vi−1 f0 . √ √ >2 (ii) Suppose ri ≤ 2 f0 . Then vi < 2k 3 f0 .

Proof. (i) vi−1 = 2k ri−1 + si−1 f1 + ti−1 g1 > f0 0 = 2k ri−1 − 2k qi−1 rfi−1 > 2 ri−1 − 2k ri−2 +ri √ √ √ f0 f0 k k k−1 2 (ri−1 − 2ri ) > 2 ( f0 − 2 ) = 2 f0 . √ f0 (ii) vi = 2k ri + si f1 + ti g1 ≤ 2k 2 f0 + 2k ri−1 < √ √ f0 k k√ k 2 2 f 0 + 2 f = 2 3 f0 . k

0

„ « m with m > n > 0 we will see that the n « „ vi with magniabove lemmas allow us to find a pair vi+1 tudes in the desired ranges (this will be explained more carefully in the next section). Two problems may arise. One is that we require both to be nonnegative; the lemmas will only guarantee that vi > 0. Second, we require that vi > vi+1 . These requirements are in order to meet the hypotheses of lemma 3 and thus assert that we have a consecutive pair in the remainder sequence for our inputs. We now provide a lemma to assist in repairing our intermediate pair, should either of these possible flaws arise. „ « 0 1 Lemma 7. Given an elementary matrix 1 −qj (this implies qj is a positive integer). Then for any integer Given a pair

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√ to obtain the pair straddling m, returning it and the corresponding transformation matrix. √ Case (ii). m ≤ w < v < 23/2 3m3/4 (in typical examples, w and v will both be close to m3/4 ). Similarly to step 1, we take l = blog 2 mc − blog2 vc (so 2l is within a factor of 2 of m/v; we will soon see why this is the appropriate value). Observe that l is roughly between one fourth and one half the bit length of m. Specifically, we have blog 2 mc/4 − 3 < to step l < blog2 mc/2 + 3. We proceed „ « « 7.„ l v 2 f2 + f 3 Step 7: This time we write with = l w 2 g2 + g 3 blog2 f2 c = blog 2 vc − l. The upper bound on log 2 v and lower √ bound on l show that f2 and g2 are no larger than O( m). This fact is required for the claim of asymptotic speed (though not for correctness). „ « f As in step 2, recursively compute HGCD 2 . As in steps g2 3 and 4 we obtain a transformation matrix S, and a„ con„ « « vj m secutive pair in the remainder sequence for , n vj+1 √ l−2 f2 then the condition of with vj > vj+1 ≥ 0. If vj ≤ 2 lemma 5(i) cannot hold, and thus lemma 6(ii) applies. So we do a single reverse Euclidean step to get the previous consecutive pair in the sequence. At this point we have a „ « x , wherein lemma 6 guarantees consecutive pair, call it y l+2 √ l−2 √ that y < 2 f2 and x > 2 f2 . Step 8: From step 7 we√know that√f2 is p within a factor √ m√ of 2 of 2−l v and hence 2l f2 ≈ 2l/2 v ≈ v = m v where the approximation from first to last is within a factor of 2 because √ each intermediate approximation is within a factor of 2. The √ inequalities√at the end of step 7 therefore imply y < 8 m and x > m/8; this was the point in selecting l as we did. Thus with a limited number of Euclidean steps, or reversals thereof, we obtain the√consecutive pair in the remainder sequence that straddles m, and the transformation matrix that gives this pair. Possibly with care we might tighten the bound on the number of forward or reverse Euclidean steps. In practice this is unimportant. One simply codes a while loop for the iterations; that it terminates in a fixed number of steps suffices to prove the claim of asymptotic speed.

Case (ii). Suppose vi+1 < 0 and vi+1 + vi „≥ 0. «First 1 0 and assume qi > 1. Then we use the matrix H = 1 1 proceed as in case (i) above, to obtain a positive pair in the correct « „ order.« This is appropriate because the product „ 0 1 1 0 is an elementary matrix so we may in1 −qi 1 1 „ « u voke lemma 3. Again we call the resulting pair , and v continue to call the transformation matrix Ri . Note that √ u − v =| vi+1 |< 2k+1 f0 . This means that a Euclidean step will bring the pair into the range claimed in step 6 below. If qi = 1 the situation is a bit more subtle. Again we use the matrix H as defined above, and again we obtain a positive pair „ « in the correct order, but now the product HRi 0 1 is which is not an elementary matrix. To correct 1 0 „ « 0 1 for this we multiply by again, giving as product the 1 0 „ « u . Thus identity matrix. This has the effect of flipping v we have used premultipliers to take us from Ri to Ri−1 , which we know is also a product of „elementary matrices. « v We have also obtained as our vector ; it has appropriu ate components except they are in the wrong order. As this is exactly the situation of case (i) above we proceed there to correct it. Case (iii). √ If vi+1 < 0 and√vi+1 + vi < 0 then either vi ≤ 2k−1 f0 or vi+1 ≤ −2k−1 f0 . In either case, √ lemmas 5(i) and 4(ii) respectively guarantee that ri ≤ 2 f0 . We will perform a reversal of a Euclidean step, obtaining the „ « „ « „ « √ m u vi−1 . As ri ≤ 2 f0 , lemma pair = = Ri−1 n vi v √ 5(ii) guarantees that v > 0 and furthermore v < 2k 3 f0 by lemma 6, so again the bounds given in step 6 will hold. If u < v then we go to case (i) above. „ « u Step 5: Perform a Euclidean reduction on . We obv „ « v tain the next consecutive pair in the remainder sew „ « m quence for , with elementary transformation matrix n „ « 0 1 Q= , where q = bu/vc, and w = u − qv. We form 1 −q the corresponding transformation matrix R = QRi . « 6: At this point we examine the values of our pair „ Step v . Lemmas 4, 5, 6, the remarks from the step 4 cases, w √ and our choice of k (implying f0 < 2 m and 2k−1 < √ √ k m) guarantee that 0 < v < 2 3 f0 < 2k+1/2 3m1/4 < √ 3/2 2 3m3/4 and u√> 2k−1 > √m/4. √ Case (i). w < m. If v ≥ m we have our pair straddling m. We √ return it along with the transforming matrix R. If v < m we do reverse Euclidean steps, updating our remainder sequence pair and transformation matrix using the √ formulas in lemma 1. Since u > m/4 and it immediately precedes v in the remainder sequence, √ we have at most five such steps before an element exceeds m (possibly one could decrease this upper bound by constructing tighter bounds in the lemmas).We perform as many such steps as is needed

5. APPLICATIONS OF THE HGCD ALGORITHM First note that the asymptotic complexity is O(nM (n)) where n is the bit size of the inputs and M (n) is the complexity of multiplying a pair of number of that size. This is well known (see the various references) and follows from the fact that we do two recursive steps on numbers no larger than roughly n/2 (see steps 1 and 7 above), along with a bounded number of multiplications, Euclidean steps, and reverses thereof. It is this speed that motivates the various applications mentioned below. The HGCD algorithm is used recursively in gcd computations. An HGCD computation followed by a Euclidean step is guaranteed to reduce the size of the inputs (in bits) by at least half. Another advantage is that one gets the corresponding multiplier matrix for free, so computation of the extended gcd is not much more costly than that of the ordinary gcd. This is important for e.g. matrix Hermite normal form or integer Gr¨ obner basis computations [11], where

234

speed of extended gcds is paramount. As a standard benchmark example we will find the gcd of a pair of consecutive large Fibonacci numbers. This and all other timings are from runs using version 5.1 of Mathematica under Linux on a 1.4 GHz Athelon processor.

testPAdicSolver[dim_Integer, csize_Integer, recoveryfunc_] := Module[ {ls1, ls2, mat, b}, mat = Table[Random[Integer, {-10^csize,10^csize}], {dim}, {dim}]; b = Table[Random[Integer, {-10^csize,10^csize}], {dim}]; {First[Timing[ls1 = pAdicSolve[mat,b,recoveryfunc]]], First[Timing[ls2 = LinearSolve[mat, b]]], ls1 === ls2, Max[Log[10., Denominator[ls1]]]}]

fibs = Fibonacci[10^7],Fibonacci[10^7+1]}; Each has about two million digits. We compute both regular and extended gcd and check that the result is plausible. Timing[gcd=Apply[GCD,fibs];] Timing[gcd2,mults}=Apply[ExtendedGCD,fibs];] mults.figs==gcd==gcd2==1

In the set of tests below input data will consist of 10-digit integers. First we try a 50 x 50 system.

{31.04 Second, Null} {40.21 Second, Null} True

testPAdicSolver[50, 10, rationalRecover] {1.35 Second, 0.62 Second, True, 517.912}

A particularly nice application of the HGCD is in recovering rational numbers from p-adic approximations. This is explained in some detail in chapter 5 of [8]. Given a prime power pk and a smaller nonnegative integer x not divisible by p, we can obtain a rational a/b equivalent to x modulo pk with both numerator and denominator smaller than the square root of the prime power. It is obtained directly from „ kthe « HGCD matrix and midp dle pair given by HGCD . In brief, we have a max „ k« „ « „ « u sj tj p . Moreover = trix Rj = with Rj sj+1 tj+1 v x p v {v, tj+1 } < pk and x ≡pk tj+1 because sj+1 pk +tj+1 x = v. Thus we have our desired rational. The below code will do this recovery given the input pair {x, pk }.

The built in method was faster by a factor between 2 and 3. The standard Euclidean algorithm of rationalRecover2 makes it about three times slower still, thus indicating that even at this low dimension most of the time might be spent in rational recovery if we use a pedestrian approach. This example takes about 17 seconds using rationalRecover3. We now double the dimension. testPAdicSolver[100, 10, rationalRecover] {11.79 Second, 9.04 Second, True, 1053.32} This time the speeds are quite close. Doubling again will show the p-adic solver well ahead. testPAdicSolver[200, 10, rationalRecover]

rationalRecover[x_, pk_] := ((#[[2, 2]]/#[[1, 2, 2]]) &)[Internal‘HGCD[pk, x]]

{93.26 Second, 173.27 Second, True, 2136.99}

For contrast we also give the standard Euclidean sequence method as well as a simple method based on lattice reduction.

We remark that most of the time is spent in finding the p-adic approximate solutions. The utility of fast rational recovery is indirectly witnessed in the above computations; were we to use a less efficient method, it would become more prominent in the timings. As it stands, the overwhelming component is now in the improvement iterations. We should note that one can use a very different iterative approach when the input matrix is well conditioned. One can solve the system numerically to sufficiently high precision using the iterative method of [9]. Then the exact result may be recovered using rationalization of a high precision approximate result. Interestingly, the technology underlying this type of rational recovery involves continued fractions; efficient computation of these is similar to the divide-andconquer approach of HGCD. A potential drawback to this method (in addition to the conditioning requirement) is that it requires an a priori precision estimate that might be quite large, or else an expensive check of correctness that could outweigh the cost of the actual construction of a solution. Quite recently a related method based on iterative refinement of numerical solutions was described in [16]. It uses rescaling of residuals and stepwise rational approximations to construct its result. At this time it appears to be the state of the art in solving linear systems over the rationals. That said, clearly there remains a need for fast rational recovery from a p-adic approximation. For example, other recent methods requiring rational recovery have been discussed in

rationalRecover2[a_, b_] := Module[ {mat, aa=a, bb=b, cc=1, dd=0, quo}, mat = {{aa,cc}, {bb,dd}}; While[Abs[aa] >= Sqrt[b], quo = Quotient[bb,aa]; {{aa,cc}, {bb,dd}} = {{bb,dd} - quo*{aa,cc}, {aa,cc}};]; aa/cc] rationalRecover3[n_, pq_] := (#[[1]]/#[[2]] &) [First[LatticeReduce[{{n,1},{pq,0}}]]] We illustrate this application by solving linear systems over the rationals, using a simple p-adic linear solver based on the method presented in [5] (the code for pAdicSolve is omitted for brevity but available from the author). To get some idea of speed we will compare to the built in LinearSolve function. The latter at this time uses a Gaussian elimination via one-step row reduction [2]. The tests we use will involve creating random linear systems of a given dimension and coefficient size in decimal digits. In the results we will show timings, a check that the results agree, and the size in decimal digits of the largest denominator in the result.

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[7] J.-G. Dumas, P. Giorgi, and C. Pernet. FFPACK: finite field linear algebra package. In ISSAC ’04: Proceedings of the 2004 International Symposium on Symbolic and Algebraic Computation, pages 119–126, New York, NY, USA, 2004. ACM Press. [8] J. von zur Gathen and J. Gerhard. Modern Computer Algebra. Cambridge University Press, New York, NY, USA, 1999. [9] K. O. Geddes and W. W. Zheng. Exploiting fast hardware floating point in high precision computation. In ISSAC ’03: Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation, pages 111–118, New York, NY, USA, 2003. ACM Press. [10] The gnu multiprecision bignum library. [11] D. Lichtblau. Revisiting strong Grbner bases over Euclidean domains. submitted, 2003. [12] R. T. Moenck. Fast computation of gcds. In STOC ’73: Proceedings of the Fifth Annual ACM Symposium on Theory of Computing, pages 142–151, New York, NY, USA, 1973. ACM Press. [13] V. Y. Pan and X. Wang. Acceleration of Euclidean algorithm and extensions. In ISSAC ’02: Proceedings of the 2002 International Symposium on Symbolic and Algebraic Computation, pages 207–213, New York, NY, USA, 2002. ACM Press. [14] D. Stehl and P. Zimmermann. A binary recursive gcd algorithm. Rapport de recherche 5050, INRIA, 2003. [15] K. Thull and C. Yap. A unified approach to hgcd algorithms for polynomials and integers. Manuscript. Available from http://cs.nyu.edu/cs/faculty/yap/allpapers.html/, 1990. [16] Z. Wan. Exactly solve integer linear systems using numerical methods. Submitted. Preprint available from http://www.eecis.udel.edu/~wan/, 2004. [17] S. Wolfram. The Mathematica Book. Wolfram Media, Champaign, IL, USA, 5th edition, 2003.

[7, 4]. Both derive speed by clever use of level 3 BLAS for modulo prime linear algebra as described in [6]. While [4] describes ways to speed this process considerably even when using the standard Euclidean method, it remains true that an asymptotically fast rational recovery is a desirable further improvement.

6.

SUMMARY

We have demonstrated a correct, asymptotically fast integer GCD algorithm based on the classical Half-GCD method. The various correction steps needed to address deficiencies caused by integer carries are, we believe, relatively simple both from the standpoint of theory and practical implementation. We apply this to solving large linear systems over the rationals, with results that scale well with dimension.

7.

REFERENCES

[1] A. V. Aho, J. E. Hopcroft, and J. D. Ullman. The Design and Analysis of Computer Algorithms. Addison-Wesley, Reading, Massachussetts, USA, 1974. [2] E. H. Bareiss. Sylvester’s identity and multistep integer-preserving Gaussian elimination. Mathematics of Computation, 22(103):565–57, 1968. [3] R. P. Brent, F. G. Gustavson, and D. Y. Yun. Fast solution of Toeplitz systems of equations and computations of Pad´e approximants. Journal of Algorithms, 1:259–295, 1980. [4] Z. Chen and A. Storjohann. A BLAS based C library for exact linear algebra on integer matrices (these proceedings). In ISSAC ’05: Proceeding of the 2005 International Symposium on Symbolic and Algebraic Computation, New York, NY, USA, 2005. ACM Press. [5] J. D. Dixon. Exact solutions of linear equations Using p-adic methods. Numerische Math., 40:137–141, 1982. [6] J.-G. Dumas, T. Gautier, and C. Pernet. Finite field linear algebra subroutines. In ISSAC ’02: Proceedings of the 2002 International Symposium on Symbolic and Algebraic Computation, pages 63–74, New York, NY, USA, 2002. ACM Press.

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Application of Wu’s Method to ∗ Symbolic Model Checking 1

Weibo Mao1

2

Chengdu Institute of Computer Applications Chinese Academy of Sciences Chengdu 610041, China

[email protected]

School of Information Science & Engineering Lanzhou University, Lanzhou 730000, China 3 Universitat ¨ Mannheim, D7, 27, 68131 Mannheim, Germany

[email protected]

ABSTRACT

failure is unacceptable: electronic commerce, highway and air traffic control systems, and many others. Usually, methods for design validation are simulation and testing. They can find errors, but cannot cover all bugs. Formal verification approaches, e.g. model checking, can provide this guarantee. Several approaches to formal verification of concurrent systems have been proposed over the years. Compared to others, symbolic model checking enjoys some advantages. It is fully automatic, and its application requires no user supervision or expertise in mathematical disciplines. In addition, It uses Ordered Binary Decision Diagrams (OBDDs) to reduce space, which enable model checking to alleviate the state explosion problem, and therefore to be able to verify large scale systems. Many tools based on symbolic model checking have been developed and been widely used in practice, for instance, SMV [5], VIS [14] and NUSMV [15]. In model checking, Kripke structure models the system, and Computation Tree Logic (CTL) formula specifies the property to be verified. By means of the manipulation of Boolean formulas, symbolic model checking uses OBDDs to represent Kripke structures and CTL formulas. However, with the system becoming larger and more complex the size of OBDDs greatly increases, and there are many concurrent systems for which OBDDs are too large to make model checking feasible. Moreover, the gap is widening between the system to be verified and the capability of symbolic model checking. Symbolic computation is very active in recent years, and has been successfully applied in many scientific fields, for example, geometry theorem proving [21, 23, 24], robotics [25], computer vision [31] and signal processing [26]. Also, it seems to have promise to bring the powerful mathematical machinery to settle the model checking problem. As an important method and tool, Wu’s method performs mechanical geometry theorem proving better than other ways in symbolic computation, and has aroused attention to symbolic computation once more. It mainly calculates a characteristic set of a finite set of polynomials, which was first discussed by J. F. Ritt [16] in the context of his work on differential algebra, which was received little attention until Wu Wen-Ts¨ un revised it independently [21] and successfully applied it to geometry theorem proving. This paper shows how Wu’s method can be applied to symbolic model checking. The main idea of the framework is that any Boolean vector set can be regarded as the common

Model checking is widely used in verifying properties of concurrent systems. As an improvement, symbolic model checking uses OBDDs to alleviate the state explosion problem. However, the size of OBDDs greatly increases with the system becoming larger and more complex. In symbolic computation Wu’s method is efficient to solve algebraic equations, and has succeeded in geometry theorem proving. In this paper, we present a framework to apply Wu’s method to symbolic model checking. Briefly, symbolic model checking is carried out by using Wu’s method to calculate the characteristic sets of polynomials that represent Kripke structures as well as CTL formulas. This is a new approach to model checking, and is a new successful application of Wu’s method.

Categories and Subject Descriptors I.1.4 [Symbolic and Algebraic Manipulation ]: Applications—algebraic applications

General Terms Algorithms, Theory

Keywords symbolic model checking, Wu’s method, characteristic set

1.

Jinzhao Wu1,2,3

INTRODUCTION

Formal verification is a very attractive approach to the verification of current systems [1, 2, 3]. Nowadays hardware and software systems are widely used in applications, where ∗Partially supported by a NKBRPC (2004CB318000), and by the National Natural Science Foundation of China (60373113).

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. ISSAC’05, July 24–27, 2005, Beijing, China. Copyright 2005 ACM 1-59593-095-7/05/0007 ...$5.00.

237

f

zeros of a finite set of polynomials in the Boolean field. For example, the zeros of the set of polynomials {x1 , x2 (x3 + 1)} are the points a1 , a2 , a3 , where a1 = 1, and at least one of a2 = 0 and a3 = 1 is satisfied. Wu’s method is a method of representing the zeros of a set of polynomials with the characteristic sets, just as OBDDs represent the true values of a set of Boolean formulas. Symbolic model checking can be carried out through representing the system by polynomials and then by finding the common zeros of the polynomials. We can use Wu’s method to perform symbolic model checking by calculating the characteristic sets of polynomials. Symbolic model checking has also the state explosion problem. In the general case, the size of OBDDs behaves fairly bad. Furthermore, OBDDs exponentially grow with variables increasing in a bad order. It seems worthwhile to investigate alternative “symbolic” representations of Kripke structures that are used for model checking. Wu’s method, if it is practical at all, would presumably allow efficient model checking of somewhat different classes of systems than OBDDs, and supplement the existent symbolic model checking approaches. An alternative representation might lead to new theoretical insights into the practicality of symbolic model checking. On the other hand, symbolic computation is successful in many fields recently, and efficient algorithms are developed. It may provide a new method to symbolic model checking other than OBDDs. The paper is structured as follows. In the next section, we introduce some basic and necessary definitions in symbolic model checking. Section 3 briefly reviews Wu’s method, and shows how to apply Wu’s method to symbolic model checking. We illustrate the ideas with an example by using Maple [32, 33] in Section 4. Section 5 is a discussion of our future work for this approach.

2.

0-edge

xi

1-edge xj xk

1

0

Figure 1: OBDDs for f = xi xj + xk

model computations of the system. A path in the structure M from a state s is an infinite sequence of states π = s0 s1 . . . such that s0 = s and R(si , si+1 ) holds for all i
0. The temporal logic CTL is a subset of modal branching time logic defined by Clarke and Emerson [6]. In CTL, temporal operators consist of A or E; followed by G (Global), F (Future), X (neXt), or U (Until). The syntax of CTL formulas is given as follows: 1. Every atomic proposition in AP is a CTL formula. 2. If f and g are CTL formulas, then so are ¬f , f ∧ g, AXf , EXf , A(f U g) and E(f U g). The other operators can be derived from these according to the following rules: f ∨ g = ¬(¬f ∧ ¬g); AFg=A(true U g); EFg=E(true U g); AGf = ¬E(trueU ¬f ); EGf = ¬A(trueU ¬f ).

SYMBOLIC MODEL CHECKING

Model checking is an automatic technique for verifying concurrent systems of finite states, which checks whether a given model has the desired properties. Generally, a Kripke structure M models the system, and a temporal logic formula φ represents a property. Model checking performs by checking whether all of the initial states of M are in the set of states that satisfy φ. In practice, Symbolic CTL model checking is mainly used. Symbolic CTL model checking uses OBDD representation for Kripke structures and CTL formulas to carry out model checking, which is usually called symbolic model checking. To find all states that OBDDs represent, fixed point theory in lattice is used to calculate the fixed point of a function. We will review this in the following, see [4, 7] for more details.

In symbolic model checking, a state is represented by a boolean formula through the labeling function in the Kripke structure. Every formula denotes a set of states therefore. For example, the formula x1 ∨ ¬x2 represents all the states in which x1 is true or x2 is false. Model checking is to check if (∀s ∈ S0 )[M, s |= p], or S0 ⊆ {s ∈ S : M, s |= p}. Here, the notation M, s |= p means that the state s in M is in the set of states satisfying p. Let P (S) be the set of all subsets of S, which forms a lattice under union and intersection. Let f : P (S) → P (S). Tarski [8] has verified that a monotonic function f has a least fixed point µx.f and a greatest fixed point νx.f , where µx.f = ∪i
0 f i (f alse) and νx.f = ∩i
0 f i (true). Here,
∪i f i = f (f i−1 ) (∪i−1 f i−1 ),

Definition 1. A Kripke structure is a tuple M = (S, S0 , R, AP, L), where S is a finite set of states, S0 ⊆ S is the set of initial states, R ⊆ S × S is a transition relation, AP is a set of atomic propositions, L : S → 2AP is the labeling function.

and ∩i f i is defined in the same way.

A Kripke structure is like a state transition graph (STG) used to model system generally. Paths in a Kripke structure

Theorem 1. For a finite Kripke structure

238

Let G = g1 , . . . , gr  be a finite sequence of polynomials g1 , . . . , gr ∈ K[x1 , . . . , xn ]. Let P = {p1 , . . . , pr } be a finite nonempty subset of K[x1 , . . . , xn ]: p1 , . . . , pr ∈ K[x1 , . . . , xn ].

M = (S, S0 , R, AP, L), there are (1) EF p = µy. (p ∨ EXy), (2) EGp = νy. (p ∧ EXy), (3) E(qU p) = µy. (p ∨ (q ∧ EXy)). The other CTL formulas can be derived from the above and the following facts:

3.1 Wu’s Method In the following, all the polynomials mentioned are in K[x1 , . . . , xn ]. A fixed ordering on the set of variables is considered. Without loss of generality, we assume that the given variables ordering is

AXf = ¬EX(¬f ); AGf = ¬EF (¬f ); AF f = ¬EG(¬f ); A(f U g) = ¬E[¬gU (¬f ∧ ¬g)] ∧ ¬EG(¬g).

x1 ≺ x2 ≺ . . . ≺ xn . Definition 2. A variable xj is said to be in f if some monomial in f with nonzero coefficient contains a positive power of xj . If xj is in f , and no variable xi with xi  xj is in f , then

By Theorem 1, the CTL operators can be characterize in terms of fixed points of appropriate functions. For simplicity, we only show how to calculate the formula EXφ, as all the calculations of fixed point for a CTL formula are based on the calculation of the formula EXy shown in Theorem 1. Kripke structures and CTL formulas are all represented by OBDDs in symbolic model checking, through Boolean functions representation of Kripke structures. OBDDs are a directed acyclic graph representation of Boolean functions, as shown in Figure 1. The 0-edge and 1-edge point to subfunctions when the variable is assigned 0 and 1, where the two terminal nodes represent the function value {0, 1} decided by variables assignments {0, 1}n . The following two rules [9] can be used to obtain a canonical form of an OBDD to a Boolean function:

class(f ) = j

cdeg(f ) = degxj (f ).

Definition 3. We say that f is of lower rank than g, if either 1. 2.

class(f ) < class(g), or class(f ) = class(g) and cdeg(f ) < cdeg(g).

Definition 4. Given a polynomial f = Id (x1 , . . . , xj−1 )xdj + + . . . + I0 (x1 , . . . , xj−1 ), with Id−1 (x1 , . . . , xj−1 )xd−1 j class(f ) = j and cdeg(f ) = d, its initial polynomial, In(f ), is defined to be the polynomial

1. Share all isomorphic subgraphs. 2. Delete all nodes whose left and right children are isomorphic.

Id (x1 , . . . , xj−1 ). We use In(P ) in the following to denote the set of the initial polynomials in P :

OBDDs are constructed by a sequence of logic operations from single-node OBDDs. The variable order has heavy impact on the size of OBDDs [13], which may exponentially augment with the number of variables increasing in an unlucky order. Since we need to calculate the CTL formula EXy, operations on OBDDs are shown to complete the calculations. Let f be the Boolean function representing the set of current states that satisfy y, and g be the Boolean function representing the transitions, both represented by OBDDs. Replace the current state variables V in f with the next state variables V
, such that f
= f (V ← V
). Then p = ∃V
(f
∧ g) represents EXy.

3.

and

If no variables exist in f , then class(f ) = 0 and cdeg(f ) = 0.

In(P ) = {In(pi ) : pi ∈ P }. Theorem 2. For any two polynomials f and g, with class(f ) = j. There exist a non-negative number α and polynomials q and r, such that In(f )α g = qf + r, where degxj (r) < degxj (f ) and α ≤ degxj (g) − degxj (f ) + 1. If r = g, then g is said to be reduced with respect to f , and r is denoted by prem(g, f, xj ) as usual. Definition 5. G is said to be an ascending set, if one of the following two conditions holds: 1. r=1 and g1 is not equal to zero; 2. r > 1, and 0 < class(g1 ) < class(g2 ) < . . . < class(gr )  n, and each gi is reduced with respect to each of the preceding polynomials gj (1  j < i).

APPLYING WU’S METHOD TO SYMBOLIC MODEL CHECKING

Wu’s method is an efficient method in symbolic computation, which finds zeros of a set of polynomials by looking for the characteristic sets. In this section, we give a brief review of the necessary results in the theory of Wu’s method, see [17, 18, 19, 20, 21, 22] for a detailed exposition. We first fix some notational conventions, which are also directly useful for our purpose of symbolic model checking. Let K[x1 , . . . , xn ] be the ring of polynomials in n variables over a field K. For the sake of simplicity, K is supposed to be an algebraically closed field. Even considering an arbitrary field K and its extension, all arguments do work out. Let degxj (f ) be the maximum degree of the variable xj in the polynomials f ∈ K[x1 , . . . , xn ].

Considering an ascending set G, a polynomial f , and using pseudodivision successively: rr = f rr−1 = prem(rr , gr , xr ) rr−2 = prem(rr−1, gr−1 , xr−1 ) .. . r0 = prem(r1 , g1 , x1 ),

239

r we obtain In(gr )αr . . . In(g1 )α1 f = i=1 qi gi + r0 , where
αi In(gi ) ri = qi gi + ri−1 . The remainder r0 is usually denoted by prem(f, G).

3.3 Translation In symbolic model checking, a Kripke structure models the concurrent system, and a CTL formula specifies a property, OBDDs are used to represent both Kripke structure and CTL formula, and all operations are done on OBDDs. To apply Wu’s method, we use polynomials to represent the Kripke structure as well as the CTL formula. Model checking is transformed to check whether zeros of the polynomials representing the initial states of the Kripke structure are in the set of zeros of the polynomials representing the states that satisfy the property. In the following, we first show how to use polynomials to represent the states and the transition relation of the Kripke structure. As the Kripke structure is represented by polynomials, we give the polynomial representation of EXφ, and use Wu’s method to calculate the characteristic sets of polynomials representing the CTL formula EXy. Further, we get the polynomials to represent the fixed point. Symbolic model checking can thus be performed by using Wu’s method. Although we can use polynomials to represent Kripke structures, whose zeros are in the field of more than two elements, all variables are the Boolean type in symbolic model checking. When performing symbolic model checking, a Kripke structure M is described in terms of n Boolean state variables. So a state of the Kripke structure is generally modeled as a vector where each component represents the state of one component of the system. A state of the structure can be viewed as a zero of a set of polynomials. For example, the state x1 = 0, x2 = 1 can be represented by {x1 , x2 + 1}. Under this interpretation, a set of polynomials over state variables denotes a set of states. For instance, the set of polynomials {x1 (x2 + 1)} represents all states in which x1 = 0 or x2 = 1. Since polynomials can be used to represent states of the system, the transition relation of Kripke structure M can also be represented by polynomials. Let a transition be

Definition 6. The ascending set G is a characteristic set of P if and only if (∀f ∈ P )[prem(f, G) = 0]. From the above definition, we can present an algorithm on how to compute a characteristic set of a set of polynomials. We call an algorithm to compute a characteristic set of a set of polynomials Wu’s method. The algorithm can be modified in various ways for better performance. However, the common algorithm in Wu’s method is doing pseudodivision successively, with respect to an ascending set. It adds the remainders to the set of polynomials successively until all the remainders are 0.

Definition 7. We use Z(P ) to denote the zero set of P : Z(P ) = {a1 , . . . , an  ∈ K n : (∀p ∈ P )[p(a1 , ..., an ) = 0]}. Theorem 3. Let G be a characteristic set of P . Then Z(P ) = (Z(G)\Z(In(G))) ∪

r


(Z(P ∪ In(gi ))).

i=1

Let G be a characteristic set of P , and Pi = P ∪ In(gi ), obviously, Z(Pi ∪ G) = Z(Pi ). The following theorem holds. Theorem 4. Given each Pi ∪ G, we calculate the corresponding characteristic set by using Wu’s method. Let {Gi : 1  i  t} be all the characteristic sets obtained that include G, and have Z(P ) =

t








(x1 , . . . , xn , x1 , . . . , xn ) ∈ R. Let a set of polynomials F represent the left argument in the parenthesis, and G represent the right argument. So F ∪ G represents the transition. For example, a transition

(x1 = 0, x2 = 1, x1 = 0, x2 = 0) can be represented by a

set of polynomials {x1 , x2 + 1, x1 , x2 }. There are other two relations “∨” and “∧” that are needed in symbolic model checking. Let P = {p1 , . . . , pn } and F = {f1 , . . . , fr } be two sets of polynomials. We define the product of the two sets of polynomials

(Z(Gi )\Z(In(Gi ))).

i=1

3.2 Preparations Although we know how Wu’ method computes a characteristic set of a set of polynomials, given two sets of polynomials P and F , we generally cannot decide whether Z(P ) = Z(F ). There will be more than one characteristic set representing the same set of zeros. For instance, when we are working modulo 2, both the characteristic sets (x1 + 1)2  and x1 + 1, x2 (x2 + 1), . . . , xn (xn + 1) define the same elements of K n , where K is a field of two elements. In order to do symbolic model checking, we need to know when two characteristic sets represent the same set of points. Let Z be a set of polynomials:

P F = {pi fj : 1  i  n, 1  j  r}. Let p and q be two formulas. Let S1 be the set of states satisfying p, and S2 be the set of states satisfying q. If two sets of polynomials P and F represent the two sets of states S1 and S2 . Then p ∨ q can be represented by P F , since S1 ∪ S2 represents the set of states satisfying p ∨ q. It stands to reason that p ∧ q can be represented by P ∪ F . We represent the possible states of the structure by the elements of the vector space K n , where K is still a field of two elements. And the transition relation is represented by points of K 2n . So we can use polynomials to represent CTL formulas. For symbolic model checking, we need to be able to describe the polynomials corresponding to, for ex-

Z = {xi (xi + 1) : 1  i  n}. Since every point of K n is a zero of Z and Z(P ) is a subset of K n , we have Z(P ) = Z(P ∪ Z). So there are some advantages to let characteristic sets contain Z. We can thus determine whether two characteristic sets represent the same set of points and every subset of K n can still be represented.

240

of polynomials in a characteristic set for the given polynomi2 als is O(sdO((n−dimI) ) ) [28, 29], where s is the number of the polynomials, n is the number of variables, and dimI is the dimension of the ideal I generated by the given polynomials. As we know, the complexity is an exponential function of n. Over the field of two elements, however, all the coefficients are 0 or 1, and when the characteristic set includes all xi (xi + 1), we can only consider the polynomials in which no variable appears with the degree greater than 1. For this case, we are not aware of the specific complexity results. It seems that the performance is fairly good for these simple polynomials. If we could optimize the algorithm in future work, it may dramatically reduce the calculations.

ample, the formula EXφ. Suppose we have a set of points Q ⊆ K n corresponding to φ. There are points x1 , . . . , xn  ∈

K n such that there exists a point x1 , . . . , xn  ∈ Q with

x1 , . . . , xn , x1 , . . . , xn  ∈ Z(T ), where T is the set of polynomials representing the transition relation. Let P be the set of polynomials representing Q. Then the CTL formula EXφ can be represented by the set of polynomials G = (P ∪ T ) ∩ K[x1 , . . . , xn ], where G stands for all the states that satisfy EXφ. However, we need to get points corresponding to EXy by calculating zeros of polynomials in practice. From the above translation, y can be represented by the set of polynomials P in K[x1 , . . . , xn ]. Let f be a map:

4. A CASE STUDY




xi → xi (1  i  n).

This section will show how Wu’s method applies to symbolic model checking by verifying a property of a rather small system. The example is a mutual exclusion protocol mutex, whose SMV code is shown in Figure 2. The system is used to model two processes, and each of them has three states. When a process is in a noncritical state (n1 or n2), it will be in trying state (t1 or t2) as the next state. If a process is in a trying state while the other is in a noncritical state, it will directly go into a critical state (c1 or c2). When two processes both go from a trying state to a critical state, they should first check the value of “turn”. If turn=1, only state1 can go into c1. Otherwise, only state2 can go into c2. They implement the protocol by the “turn” variable as the mutually exclusive signal. After critical state, a process will be in a noncritical state again. In other conditions, a process will remain in the same state. For the “turn” variable, if state1=n1 and state2=t2, its value will be 2 in the next state. If state1=t1 and state2=n2, the value of “turn” will be 1. It will not change other than these cases. First, we need to encode the state variables. We use ten Boolean state variables for the original variables. Figure 3 shows how we partition the variables, where x1 = 0 means turn = 1. Because state1 has three different evaluations, we use two Boolean variables x2 , x3  to express it. We set the pair of bits to 0, 0 for n1, to 0, 1 for t1, and to 1, 0 for c1. For state2, we set 0, 0 for n2, 0, 1 for t2, and 1, 0 for c2. Then, we translate the transition relation to polynomials. Our approach is to find a set of polynomials whose common zeros correspond to pairs of states in which the appropriate assignments are made. We first consider state1. Line (1) shows that, if state1=n1 and state2=t2, state1 will be t1 in the next state. So we need to find a set of polynomials representing the points with

x2 = 0, x3 = 0, x4 = 0, x5 = 1, x2 = 0, and x3 = 1. Then

the set of polynomials is: F1 := {x2 , x3 , x4 , x5 +1, x2 , x3 +1}.

Then EXy is represented by G = (f (P ) ∪ T ) ∩ K[x1 , . . . , xn ]. As the above formula presents, the calculation of EXy just behaves like OBDDs. When computing EXy, OBDDs have two operations, replacing the current state variables with the next state variables and existentially quantifying the next state variables. Here, we give a map that the variables in y are mapped to the next state variables, then get the corresponding results in K[x1 , . . . , xn ]. So now the problem is to find Z(G). Now let  be a monomial order induced by the variables ordering satisfying the property that any
monomial containing one of the xi is greater than the monomial containing only x1 , . . . , xn . By using Wu’s method, Gy can be obtained, which is the set of the characteristic sets of f (P ) ∪ T . It can be shown that Gy ∩ K[x1 , . . . , xn ] is the set of characteristic sets representing Z(G) as long as we produce a suitable monomial order, for example, the graded lexicographic order. Theorem 1 shows that a fixed point can be derived by computing EXy, and from the above discussion, we know how to calculate EXy by means of the manipulation of polynomials. Using EF p as an example, we iteratively calculate the polynomials representing p ∨ EXy to get the fixed point. Hence, model checking conducts all operations on polynomials. We repeatedly use Wu’s method to calculate the characteristic sets representing the common zeros of polynomials, and get the polynomials representing the fixed point in the end, which means that the set of all states satisfies the property of the system. When obtaining the set of polynomials representing the fixed point, we let the initial states be represented by polynomials too, from which we obtain the characteristic sets. So we can decide if the model has the property by checking whether all the zeros of the characteristic sets satisfy the polynomials representing the fixed point.

MODULE main

3.4 Complexity When calculating characteristic sets of polynomials, it is natural to measure the number of polynomials, the maximum degree of the polynomials, and the time cost. Given a polynomial set P , we are interested in these measures for a characteristic set of P , as well as for the intermediate sets constructed in the procedure of finding a characteristic set. In general, the performance is bad. Given polynomials of degree less than or equal to d, the general upper bound degree

VAR state1: {n1, t1, c1}; ASSIGN init(state1) := n1;

241

Similarly, from lines (2)-(6) we get the sets of polynomials:

next(state1) := case (1) (state1 = n1) & (state2 = t2): t1; (2) (state1 = n1) & (state2 = n2): t1; (3) (state1 = n1) & (state2 = c2): t1; (4) (state1 = t1) & (state2 = n2): c1; (5) (state1 = t1) & (state2 = t2) & (turn = 1): c1; (6) (state1 = c1): n1; (7) 1: state1; esac;







F2 := {x2 , x3 , x4 , x5 , x2 , x3 + 1}





F3 := {x2 , x3 , x4 + 1, x5 , x2 , x3 + 1}





F4 := {x2 , x3 + 1, x4 , x5 , x2 + 1, x3 }





F5 := {x2 , x3 + 1, x4 , x5 + 1, x1, x2 + 1, x3 }





F6 := {x2 + 1, x3 , x2 , x3 }. For line (7), There are two conditions under which all the above guards fail:

VAR

(state1=t1, state2=c2), and (state1=t1, state2=t2, turn=2),

state2: {n2, t2, c2};

where the value of state1 in the current is equal to the next state. Moreover, the value of state1 is t1 in both conditions. Each of the conditions represents the points with:

ASSIGN init(state2) := n2;







{x2 = 0, x3 = 1, x4 = 1, x5 = 0, x2 = 0, x3 = 1}

next(state2) := case (state2 = n2) & (state1 = t1): t2; (state2 = n2) & (state1 = n1): t2; (state2 = n2) & (state1 = c1): t2; (state2 = t2) & (state1 = n1): c2; (state2 = t2) & (state1 = t1) & (turn = 2): c2; (state2 = c2): n2; 1: state2; esac;







{x2 = 0, x3 = 1, x4 = 0, x5 = 1, x1 = 1, x2 = 0, x3 = 1}, and the corresponding two sets of polynomials are:





F7 = {x2 , x3 + 1, x4 + 1, x5 , x2 , x3 + 1}





F8 = {x2 , x3 + 1, x4 , x5 + 1, x1 + 1, x2 , x3 + 1}. When state1 is changing from the current state to the next state, the product Fs1 = F1 F2 F3 F4 F5 F6 F7 F8 represents the

points x1 , . . . , x5 , x1 , . . . , x5  with the appropriate assignments to corresponding variables, by the definition of the product of sets of polynomials. Similarly, when considering the state variables state2 and turn, we get Fs2 and Fturn . Let

VAR turn: {1, 2};













ASSIGN

Z = {x1 (x1 + 1), . . . , x5 (x5 + 1), x1 (x1 + 1), . . . , x5 (x5 + 1)}.

–init(turn) := 1;

Then the transition relation of the system can be represented by T = Fs1 ∪ Fs2 ∪ Fturn ∪ Z. Since we have obtained the polynomial representation of the transition relation, we can check the property by using Wu’s method. The property to be checked is

next(turn) := case (state1 = n1) & (state2 = t2): 2; (state2 = n2) & (state1 = t1): 1; 1: turn; esac;

EF (state1 = c1 ∧ state2 = c2). Let φ = (state1 = c1 ∧ state2 = c2), and the corresponding set of polynomials is P = {x1 (x1 + 1), x2 + 1, x3 , x4 + 1, x5 }. We want to find the least fixed point of λy.φ ∨ EXy, namely µy.(φ ∨ EXy). To find the zeros corresponding to EXy, where y is represented by Fy , a map f is needed:

SPEC EF((state1 = c1) & (state2 = c2)) Figure 2: SMV program for mutex




f : xi → xi (1  i  5). Then we calculate the set of the characteristic sets of T ∪ f (Fy ), denoted Gy , and (Gy ∩ K[x1 , . . . , x5 ])P corresponds to the formula φ ∨ EXy. In this fashion, we can calculate the least fixed point of λy.φ ∨ EXy. We run the example by using Maple. To learn more about the implementation of characteristic sets in Maple, we refer the reader to [27, 28, 30]. Our algorithm is based on the implementation. Let G be the set of polynomials corresponding to the least fixed point of λy.φ ∨ EXy, then G = {x1 (x1 + 1), x2 + 1, x3 , x4 + 1, x5 }. The initial states are represented by the set of polynomials Finit = {x1 (x1 + 1), x2 , x3 , x4 , x5 }. There is a zero of Finit that does

⎧ x1 ⎨ turn : state1 : x2 , x3  Current State ⎩ state2 : x , x  4 5 ⎧
⎪ x1 ⎨ turn :

N ext State state1 : x2 , x3  ⎪ ⎩ state2 : x
, x
 4 5 Figure 3: State variables

242

orem proving. It might provide new ways to understand and abstract system to be checked, or represent the constraints of the interface between a subsystem and its environment. We may improve model checking methods from the results in these directions.

not satisfy G. Therefore we conclude that the specification EF φ is false. The example is runned on a PC with a 1.8 GHz Pentium 4 and 256 MB of memory. It took about 8 seconds and 907 KB of memory when applying Wu’s method. On the same machine, SMV took approximately 0.1 seconds and 981 KB. SMV builded OBDDs directly from the code in Figure 2, while using Wu’s method, we manually translated this code into polynomials. It took less memory to do model checking when using our approach. However, This is only the first and initial attempt, and the algorithm is not optimized. From long views, Wu’s method is a promising way to settle the state explosion problem. Furthermore, Wu’s method can be more efficient if the algorithm is specilized for the special field. On the other hand, Maple is however a software for universal symbolic computation. The algorithm and data structure used here is for general polynomials, not optimized for the polynomials from symbolic model checking. For this special field and special polynomials, more efficient algorithm may be found.

5.

6. ACKNOWLEDGMENT We are grateful to our colleagues in the Institute of Computer Applications, Chinese Academy of Sciences for helpful discussions about Wu’s method. We would also like to thank the anonymous referees for providing us very valuable suggestions on the paper.

7. REFERENCES [1] C. Kern and M. Greenstreet. Formal Verification in Hardware Design: A Survey. ACM Trans. on Design Automation of Electronic Systems, 4(2), 123–193, 1999. [2] D. Kroening and W. J. Paul. Automated Pipeline Design. In DAC 2001 conference proceedings, 810–815, ACM Press, 2001. [3] G. J. Holzmann. The Model Checker SPIN. IEEE Trans. on Software Engineering, 23(5), 279–295, 1997. [4] K. L. McMillan. Symbolic Model Checking. Kluwer Academic Publishers, 1993. [5] K. L. McMillan. The SMV System. http://www-2.cs. cmu.edu/∼modelcheck/smvmanual.ps, 2000. [6] E. M. Clarke and E. A. Emerson. Design and Synthesis of Synchronization Skeletons Using Branching Time Temporal Logic. In Logic of Programs: Workshop, 52–71, Springer-Verlag, 1981. [7] E. M. Clarke, O. Grumberg and D. A. Peled. Model Checking. MIT Press, 1999. [8] A. Tarski. A Lattice-Theoretical Fixpoint Theorem and its Applications. Pacific J. Math., 5, 285–309, 1955. [9] R. E. Bryant. Graph-Based Algorithms for Boolean Function Manipulation. IEEE Trans. on Comp., 35(8), 677–691, 1986. [10] R. Bryant and Y. A. Chen. Verification of Arithmetic Functions with Binary Moment Diagrams. In Proc. of the 32nd ACM/IEEE conference on Design automation, 535–541, ACM Press, 1995. [11] R. Rudell. Dynamic Variable Ordering for Ordered Binary Decision Diagrams. In Proc. of the 1993 IEEE/ACM int’l conf. on Computer-aided Design, 42–47, IEEE Computer Society Press, 1993. [12] B. Bollig, I. Wegener. Improving the Variable Ordering of OBDDs Is NP-Complete. IEEE Trans. on Comp., 45(9), 993–1002, 1996. [13] B. Yang, et al. A Performance Study of BDD-Based Model Checking. In 2nd Int’l Conf. on Formal Methods in Computer-Aided Design (FMCAD’98), 255–289, Springer-Verlag, 1998. [14] R. K. Brayton, et al. VIS: A System for Verification and Synthesis. In Proc. 8th International Conference on Computer-Aided Verification (CAV’96), 428–432, Springer-Verlag, 1996. [15] A. Cimatti, et al. NuSMV 2: An OpenSource Tool for Symbolic Model Checking. In Int’l Conf. on

CONCLUDING REMARKS

We have presented a framework to apply Wu’s method to symbolic model checking. This is a new approach to the existent symbolic model checking approaches, and gives engineers a new guidance for verifying properties of concurrent systems. It may also give new theoretical insights into symbolic model checking, particularly in overcoming the state explosion problem. There are nevertheless some aspects that may be improved. We manually translate the SMV code into polynomials, although this is not practical in symbolic model checking. In the future research, an ideal environment is to develop a tool that directly works when given in the SMV language. Another aspect is that Wu’s method spends the main calculation on the characteristic set computation. It is an anticipant that an independent software can calculate the characteristic set more efficiently. Several directions we proposed here might be generalized. The variable order has heavy impact on the size of OBDDs. An unfortunate variable order leads to state explosion. Dynamic variable reordering [11] is usually used to reduce the size. However, finding the best variable order is NPComplete [12]. There also exists the problem of variable order when applying Wu’s method [27]. A suitable order may dramatically reduce the calculations. However, their relations are not legible. It seems that a good variable order for OBDDs is also good for Wu’s method. Generally, variable order is chosen at the beginning of Wu’s method, without algorithms like dynamic variable reordering. And in case study, we use polynomials over Boolean field to represent states and transitions, as OBDDs have done. This is advantageous to calculation and representation, but with the price of many variables. It might perform model checking over the field of more than two elements, since diagrams can represent not only one or zero, for example, multi-value decision diagrams or BMD [10]. It is difficult to predict exactly to what extent the results in algebra can be applied to symbolic model checking. But some directions can be tried. As there is a rich collection of solutions to solving polynomials, methods for membership problem of ideals, symbolic model checking is more like the-

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[21]

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Computer-Aided Verification (CAV 2002), 27–31, Springer-Verlag, 2002. J. F. Ritt. Differential Algebra. Amer. Math. Sco., New York, 1950. B. Mishra. Algorithmic Algebra. Springer-Verlag, 1993. J. von zur Gathen and J. Gerhard. Modern Computer Algebra. Cambridge University Press, 1999. J. Grabmeier, E. Kaltofen and V. Weispfenning. Computer algebra handbook. Springer-Verlag, 2003. W. T. Wu. On the Decision Problem and the Mechanization of Theorem Proving in Elementary Geometry. Scientia Sinica, 21, 157–179, 1978. W. T. Wu. Basic Principles of Mechanical Theorem Proving in Geometries. J. of Automated Reasoning, 2(4), 221–252, 1986. W. T. Wu. Some Remarks on Characteristic-Set Formation. Math. Mech. Res. Preprints 3, 27–29, 1989. S. C. Chou. Proving and Discovering Theorems in Elementary Geometries Using Wu’s Method. PhD thesis, Department of Mathematics, University of Texas at Austin, 1985. S. C. Chou. Mechanical Geometry Theorem Proving. Reidel Publishing, 1988. D. Manocha. Algebraic and Numeric Techniques for modeling and Robotics. PhD Thesis, Computer Science Division, University of California, Berkeley, 1992.

[26] Hyungju Park. Symbolic Computation and Signal Processing. J. Symbolic Computation, 37(2), 209–226, 2004. [27] D. M. Wang. An Implementation of the Characteristic Set Method in Maple. RISC-Linz Series, No. 91-25.0, Johannes Kepler University, Austria, 1991. [28] G. Gallo and B. Mishra. Efficient Algorithms and Bounds for Wu-Ritt Characteristic Sets. In Proc. MEGA’90, 119–142, Birkh¨ auser, 1991. [29] G. Gallo and B. Mishra. Wu-Ritt Characteristic Sets and Their Complexity. DIMACS series in Discrete Mathematics and Theoretical Computer Science, 6, 111– 136, American Mathematical Society and Association of Computing Machinery, 1991. [30] Z. J. Liu and D. D. Lin. An Approach Towards the Implementation of Characteristic Set Method. Sys. Sci. and Math. Sci., 12 Suppl., 88–95, 1999. [31] X. S. Gao, X. R. Hou, J. L. Tang and H. F. Chen. Complete Solution Classification for the Perspective-Three-Point Problem. IEEE Tran. on PAMI, 25(8), 930–943, 2003. [32] A. Heck. Introduction to Maple. Springer-Verlag, 1996. [33] R. M. Corless. Essential Maple 7: An Introduction for Scientific Programmers. Springer-Verlag, 2002.

244

Probabilistic Algorithms for Computing Resultants Michael Monagan∗ Centre for Experimental and Constructive Mathematics Simon Fraser University, Burnaby, B.C. V5A 1S6, CANADA [email protected]

ABSTRACT

of A are repeated n times in the first n rows, and the coefficients of B are repeated m times in the last m rows. Thus the resultant is an element of the ring R.

Let A and B be two polynomials in [x, y] and let R = resx (A, B) denote the resultant of A and B taken wrt x. In this paper we modify Collins’ modular algorithm for computing R to make it output sensitive. The advantage of our algorithm is that it will be faster when the bounds needed by Collins’ algorithm for the coefficients of R and for the degree of R are inaccurate. Our second contribution is an output sensitive modular algorithm for computing the monic resultant in [y]. The advantage of this algorithm is that it is faster still when the resultant has a large integer content. Both of our algorithms are necessarily probabilistic. The paper includes a number of resultant problems that motivate the need to consider such algorithms. We have implemented our algorithms in Maple. We have also implemented Collins’ algorithm and the subresultant algorithm in Maple for comparison. The timings we obtain demonstrate that a good speedup is obtained.





am−1

...

a0

0

...

0

0

am

am−1

...

a0

...

0

..

..

..

.. .





















.. . 







.

.

.




















S=

0

...

0

am

am−1

...

a0

bn

bn−1

...

b0

0

...

0

0

bn

bn−1

...

b0

...

0

..

..

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.

.. .

...

b0

































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.

.







0

...

0

bn

bn−1

In this paper we are interested in computing the resultant when the coefficients of A and B are polynomials over the integers. In particular we focus on the bivariate case where A and B are elements of [x,y] and R ∈ [y]. Because [y] is an integral domain, the resultant may be computed using fraction-free algorithms. The best fraction-free algorithm is the subresultant algorithm of Brown and Traub [1]. It is a fraction-free modification of Euclid’s algorithm. It requires O(mn) arithmetic operations in [y]. For [y], the fastest method is the modular method of Collins [2]. Collins’ method computes the resultant modulo a sequence of primes S = p1 , p2 , p3 , .... For each prime p ∈ S it computes the resultant at y = α0 , y = α1 , ..., ∈ p. The resultant R ∈ [y] is reconstructed from these images in p using polynomial interpolation and Chinese remaindering. This reduces the problem to computing many resultants in p[x]. The following property (see Ch. 9 of [6] for a proof),

Categories and Descriptors: I.1.2 [Symbolic and Algebraic Manipulation]: Algorithms – Algebraic algorithms. General Terms: Algorithms Keywords: Sylvester’s resultant, polynomial resultants, modular algorithms, probabilistic algorithms.

1.

am 

INTRODUCTION

Let A = am xm + am−1 xm−1 + ... + a0 and B = bn xn + bn−1 xn−1 + ... + b0 be two non-zero polynomials in x over a commutative ring R of degrees m and n respectively. Let R = resx (A, B) denote the resultant of A and B taken with respect to x. If m = 0 the resultant is an 0 . If n = 0 the resultant is bm 0 . If m > 0 and n > 0, the resultant is the determinant of Sylvester’s matrix. Sylvester’s matrix is the following m + n by m + n matrix S over R. The coefficients

C resx (A, B) = (−1)mn bm−deg resx (B, C), n

∗

Supported by the MITACS NCE of Canada and NSERC of Canada

where C is the remainder of A divided by B, means we can modify the ordinary Euclidean algorithm to compute the resultant in p[x] using O(nm) arithmetic operations in p. Let R = cd y d + cd−1 y d−1 + ... + c0 . Let h be the height of R, that is, h = max(|cd |, |cd−1 |, ..., |c0 |). To apply Collins’ method, one needs a degree bound D on the degree of the resultant, i.e., D ≥ d and, a coefficient bound H on the height of the resultant, i.e., H ≥ h. The algorithm will use sufficiently many primes pi such that Πpi > 2H. Here, the factor of 2 is to allow for both positive and negative coefficients in . For each prime p it will need D + 1 evaluation

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. ISSAC’05, July 24–27, 2005, Beijing, China. Copyright 2005 ACM 1-59593-095-7/05/0007 ...$5.00.

245

Collins’ Algorithm

points from p. Suppose we use 31 bit primes on a 32 bit machine. Recall that the prime number theorem states that the number of primes < x is asymptotically x/ ln x. Thus there are approximately (231 /31 − 230 /30)/ ln 2 = 48.3 million 31 bit primes. This means we can reconstruct integers in R of size approximately 1.5 billion bits using 31 bit primes. If we do this then the modular resultant algorithm will use M = O(log 231 H) 31 bit primes. Suppose the cost of computing one resultant of A(αj ) mod pi and B(αj ) mod pi is bounded by C. Then the cost of Collins’ modular resultant algorithm, assuming classical (i.e. quadratic) algorithms for interpolation and Chinese remaindering, is O(CM (D + 1)) + O(M (D + 1)2 ) + O(M 2 (D + 1)) = O(M D(C + M + D)). These three contributions are for the M (D + 1) modular resultants, M interpolations of D + 1 points, and applying the Chinese remainder theorem to at most D + 1 coefficients, respectively. Observe that any over estimate of either the degree bound or the coefficient bound will affect the cost of the algorithm proportionately. Thus for a practical implementation of Collin’s algorithm, we need to consider how to obtain good bounds. One can obtain a bound on d = deg y R from Sylvester’s matrix using the rows or the columns. That is, m+n

The correctness of the algorithm follows from noting that if p is not a bad prime and α is not a bad evaluation, then R(α) mod p = resx (A(α) mod p, B(α) mod p).

Algorithm CRES Input A, B ∈ [x, y]\{0} of degree m and n resp. Input D ≥ L ≥ 0 satisfying D ≥ d and L ≤ l. Input H ≥ h = max(1, |cd |, |cd−1 |, ..., |cl+1 |, |cl |). Output R = resx (A, B) ∈ [y].

n+m

max degy Si,j and Dcol = i=1

Definition 1. Let A = am xm + ... + a0 and B = bn xn + ... + b0 be non-zero polynomials in [y][x] of degrees m and n. A prime p is said to be bad if am ≡ 0 mod p or bn ≡ 0 mod p. Similarly, an evaluation point α ∈ p is said to be bad if am (α) ≡ 0 mod p or bn (α) ≡ 0 mod p.

m+n m+n

Drow =

In the specification of the algorithm, R = resx (A, B). So R = 0 or R = cd y d + cd−1 y d−1 + ... + cl+1 y l+1 + cl y l for some d ≥ l ≥ 0 and cd cl 6= 0. We call l the low degree of the resultant and cl the trailing coefficient. The algorithm assumes a degree bound D ≥ d and also a low degree bound 0 ≤ L ≤ l. A low degree bound L can also be obtained directly from Sylvester’s matrix. In our presentation of Collins’ algorithm we improve the efficiency when a nontrivial low degree bound L is known. For if R(y) is divisible by y L then D − L + 1 evaluation points are sufficient.

j=1

max degy Si,j j=1

i=1

both bound d. We also have the Bezout theorem which says that the number of roots of R(y) is bounded by Dbez = deg A × deg B where deg A denotes the total degree of the polynomial A in x and y. The Dbez bound is better than Dcol and Drow when the inputs are dense. In our experience, the bound Dcol based on the columns of S is usually better, often by a factor of two than Drow , though the row bound can be better, e.g., if m  n. We compute all three and take D = min(Drow , Dcol , Dbez ). 0 Let S 0 be the matrix of integers where Si,j is the one-norm of Si,j , i.e., the sum of the absolute value of the coefficients of Si,j . In [7], Goldstein and Graham prove that Hadamard’s bound on det(S 0 ) bounds the height of the coefficients of

1 Initialize M = 1, ∆ = D − L. 2 Initialize S to the set of primes such that Πp∈S (p) > 2H, and, for each prime p ∈ S, p > ∆+deg y A+degy B and p is not a bad prime. 3 REPEAT 3.1 Choose the next prime p from S. Set Ap = A mod p and Bp = B mod p. 3.2 Set N = ∆ and choose N + 1 distinct non-zero evaluation points α0 , α1 , ..., αN from p such that am (αi ) 6≡ 0 mod p and bm (αi ) 6≡ 0 mod p.

m+n 0 2 det(S). Hadamards bound is Πm+n i=1 j=1 Si,j . One can compute Hadamard’s bound along the rows or down the columns of S 0 and use the smaller of the two bounds. Again, we find that the bound obtained using the columns of S 0 is usually better than that obtained using the rows. How good are these bounds? If one fixes the degree in y of all coefficients of A and B in x and chooses random integers for the integer coefficients of A and B from a sufficiently large set, say 10 digit random integers, then the degree bound will be exact and the coefficient bound will be almost tight. If the bounds for h and d could off by a factor of 2 at most, then we would be happy. It turns out, however, that there are real examples where these bounds can be arbitrarily far off. In those cases, the subresultant algorithm can be better, sometimes much better, than the modular resultant algorithm. This makes algorithm selection difficult. To motivate the need for an improved modular algorithm, we need to study some real problems. We will consider four problems in section 2. In section 3 and 4 we give two new probabilistic algorithms. We end the introduction with a complete description of the modular resultant algorithm of Collins [2] for inputs A, B ∈ [x,y].


3.3 FOR i = 0, 1, ..., N DO



Compute ri ∈ p the resultant of Ap (αi ) and Bp (αi ) modulo p using the Euclidean algorithm and set ri = ri /αL i mod p. 3.4 Interpolate r ∈ Set r = y L r.

p[y]

from (αi , ri ) for i = 0..N .

¯ = r. 3.5 IF M = 1 then set M = p, R 3.6 ELSE (apply the Chinese remainder theorem) ¯ mod M for C. Solve C ≡ r mod p and C ≡ R ¯ Set M = pM, R = C. UNTIL M > 2H. ¯ in the symmetric range for 4 Put the coefficients of R ¯ M . Output R. Remark: We require that p > ∆ so that there are sufficiently many points for interpolating the resultant in p[y]. The requirement that p > ∆ + deg y am + degy bn allows for the possibility that there could be as many as deg y am + degy bn bad evaluation points which must be avoided.

246

n 3 4 5 6 7 8

Remark: The set of primes S is usually taken to be the biggest primes which the hardware of the machine supports. Thus if we are using 31 bit primes on a 32 bit machine, we would start with the largest such prime and simply count down towards 0 skipping primes which are bad. Remark: If y divides neither A nor B, the low degree bound L computed from the rows of Sylvester’s matrix must be 0, hence, a non-trivial bound for l can only come from the columns of Sylvester’s matrix. If y divides A or B then one ¯ B = y j B) ¯ = y ni+mj resx (A, ¯ B). ¯ should apply resx (A = y i A,

2.

D 98 430 1798 7350 29718 119510

l 49 225 961 3969 16129 65033

d 73 289 1121 4353 17025 67073

M 24 97 383 1501 5894 23262

m 7 19 48 115 269 615

depends on the size of the resultant R. We detail this algorithm, algorithm PRES, in section 2. We have implemented it in Maple [9]. Below is a timing comparison comparing it with Collins’ algorithm and the subresultant algorithm.

HOW GOOD ARE THE BOUNDS?

We first look at four resultant problems to investigate how good the degree bounds and height bounds for the resultant are. If they are not good then the obvious line of attack would be to look for better bounds. The examples, however, tell us that it will be impossible to obtain accurate bounds for real problems in general.

n 3 4 5 6 7 8

Example 2.1 (Cyclotomic Polynomials) Let Φn (x) ∈ [x] denote the n’th cyclotomic polynomial. For 2 ≤ i < j ≤ 200 we find that if i does not divide j then the resultant of Φi (x) and Φj (x) is 1! Let Si,j be the Sylvester matrix for Φi (x) and Φj (x). Hadamard’s bound for Si,j is a moderately large integer. For i = 197 and j = 199 Hadamard’s bound is > 10427 using the columns of Si,j and > 10452 using the rows of Si,j . This example shows that the height bound may be arbitrarily far off. It tells us also that it will be impossible to obtain accurate bounds in general without actually computing R. Furthermore, for i = 197, j = 199, the remainder of Φj (x) divided Φi (x) is x + 1. This means the subresultant algorithm will be much faster than Collins’ modular resultant algorithm.

Subres 0.01 0.49 21.3 1359 143,514 –

Collins .02 .32 13.2 688.8 41,354 –

PRES .01 .06 .57 7.6 120 2,092

factor 0.01 0.01 0.05 0.20 2.05 19.9

The timing data (in CPU seconds) was obtained using a 64 bit compile of Maple on an AMD Opteron 248 processor running at 2.2 GHz. On such a machine, Maple uses 31.5 bit primes. We see that Collins’ algorithm is not much faster than the subresultant algorithm. Algorithm PRES is over 300 times faster than Collins’ algorithm for n = 7. The timings in the last column are for factoring the resultant. They are included to emphasize that the hard part of this problem is computing the resultant. Example 2.3 (Primary Decomposition of Ideals) Let I be a zero-dimensional ideal in [x, y, z, ...]. One way to compute the primary decomposition of I is to first compute a Gr¨ obner basis G for I using a lex ordering. Suppose the Gr¨ obner basis is of the form

Example 2.2 (The Logistic Map) Let f (x) = ax(1 − x) for parameter a. Let g(x) = f (n) (x). The bifurcation points of f are roots of the polynomial r = resx (g(x) − x, g 0 (x) + 1) ∈ [a]. For example, for n = 2,




G = {g1 (x), g2 (x, y), g3 (x, y, z), ...}.

g(x) − x = f (f (x)) − x 2 2

2

One would then factor g1 ∈ [x]. Suppose g1 (x) is irreducible and g2 is not linear in y. One would then factor g2 modulo g1 . To factor g2 modulo g1 one may use Trager’s algorithm [12]. Trager’s algorithm will begin by factoring the polynomial r(y) = resx (g1 (x), g2 (x, y)) ∈ [y]. If r(y) is square-free, then, for each irreducible factor fi of r, Trager’s algorithm computes hi = gcd(fi , g2 ) modulo g1 , an irreducible factor of g2 . Consider the following example from the POSSO test suite [11] where I ⊂ [x, y, z, t, u] is generated by

= −x(ax + 1 − a)(a x − a x − ax + a + 1), 2

0

L 37 148 571 2202 8569 33664

3

2




3

g (x) + 1 = 1 + (1 − 2x)a + (−4x + 6x − 2x)a . The resultant r = −a9 (a2 + 1)(a2 − 4a + 5)(a2 − 2a − 5)2 . Notice the high power of a dividing the resultant even though neither input is divisible by a. In the following table we compare the bounds for the degrees and heights with the actual values for n = 3, 4, 5, 6, 7, 8. Columns l and d are the actual low degree and degree of r in a. Columns L and D are the best low degree and degree bounds for a. Column m is the number of decimal digits of the height h of r and column M is the number of digits of the bound H for the height of r. Observe that for n = 8 the height bound is off by a factor of over 38, and the number of points necessary to interpolate r(a)/al , namely d − l + 1, is 42 times less than the degree bounds D − L + 1 yield. Thus if we knew l, d and m the modular algorithm would run over 1000 times faster! These examples motivate us to design an output sensitive modular resultant algorithm which will be probabilistic, that is, it will output R with controllable high probability, and, the number of primes and number of evaluation points used




−1 + 2x2 − 2y 2 + 2z 2 − 2t2 + 2u2 , −1 + 2x3 − 2y 3 + 2z 3 − 2t3 + 2u3 , −1 + 2x4 − 2y 4 + 2z 4 − 2t4 + 2u4 , −1 + 2x5 − 2y 5 + 2z 5 − 2t5 + 2u5 , −1 + 2x6 − 2y 6 + 2z 6 − 2t6 + 2u6 . We find that with u < t < x < y < z, G = {g1 (u), g2 (u, t), g3 (z, u), g4 (y, t, u), g5 (x, z, u)}. g1 (u) is irreducible of degree 36 with 14 digit coefficients, g2 (u, t) is quadratic in t, of degree 35 in u with 112 digit coefficients, g3 (u, z) is quadratic in z with 113 digit coefficients, g4 (y, t, u) is linear in y and t and g5 (x, z, u) is linear in x and z. The resultant r = resu (g1 , g2 ) has degree 72 in t and 3966 digit coefficients.

247

The bounds are good. The bound D on the degree is 72 and the bound H on the coefficients of r is a 4485 digit integer. However, r = cr 0 where c is a 3931 digit integer content and r0 is a polynomial of degree 72 with 36 digit coefficients! Similarly, resu (g1 , g3 ) also has a large integer content. In the context of Trager’s algorithm, all we need is r 0 . This motivates us to consider a modular resultant algorithm that computes r 0 not r. We do this via computing the monic resultant r˜ = r/lcy r(y) using a modular algorithm which uses rational number reconstruction [13]. The algorithm, algorithm MRES, is presented in section 4. On this example, we find that the time to compute r using Collins’ algorithm is 7.5s and the time to compute r 0 using algorithm MRES is 0.16s for a speedup of a factor of 47. More significantly, the time Maple takes to compute the entire primary decomposition of I is reduced from 39.1s to 1.7s! The following example is from Kotsireas [4]. Let I ⊂ [t, u, v, x, y, z] be generated by

integer content of f (x, y) and column h is the length of the height of the primitive part of f (x, y).

(t − u)(x − y) − 2z + 2, (t − u)(x + y − 2z) − 2(x − y), (t − u)(t − v) − 2t − 2u + v + 1, x2 t3 − 1, y 2 u3 − 1, and z 2 v 3 − 1.

Observe that the degree bounds are accurate. Observe that the height of the resultant f (x, y) is a factor of 15 to 40 times longer than the height of the primitive part of the resultant. This is another example where a monic resultant algorithm that is output sensitive will help. The last column N shows the number of terms in the resultant f (x, y) indicting how sparse it is. We infer N = 1 + m + n = 1 + (degx f + degy f )/2.

m 11 11 11 11 11 11 13 15 17 19 21 23

n 13 15 17 19 21 23 15 17 19 21 23 25

D (26,22) (30,22) (34,22) (38,22) (42,22) (46,22) (30,26) (34,30) (38,34) (42,38) (46,42) (50,46)

d (26,22) (30,22) (34,22) (38,22) (42,22) (46,22) (30,26) (34,30) (38,34) (42,38) (46,42) (50,46)

M 276 346 416 466 516 566 381 501 637 789 958 1144

m 168 195 221 249 275 302 230 301 382 472 583 883

c 159 184 209 235 260 295 219 289 368 457 556 665

h 9 11 12 14 15 17 11 12 14 15 17 18

N 25 27 29 31 33 35 29 33 37 41 45 49




With u < v < x < y < t, the lex Gr¨ obner basis has two polynomials g1 (u) and g2 (v, u) where g1 (u) is of degree 51 in u with 18 digit coefficients and g2 (v, u) is of degree 2 in u with 446 digit coefficients. On this example r = resu (g1 , g2 ) has degree 102 in v with 22665 digit coefficients. Again, the bounds are good. The bound D = 102 and the bound H on the coefficients of R is 23641 digits. Again r = cr 0 where c is a large integer, a 23607 digit integer and r 0 has 33 digit coefficients. We find that the time using our Maple implementation of Collins’ algorithm is 219.6s, algorithm MRES takes 0.31s for a gain of a factor of 708.

3. A PROBABILISTIC ALGORITHM Algorithm PRES below chooses a prime p from a suitably large set of primes S. It then computes the first image r1 ∈ p[y] using the degree bounds D and L as in algorithm CRES. Thus this first image uses D−L+1 evaluation points. Let d1 = degy r1 and l1 be the low degree of r1 in y. To make the algorithm output sensitive, one could assume d1 = d and l1 = l and proceed to use d1 − l1 + 1 evaluation points for the subsequent primes and terminate when the result of the Chinese remaindering does not change for several iterations, say K = 10 iterations. This will not work if d1 < d or l1 > l. We begin with a definition.

Example 2.4 (SYZYGY Polynomials) For positive integers m, n, the parametrization of the syzygy figures in 2 is given by x(t) = sin(mt) and y(t) = cos(nt). We can construct the equation f (x, y) = 0 for the curve by first expanding multiple angles using

Definition 2. Let R = cd y d + ... + cl y l where d ≥ l ≥ 0 and cd cl 6= 0. A prime p is said to be unlucky if p|cd or p|cl .

sin(2t) = 2 sin(t) cos(t) and cos(2t) = 2 cos(t)2 − 1,

Observe that for a given input A and B, the number of unlucky primes is finite.

applying the rational parametrization sin(t) = 2t/(1 + t2 ),

cos(t) = (1 − t2 )/(1 + t2 ),

Example 3.1 (Unlucky Primes) Consider A = x4 + a3 y 4 x2 + a0 y 2 and B = x − 1 in [y][x]. Then R = resx (A, B) = a3 y 4 + a0 y 2 + 1. Thus any prime dividing a3 is unlucky.

and then computing the resultant f = rest ((1 + t2 )m (x − sin(mt)), (1 + t2 )n (y − cos(nt)) in

[x,y]. For x(t) = sin(3t), y(t) = cos(5t) we obtain 10

8

Observe that the leading coefficient of R in example 3.1 does not depend on the leading coefficients of A and B. In general, it will not be possible to efficiently compute cd or cl from A and B to detect unlucky primes. Thus unlike bad primes, we cannot efficiently avoid unlucky primes in advance. Suppose the first prime p1 is unlucky. Our idea to detect this is as follows. When we compute the second image, r2 , the resultant of A and B modulo p2 , instead of using d1 −l1 + 1 evaluation points, we will use δ > 0 additional evaluation points. If d1 = d and l1 = l then when we interpolate the resultant modulo p2 , it must have degree d2 ≤ d and l2 ≥ l where d2 and l2 are the degree and low degree of the second image. If d1 < d or l1 > l then when we interpolate r2 ,

6

f (x, y) = 70368744177664(256 x − 640 x + 560 x + 16 y 6 − 200 x4 − 24 y 4 + 25 x2 + 9 y 2 − 1) Observe that f (x, y) has a large integer content relative to the size of the height of the primitive part of f (x, y). Observe also that f (x, y) is sparse, the monomials are even powers in x or in y only. We compute the following data for selected m and n. Column D is the degree bounds in x and y for the resultant. Column d is the actual degrees in x and y of the resultant. Column M is the length of the height bound on the resultant, column m is the length of the actual height of f (x, y), column c is the length of the

248

then the algorithm will output y 4 + 1. The prime p1 is not unlucky. The algorithm fails because it stabilizes too early.

provided p2 is not also unlucky, r2 will probably have degree d2 = d1 + δ. If d2 > d1 then we detect that p1 was unlucky and we will restart the algorithm with a new prime. By requiring that the algorithm needs K primes of agreement before it can terminate, there are at least K chances that the algorithm can identify an unlucky prime in this way. The timings for algorithm PRES reported in section 2 assumed 31.5 bit primes on a 64 bit machine, K = 5, and δ = 1. We present the algorithm.

Example 3.3 (Unlucky Prime Failure). Consider A = x4 + ay 4 x2 + y 2 and B = x − 1 in [y][x] where a = p1 p2 × ... × p11 . The resultant R = resx (A, B) = ay 4 + y 2 + 1. If we call algorithm PRES with K = 10 then it outputs y 2 + 1. The algorithm stabilizes before detecting that p1 is unlucky.

Algorithm PRES

First we argue that the algorithm must terminate in finite time. If the first prime p1 is not unlucky, it must terminate K primes after M > 2h where h is the height of R. It could terminate earlier as in example 3.2 with an incorrect output. If p1 is unlucky then the algorithm must either terminate early as in example 3.3 or it must eventually detect that the first prime was unlucky and restart. It may restart with another unlucky prime. But it must either terminate with an incorrect output or eventually restart with a prime which is not unlucky since the number of unlucky primes is finite. There are two useful measures for the probability that algorithm PRES fails. The first assumes the coefficients of R modulo a prime p are uniformly distributed on [0, p). This assumption will be true asymptotically over the set of all inputs A, B of bounded size. The second is an adversarial approach. Allow an adversary to choose R to maximize the probability of failure. In both cases we bound the probability of failure and then choose S, K and δ so that this probability of failure is low. The first measure is optimistic, the second pessimistic. The performance of the algorithm on real data will lie between the two. First note that if R = 0 the algorithm always outputs 0. Thus from now on assume R = cd y d + ... + cl y l with cd cl 6= 0.

Input A, B ∈ [x,y]\{0} of degree m and n resp. Input D ≥ L ≥ 0 satisfying D ≥ d and L ≤ l. Input H ≥ h = max(1, |cd |, |cd−1 |, ..., |cl+1 |, |cl |). Input K ≥ 1 (number of primes of agreement). Output R = resx (A, B) ∈ [y]. 1 Initialize M = 1, ∆ = D − L, δ = 0, l = L, d = D. 2 Initialize S to a set of primes such that Πp∈S p  2H, and for each p ∈ S, we have p > ∆ + degy A + degy B and p is not a bad prime. 3 REPEAT 3.1 Choose a new prime p from S. 3.2 Set N = ∆+δ and choose N +1 distinct non-zero evaluation points α0 , α1 , ..., αN from p at random such that am (αi ) 6≡ 0 mod p and bm (αi ) 6≡ 0 mod p. 3.3 FOR i = 0, 1, ..., N DO Compute ri ∈ p the resultant of A(αi ) and B(αi ) modulo p using the Euclidean algorithm and set ri = ri /αli mod p. 3.4 Interpolate r ∈ Set r = y l r.

p[y]

Unlucky primes: Let p1 , p2 , ... be the sequence of primes chosen in step 3.1. Assuming that the coefficients of R modulo p are uniformly distributed on p, the Pr(p1 |cd ) = 1/p1 and Pr(p1 |cl ) = 1/p1 , hence, the Pr(p1 is unlucky) < 2/p1 .

from (αi , ri ) for i = 0..N.

3.5 IF δ = 0 set the bounds: 3.5.1 If r = 0 set d = L − 1, l = L otherwise set d = degy (r) and l to the low-degree of r in y. ¯ = r, j = 0. 3.5.2 Set δ = 1, ∆ = d − l, M = p, R

Premature termination with a wrong output: Suppose p1 is not unlucky. Then for each subsequent prime p, the algorithm uses sufficient points to interpolate R modulo p thus at the end of each iteration of step 3 we have ¯ mod M. The algorithm can output an incorrect anR≡R ¯ = R, we have R ¯ ≡ swer if at some iteration j before R R mod p for K consecutive primes pj+1 , pj+2 , ..., pj+K . That ¯ − R where is, at the end of iteration j + K we have P |R ¯ 6= R then R ¯ differs from P = pj+1 × pj+2 × ... × pj+K . If R R in at least one coefficient, say c¯i 6= ci . Again, assuming the coefficients of R modulo p are uniformly distributed on ci ≡ ci mod P ) = 1/P. But the algorithm could p, the Pr(¯ terminate prematurely at any iteration j > K for this reason. Suppose the algorithm uses 31 bit primes. Then it will need no more than dlog 231 he 31 bit primes to reconstruct R. Hence the expected probability of failure when p1 is not unlucky is < dlog 231 he/231K.

3.6 ELIF degy (r) > d then – restart the algorithm Initialize M = 1, ∆ = D − L, δ = 0, l = L, d = D. 3.7 ELSE apply the Chinese remainder theorem: ¯ mod M for C 3.7.1 Solve C ≡ r mod p and C ≡ R in the symmetric range. ¯ then set j = j + 1, M = p × M 3.7.2 If C = R ¯ = C. ELSE set j = 0, M = p × M, R UNTIL j = K. ¯ 4 Output R. ¯ 6= R we say that the algorithm If the algorithm outputs R “fails”. We give two examples which illustrate the two ways in which an adversary, who knows the sequence of primes p1 , p2 , p3 , ... can make the algorithm fail.

Remark: In the above argument, if the algorithm fails then ¯ − R, that is, P must divide c¯i − ci for all i = l, l + P |R 1, ..., d. If R has d − l + 1 non-zero coefficients then one may ¯ − R) < 1/P d−l+1 . be tempted to argue that the Pr(P |R However this assumes that the coefficients ci are of the same length. In practice, it is often the case that the coefficients

Example 3.2 (Likely Failure Case). Consider A = x4 +ay 2 x2 +y 4 and B = x−1 in [y][x] where a = p1 p2 × ... × p11 . The resultant R = resx (A, B) = y 4 + ay 2 + 1. However, if we call algorithm PRES with K = 10

249

cl , cl+1 , ..., cd are of quite different lengths. In particular, cl , the trailing coefficient of R, is often the largest.

Pr(R(α0 ) = R(α1 )) =

Worst case bound: An adversary can, however, construct inputs such that the resultant R = cd y d + ... + cl y l with cd = p1 , p3 , ... and cl = p2 , p4 , ... where the primes p1 , p2 , ... are chosen from S. On such inputs the probability of hitting unlucky primes is high. Similarly, an adversary can construct inputs such that the resultant R = cd y d + ... + ci y i + ... + cl y l with ci = p1 p2 p3 × ... as in example 3.2. In this case, even if p1 is not unlucky, the probability of terminating too early is higher. The solution to both problems is to choose S so that the probability that any coefficient of R vanishes is low. Construct S, a set of primes, satisfying the conditions in step 2 with Πp∈S p > H 4 where H is the height bound. Now, if the algorithm chooses primes at random from S, then

= v∈ p

≤

CASE d1 = 1, l1 = 0, that is, r1 is linear and l = 0. Step 3.5.1 sets ∆ = 1 so that N = ∆ + δ = 2, hence, the algorithm uses three non-zero evaluation points α0 , α1 and α2 to interpolate R(y) in step 3.4. Let

Pr( p is unlucky ) = Pr( p|cd or p|cl ) < 1/2. To make the probability that that p is unlucky less than 2k we much choose S with Πp∈S p > H 4k .

r = a + b(y − α0 ) + c(y − α0 )(y − α1 )

Unlucky Prime Detection

where a = R(α0 ) and b = (R(α1 ) − R(α0 ))/(α1 − α0 ) and

We now consider the case where p1 is unlucky. The algorithm computes at least K more images and thus makes at least K attempts to detect if p1 is unlucky. We will bound the expected probability that the algorithm will fail to discover that p1 is unlucky in K iterations assuming δ = 1 additional evaluation points are used. We will assume, conservatively, that if a subsequent prime p is unlucky, the algorithm will fail to detect that p1 is unlucky. Thus is the cases considered below, we assume that the K subsequent primes are not unlucky.

c=

R(α2 ) − a − b(α2 − α0 ) . (α2 − α0 )(α2 − α1 )

It fails to identify p1 is unlucky if and only if c = 0, that is, R(α2 ) − R(α0 ) =

α2 − α 0 (R(α1 ) − R(α0 )). α1 − α 0

Because α0 , α1 and α2 are non-zero, distinct, and random, the fraction (α2 − α0 )/(α1 − α0 ) is uniformly distributed on [2, p − 1]. If R(α2 ) = R(α0 ) then the algorithm fails to identify p1 is unlucky iff also R(α1 ) = R(α0 ). This happens with probability at most d2 /(p − 1)2 . If R(α2 ) 6= R(α0 ) then if R(α1 ) = R(α0 ) then the algorithm identifies p1 is unlucky with probability 1, otherwise it fails to identify p1 is unlucky with probability at most 1/(p−2). Thus the total probability of failure is at most d2 /(p − 1)2 + 1/(p − 2).

CASE r1 = 0, that is, p1 |R(y). Step 3.5.1 sets ∆ = −1 so that N = ∆ + δ = 0, hence, the algorithm uses one non-zero evaluation point α0 to interpolate R(y)/y L . It fails to identify p1 is unlucky if and only if R(α0 )/αL 0 ≡ 0 mod p, that is, if and only if α0 is a root of R(y)/y l = cd y d−l + ... + cl . On average, a polynomial of degree n over p has exactly one root. The worst case occurs when R(y) = y n − 1 and n|p − 1 which has exactly n distinct roots. Thus we have

CASES d1 = d − 1, l1 = l and d1 = d, l1 = l + 1. Consider the case d1 = d − 1, l1 = l, that is, the degree estimate is off by 1 but the low degree l1 is correct. In step 3.4 because we are using δ = 1 additional evaluation point, we still have sufficient points to interpolate R/y l . Thus

Pr(R(α0 ) = 0) ≤ (d − l)/(p − 1) ≤ (D − L)/(p − 1).

d−l r = R/y l mod ΠN + ... + cl mod p. i=0 (y − αi ) = cd y

CASE l1 = d1 = 0, that is, r1 is a constant. Step 3.5.1 sets ∆ = 0 so that N = ∆ + δ = 1, hence, the algorithm uses two non-zero evaluation points α0 and α1 to interpolate R(y) in step 3.4. Let r = a + b(y − α0 ) where a = α0 and b = (R(α1 ) − R(α0 ))/(α1 − α0 ). The algorithm fails to identify p1 is unlucky if and only if b = 0, that is, R(α1 ) ≡ R(α0 ) mod p. We claim that for 0 < α0 6= α1 < p, the Pr(R(α0 ) ≡ R(α1 ) (mod p)) < d/p where d = deg y (R). To show this let 6= 0 : R(α) ≡ v

d2 p−1 1 d−1 d . − < < . (p − 1)2 d p−1 p−1 p

This maximum is achieved with R(y) = y d − 1 and d|(p − 1). To summarize, if R(y) = p1 (y d − 1) and d|(p − 1), p1 is unlucky and the probability that algorithm PRES fails to detect this is less than d/p.

and

p, α

c2v 1 − (p − 1)2 p−1

where 1/(p − 1) accounts for α0 6= α1 . This probability is maximized when the cv s are maximized. Thus since 0 ≤ cv ≤ d it is

Pr( ci ≡ 0 mod p ) < 1/4

cv = |{α ∈

Pr(R(α0 ) = v) and Pr(R(α1 ) = v) v∈ p

Because p is not also unlucky (our assumption), then the algorithm will identify p1 was unlucky with probability 1. Now suppose d1 = d, l1 = l + 1, that is, the low degree estimate is off by 1. Again, in step 3.4, because we are using one additional evaluation point, we have sufficient points to interpolate R/y l but the algorithm interpolates the rational function r = R/y l+1 mod ΠN i=0 (y − αi )

(mod p)}|.

where N = d − l, R/y l = cd y d−l + ... + cl+1 y + cl and

cv = p − 1 and 0 ≤ cv ≤ d. Because 0 < α0 6= α1 < p Now are chosen at random, we have

d−l + ... + (−1)d−l A ΠN i=0 = y



250

4. A MONIC RESULTANT ALGORITHM

where A = (−1)d−l α0 α1 ...αN . Computing r over we find that r = cAl y d + .... Since αi are chosen from 0 < αi < p, i.e., the αi are non-zero, A is also non-zero. Because p is not unlucky (our assumption), then p does not divide cl , hence, the algorithm will identify that p1 was unlucky with probability 1.

Let R = cd y d + cd−1 y d−1 + ... + cl+1 y l+1 + cl y l . Algorithm MRES below outputs the monic resultant R/cd with high probability. It uses Wang’s rational number reconstruction (see [13, 3, 10]) to recover the rational coefficients of R/cd from R/cd modulo M where M is a product of primes.

Algorithm MRES

CASE d1 = d − 1, l1 = l + 1. That is, the degree estimate and the low degree estimate are both off by 1. This time in step 3.4, we have one too few points to interpolate the rational function R/y l+1 , i.e., N = d − l − 1. Again, we interpolate

Input A, B ∈ [x, y]\{0} of degree m and n resp. Input D ≥ L ≥ 0 satisfying D ≥ d and L ≤ l. Input H ≥ h = max(1, |cd |, |cd−1 |, ..., |cl+1 |, |cl |). Input K ≥ 1 (number of primes of agreement.) Output R = resx (A, B) ∈ [y].

r = R/y l+1 mod ΠN i=0 (y − αi )

1 Initialize M = 1, ∆ = D − L, δ = 0, l = L, d = D.

where R/y l = cd y d + ... + cl+1 y + cl and ΠN i=0 (y − αi ) = y d−l−1 − ... − A where A = (−1)d−l α0 α1 ...αN . Over , r = (cd −

2 Initialize S to a set of primes such that Πp∈S  H 2 , and for each p ∈ S, we have p > ∆ + degy A + degy B and p is not a bad prime.

cl d−l−1 )y + .... A

3 REPEAT

Thus the algorithm fails to identify that p1 is unlucky if and only if Acd − cl ≡ 0 mod p. Now since the algorithm chooses αi from 0 < αi < p at random A is non-zero and it is uniformly distributed on (0, p). Since p is not unlucky (our assumption), cd 6≡ 0 mod p and and cl 6≡ 0 mod p. Thus Pr(Acd − cl ≡ 0 mod p) = 1/(p − 1).

3.1 Choose a new prime p from S. Set Ap = A mod p and Bp = B mod p. 3.2 Set N = ∆+δ and choose N +1 distinct non-zero evaluation points α0 , α1 , ..., αN from p at random such that am (αi ) 6≡ 0 mod p and bm (αi ) 6≡ 0 mod p.

CASE d1 = d − 2, l1 = l. That is, the degree estimate is off by 2 and the low degree l1 is correct. If δ = 1 then in step 3.4 we have 1 too few points to interpolate R/y l , i.e., N = d − l − 1. The value r interpolated in step 3.4 satisfies

3.3 FOR i = 0, 1, ..., N DO Compute ri ∈ p the resultant of Ap (αi ) and Bp (αi ) modulo p using the Euclidean algorithm and set ri = ri /αli mod p. 3.4 Interpolate r ∈ Set r = y l r.

r = R/y l mod ΠN i=0 (y − αi ) where R/y l = cd y d−l + cd−1 y d−l−1 + ... + cl and

p[y].

3.5 IF δ = 0 set the bounds: 3.5.1 If r = 0 set d = L − 1, l = L otherwise set d = degy (r) and l to the low-degree of r in y. ¯ = r, G = 3.5.1 Set δ = 1, ∆ = d − l, M = p, R F AIL, j = 0.

we have

r = (cd−1 − cd C)y d−l−1 + .... Because N > 0, i.e. we have at least two points, C is still almost uniformly distributed on [0, p). For all 0 < x, y < p we have by symmetry

3.6 ELIF degy (r) > d then – restart the algorithm Initialize M = 1, ∆ = D − L, δ = 0, l = L, d = D, G = F AIL.

Pr(C ≡ x mod p) = Pr(C ≡ y mod p).

3.6b ELIF degy (r) < d then GOTO 3.1. 3.7 ELSE apply the Chinese remainder theorem: ¯ mod M for C. 3.7.1 Solve C ≡ r mod p and C ≡ R ¯ 3.7.2 Set M = p × M, R = C.

By computer experiment, we find that Pr(C = 0) − Pr(C 6= 0) = ±

from (αi , ri ) for i = 0..N .

3.4b If r 6= 0 then make r monic in

d−l−1 − Cy d−l−2 − ... − A ΠN i=0 = y

where C = α0 + α1 + ... + αN . Computing r over

p[y]

1 . (p − 1)N

3.8 IF G 6= F AIL and G ≡ r mod p THEN set j = j + 1 ELSE set j = 0, G = F AIL.

Therefore C is almost uniformly distributed on [0, p), thus Pr(cd−1 − cd C) ≡ 0 mod p) ∼ 1/p. We conclude that if p is not also unlucky, the probability of not detecting this is ∼ 1/p in this case. To summarize, if p1 is unlucky, and p2 , ..., pK are not unlucky, then the probability that the algorithm fails to detect that p1 is unlucky is decreasing from (d − l)/(p − 1) to 1/p as degy r1 increases. The maximum failure probability depends on the difference d − l. It is bounded by (D − L)K /P where P is the product of the K check primes. This maximum failure probability can be reduced by choosing δ > 1. Then the maximum failure probability is bounded by (D − L)δK /P δ .

3.9 IF G = F AIL then apply rational reconstruction ¯ modulo M to obtain G. to the coefficients of R UNTIL j = K. 4 Clear the fractions in G. Output G. The first difference between algorithm MRES and PRES is that MRES makes the images monic in step 3.4b so that R/cd modulo M is reconstructed in step 3.7. A second difference is the addition of step 3.6b. If the degree of an image

251

we are using we can more easily use an asymptotically fast interpolation algorithm when D − L + 1 is very large.

is too low, this means the current prime p divides cd and we cannot reconstruct the rational coefficients using this image. A third difference is that we do not put the image in the symmetric range in 3.7. The treatment of negative rational coefficients is handled by rational reconstruction. The main difference is the use of rational number reconstruction in step 3.9. If this succeeds in step 3.9, the test in step 3.8 tests in the subsequent iterations whether the current image r is consistent with G the result of the rational reconstruction. The algorithm terminates when we have K consecutive primes of agreement. The rational number reconstruction in step 3.9 should be done in such a way that it will fail with high probability when M is not large enough yet to recover the coefficients of R/cd . Otherwise rational reconstruction may dominate the cost of the algorithm. How to do this is described in [10]. One may trivially modify the bounds used by Wang’s rational reconstruction (for example, by using (M − 1)/8 instead of the default bounds (M − 1)/2 for the numerators and denominators) or use the rational reconstruction algorithm of Monagan in [10] to force it to fail with high probability when M is not large enough to reconstruct R/cd . Suppose the resultant R = ay +b ∈ [y] where gcd(a, b) = 1 and a and b have the same length. To reconstruct the monic polynomial y+b/a using rational reconstruction, algorithm MRES will need approximately twice as many primes as algorithm PRES. This is the reason for the H 2 in step 2. This suggests a hybrid algorithm, where we attempt to reconstruct both R and R/cd , will be best. The argument that algorithm MRES must always terminate is the same as was made for algorithm PRES. We argue that the failure probability of algorithm MRES is no worse than twice that of algorithm PRES. This is because the only essential difference is the use of rational reconstruction, and, as we have just remarked, this may require up to twice the number of primes.

5.

Acknowledgement We acknowledge the contribution made by the following people who noticed various inefficiencies in Maple’s implementation of Collins’ algorithm in Maple: Chris Smith, Roman Pearce, and Greg Fee. We also wish to thank the two anonymous referees who contributed many useful remarks and suggestions for improving the paper.

6. REFERENCES [1] W. S. Brown, J. F. Traub. On Euclid’s Algorithm and the Theory of Subresultants. JACM 18(4), 505–514, 1971. [2] George Collins. The Calculation of Multivariate Polynomial Resultants. JACM 18(4), 515–532, 1971. [3] G. E. Collins and M. J. Encarnacion. Efficient Rational Number Reconstruction. J. Symbolic Comp. 20, pp. 287–297, 1995. [4] Karin Gatermann, “Polynomial Systems with Symmetry”, talk presented at “Solving Systems of Equations”, September 14-18, 1998, MSRI. [5] J. von zur Gathen and J. Gerhard. Modern Computer Algebra. Cambridge University Press, UK, 1999. [6] K. O. Geddes, S. R. Czapor, and G. Labahn. Algorithms for Computer Algebra. Kluwer Academic Publ., Boston, Massachusetts, USA, 1992. [7] A. Goldstein, G. Graham. A Hadamard type bound on the coefficients of a determinant of polynomials. SIAM Review, 16, pp. 394–395, 1974. [8] E. Kaltofen and B. Trager. Computing with polynomials given by black boxes for their evaluations: Greatest common divisors, factorization, separation of numerators and denominators. J. Symbolic Comp. 9, pp. 301–320, 1990. [9] M. B. Monagan, K. O. Geddes, K. M. Heal, G. Labahn, S. M. Vorkoetter, J. McCarron, P. DeMarco. Maple 9 Introductory Programming Guide Waterloo Maple, 2003. ISBN: 1-894511-43-3. [10] M. B. Monagan. Maximal Quotient Rational Reconstruction: An Almost Optimal Algorithm for Rational Reconstruction. Proceedings of ISSAC ’2004, ACM Press, pp. 243–249, 2004. [11] Posso test suite: http://www-sop.inria.fr/saga/POL/ [12] B. Trager. Algebraic Factoring and Rational Function Integration. Proceedings of SYMSAC ’76, ACM Press, pp. 219–226, 1976. [13] P. S. Wang, M. J. T. Guy, J. H. Davenport. p-adic Reconstruction of Rational Numbers. SIGSAM Bulletin, 16(2), 1982. [14] R. Zippel. Interpolating Polynomials from their Values. J. Symbolic Comp. 9(3), 375–403, 1990.

CONCLUDING REMARKS

Examples 2.1 and 2.2 tell us that the modular resultant algorithm of Collins may perform poorly when the bounds for the coefficients and degrees are off. Example 2.4 gives a second reason, namely, if the resultant is sparse and in several variables. These realities make the choice between using the subresultant algorithm, a determinant based algorithm, or the modular resultant algorithm awkward. Algorithm PRES solves the first problem. For sparse resultants one might consider using a sparse interpolation algorithm. See [14, 8]. Examples 2.3 and 2.4 tell us that the height of the primitive part of the resultant can be much smaller than the height of the resultant even when the inputs are primitive. Algorithm MRES solves this problem. An alternate design of algorithm PRES and algorithm MRES would be to incrementally interpolate the first image instead of using the bounds D and L. The main reason we chose not to do this is that incremental interpolation is significantly more expensive, especially if one does not know l. This can make the performance of the algorithm poor for the normal case when the degree bounds are good. Another practical reason was that on our test problems, the time spent computing the first image using D and L was not the main cost. Also, by knowing in advance how many points

252

Generalized Normal Forms and Polynomial System Solving Bernard Mourrain Projet GALAAD, INRIA 2004 Route des Lucioles, BP 93, 06902 Sophia Antipolis, Cedex (France)

[email protected]

ABSTRACT

Resultant-based method (see eg. [6], [2]) rely on linear algebra steps, and are stable against a small deformations of the input, but suffer from restriction both in terms of genericity and number of equations of the input system. Gr¨ obner basis techniques (see eg. [4]) compute a basis of the quotient algebra A of the polynomial ring by the ideal generated by any input system, but unfortunately suffer from unavoidable unstability: the monomial ordering attached to the Gr¨ obner basis imposes some pivoting in the underlying matrices, even if the pivot coefficient is small. This can, lead to artificial unwanted singularities in the representation of the quotient ring (compare for instance the degree reverse lexicographic Gr¨ obner basis of p1 = ax21 + bx22 + ε1 x1 x2 , 2 2 p2 = cx1 + dx2 + ε2 x1 x2 with ε1 = ε2 = 0 and ε1 = 0, ε2 = 0). Intermediate approaches such as H-bases have also been studied [12], [13]. The stopping criterion involved in such methods require the computation of generators of the syzygies of the highest degree terms of the polynomials. Though it also yields a basis of the quotient ring A, without any apriori knowledge, practically speaking, it also suffers from the swelling of the size of the linear systems to be solved. The method we propose generalizes resultant constructions and Gr¨ obner bases, taking advantage of the numerical stability of resultant constructions and of the efficiency of Gr¨ obner bases, by constructing efficiently, the normalizing rules. As for H-bases, we use a graduation of the polynomial ring, but the construction is optimized in the spirit of [17]. We obtain border bases in the sense of [10] (see also [16]) and describe here a new incremental and efficient method to obtain them. As in [8] or [7], our approach is based on linear algebra tools. Our normal form construction yields a (monomial) basis B of A, an algorithm to project the ring of polynomials K[x] onto B. From this normal form algorithm, we deduce the table of multiplication by any element in A. From these tables, we deduce, either a numerical approximation of the roots by eigenvector computation, [1], [15], [21] or a rational univariate representation [20], [9] if we want an exact encoding of these roots. In the next section, we gives the notations, used in section 3 to prove the stopping criteria for generalized normal forms. In section 4, we describe the complete algorithm and prove its correctness. Finally, we show the efficiency of our implementation and its numerical behavior, on classical polynomial benchmarks. More details on the results of this paper can be found in [19].

This paper describes a new method for computing the normal form of a polynomial modulo a zero-dimensional ideal I. We give a detailed description of the algorithm, a proof of its correctness, and finally experimentations on classical benchmark polynomial systems. The method that we propose can be thought as an extension of both the Gr¨ obner basis method and the Macaulay construction. We have weaken the monomial ordering requirement for Gr¨ obner bases computations, which allows us to construct new type of representations for the quotient algebra. This approach yields more freedom in the linear algebra steps involved, which allows us to take into account numerical criteria while performing the symbolic steps. This is a new feature for a symbolic algorithm, which has a huge impact on the practical efficiency.

Categories and Subject Descriptors G.0 [Mathematics of Computing]: General

General Terms Algorithms, application

Keywords Multivariate polynomial, normal form, quotient algebra, rootfinding, symbolic-numeric computation

1.

INTRODUCTION

Solving polynomial systems is the cornerstone of many applications in domains such as robotics, geometric modeling, signal processing, chromatology, structural molecular biology . . . In these problems, the system has often finitely many solutions and the equations often appear with approximate coefficients. In this paper, we present a new method for solving such zero-dimensional polynomial systems, based on linear algebra tools. It combines features from resultantbased methods and from Gr¨ obner basis computation.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. ISSAC’05, July 24–27, 2005, Beijing, China. Copyright 2005 ACM 1-59593-095-7/05/0007 ...$5.00.

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2.

NOTATIONS

Definition 2.3. A rewriting family F for a monomial set B is a set of polynomials F = {fi }i∈I such that : supp(fi ) ⊂ B + , fi has exactly one monomial γ(fi ) (also called the leading monomial of fi ) in ∂B, if γ(fi ) = γ(fj ) then i = j,

Let K be an effective field. The ring of n-variate polynomials over K will be denoted by R, R = K[x] = K[x1 , . . . , xn ]. We consider n-variate polynomials f1 , . . . , fs ∈ R. Our goal is to solve the system of equations f1 = 0, . . . , fs = 0 over the algebraic closure K of K. These polynomials generate an ideal of K[x] that we call I. The quotient of K[x] modulo I will be denoted by A. From now on, we suppose that I is zero dimensional so that A is a finite dimensional K-vector space. The roots, with coordinates in the algebraic closure of K, will be denoted by ζ1 , . . . , ζd , with n ζi = (ζi,1 , . . . , ζi,n ) ∈ K . We recall some of the definitions stated in [16], [17], [18] and add a few more that we will need in the sequel. For any subset S of R, we denote by S + the set S + = S ∪ x1 S ∪ · · · ∪ xn S, ∂S = S + \S. The support supp(p) of a polynomial p ∈ K[x] is the set of monomials appearing with non-zero coefficients in p. For a subset S of another set S
, we will denote by S c the set-theoretical complement of the set S in S
. Given a set S of elements of K[x], we denote by S the K-vector space spanned by the elements of S. Finally, we denote the set of all the monomials in the variables x = (x1 , . . . , xn ) by M. A set of monomials B is said to be connected to 1, if and only if, for every monomial m in B, there exists a finite sequence of variables (xij )j∈[1,l] such that 1 ∈ B, Πj=1...l
xij ∈ B, ∀l
∈ [1, l] and Πj∈[1,l] xij = m. A set of monomials B is stable by division, if for any m ∈ B and any variable xi such that m = xi m
, we have m
in B. Remark that a set B stable by division, is connected to 1.

Remark that the elements of F can be seen as rewriting rules for the initial monomial using monomial of B. Definition 2.4. A reducing family F of degree λ ∈ Λ for a set B is a set of polynomials such that: F is a rewriting family for B, ∀m ∈ ∂B of degree at most λ, ∃f ∈ F | γ(f ) = m. For the set B = {1, x0 , x1 , x0 x1 }, the set of polynomials F = {x20 − 1, x21 − x1 , x20 x1 − x1 , x21 x0 − x1 } is a reducing family of degree 3. Notice that a reducing family of degree λ for a set B (connected to 1) allows to rewrite the monomials of B + λ , modulo F as elements of Bλ . This leads in fact, to the definition of the linear projection RF , associated to a reducing family for a set B connected to 1. Definition 2.5. Given a reducing family of degree λ ∈ Λ for a set B connected to 1, we define the linear projection RF : B + λ → Bλ such that ∀m ∈ Bλ , RF (m) = m, ∀m ∈ ∂Bλ , RF (m) = m − f ; where f ∈ F is the unique member of F such as m = γ(f ). We extend this construction to B + λ by K-linearity. In the sequel, we will make a heavy use of multiplication operators by one variable that we define as follows: Mi,λ : Bλ− → b →

Definition 2.1. Let Λ be a monoid with a good total order ≺, such that:

The subscript λ is redundant as soon as we know that F is a reducing family of degree λ, and we will omit this subscript in the sequel.

∀α, β, γ ∈ Λ, α ≺ β ⇒ γ + α ≺ γ + β A (Λ, ≺)-graduation of K[x] L is the decomposition of K[x] as the direct sum: K[x] = λ∈Λ K[x][λ] , with the following property:

Definition 2.6. Let F = {f1 , . . . , fs } be a polynomial set, we denote by Fλ the vector space:

∀f ∈ K[x][α] , g ∈ K[x][β] ⇒ f.g ∈ K[x][α+β] .

Fλ = {xα fi | Λ(xα fi ) ≤ λ}.

We denote by degree of f or degΛ (f ), or deg(f ) (when no confusion is possible) or Λ(f ), the following element of Λ: M Λ(f ) = degΛ (f ) = min{λ ∈ Λ | f ∈ K[x][λ
] }. L

Bλ RF (xi b).

Obviously, we have Fλ ⊂ (F )λ where (F ) is the ideal generated by F . Next we introduce a definition, which is weakening the notion of monomial ordering for Gr¨ obner basis: Definition 2.7. We say that a function, γ : K[x] → M (M is the set of all monomials in the unknowns x1 , . . . , xn ), is a choice function refining the graduation Λ, if for any polynomial p, γ(p) is a monomial such that γ(p) ∈ supp(p), if m ∈ supp(p), m = γ(p) then γ(p) does not divide m, and Λ(γ(p)) = max{Λ(m), m ∈ supp(p)}.

λ
≺λ


For any set V ⊂ K[x], let Vλ = λ
≺λ K[x][λ ] ∩ V . For any λ ∈ Λ, let λ+ = min{λ
∈ Λ; K[x]+ λ ⊂ K[x]λ
} and let λ− = max{λ
∈ Λ; K[x]+ λ
⊂ K[x]λ }. In order not to be confused between different notions of degree, for any monomial m of K[x], we define the size of m, denoted by |m|, to be the integer d such that m = xi1 · · · xid . Another classical graduation is the one associated with a monomial ordering, where Λ = Nn and ≺ is a monomial order (see [5][p. 328]) such that for all α = (α1 , . . . , αn ) ∈ αn 1 Nn , we have K[x][α] = K xα 1 · · · xn .

Example 2.8. In the following, we consider the Macaulay 1 choice function γ, such that for all p ∈ K[x], γ(p) = xα 1 ··· n xα satisfies, degN (γ(p)) = max{degN (m); m ∈ supp(p)} = n d, and ∃i0 st. αi0 = max{degxi (m), m ∈ supp(p) and , degN (m) = d; i = 1, . . . , n}.

Definition 2.2. We say that Λ is a reducing graduation if Λ is a graduation and if we have the property: for all monomials m, m
∈ M, such that m
divides strictly m, Λ(m
) ≺ Λ(m), Λ(m
) = Λ(m).

The monomial returned by the choice function has the same name as the leading monomial of an element of a reducing family. This is intended, as we will define a reducing family on the behalf of choice functions, and in this framework the two will coincide. Hereafter if S = {p1 , . . . , ps } is a polynomial set, then we denote by γ(S) the set: γ(S) = {γ(p1 ) . . . γ(ps )}.

Both graduations induced by the classical degree and a monomial ordering are reducing graduation. Hereafter, we will denote by Λ a reducing graduation.

254

e Proof. Remark that the way we define it, makes RF inherently a linear multivalued application. Hence to prove the theorem we have first to show that under the above hye becomes a well defined application, and next potheses, RF that this well defined linear application coincide with the projection P of E on B along S. Remark also that E is obviously stable by monomial division: if all the possible decompositions of m as a product of variables are such that RF (xi0 · · · RF (xik )) is defined, then a fortiori if m
is a divisor of m, this property is true for m
. Let us show that the extension process defines an application. Let m = xi0 m
= xi1 m

with i0 = i1 and m, m
, m

∈ E, then there exists m


∈ E (since E is stable by monomial division) such that m = xi0 xi1 m


. As e e e (m
), RF (m

), RF (xi0 m
), m, m
, m

, and m


are in E, RF e

e


RF (xi1 m ), and RF (m ) are defined and we have:

Definition 2.9. Let γ be a choice function refining a graduation Λ. For any polynomials p1 , p2 ∈ K[x], let the C-polynomial relative to γ and (p1 , p2 ) be C(p1 , p2 ) =

lcm(γ(p1 ), γ(p2 )) lcm(γ(p1 ), γ(p2 )) p1 − p2 . γ(p1 ) γ(p2 )

Let the C-degree of (p1 , p2 ) be the degree of (lcm(γ(p1 ), γ(p2 )) /γ(p1 ))p1 , and let the leading monomial of the pair (p1 , p2 ) be lcm(γ(p1 ), γ(p2 )). This is almost the same definition as a S-polynomial [4] when γ is a monomial ordering. We however need a new name to underline that now γ may not be a monomial ordering (i.e. a total order compatible with monomial multiplication). As we will see in the next section, the C-polynomials express commutation conditions for the Mi,λ .

3.

e e (xi0 RF (m
)) RF e e RF (xi1 RF (m

))

GENERALIZED NORMAL FORM CRITERION

e e e = RF (xi0 RF (xi1 RF (m


))), e e e = RF (xi1 RF (xi0 RF (m


))).

Let F = {f1 , . . . , fs } be a polynomial system and let I be the ideal generated by F . Remember that Fλ (the K-vector space spanned by the monomial multiples of the fi , xα fi of degree ≤ λ ∈ Λ) is included in Iλ . Thus, when Iλ = Fλ we can define a normal form modulo I, up to the degree λ as the projection of K[x]λ along Fλ onto a supplementary space Bλ . Hereafter, we consider a set B of monomials, containing 1. Let F be a rewriting family, and let H = {m, ∃p ∈ F, γ(p) = m} be the set of their leading monomials then, obviously F allows us to define the projection RF of B ∪ H on B along F . However we may extend this projection using the following extension process:

The commutation condition guarantees that the two quantie ties are equal, so that the definition of RF does not depend on the way to write m as a product of variables. e Next we have to show that RF and P coincide on their common set of definition. We do it by induction on the size of the monomials: e (1) = P (1) = 1 (since 1 ∈ B). For It is true that RF any monomial m = 1 in E, the property of connectivity of B and the definition of E gives us: ∃m
∈ E and i0 ∈ e (m
) is defined, so that [1, n] such that m = xi0 m
and RF we have:

Definition 3.1. Let F be a rewriting family, let m be a monomial such that there exists m
∈ ∂B and there exists r integers i1 , . . . , ir ∈ [1, n]r , m = xi1 · · · xir m
. We define e (m) by induction on k, as follows. RF e e • if r = 0, RF (m
) is defined as RF (m
) = RF (m
) = m
−f where f ∈ F is such that γ(f ) = m
. e e • ∀r ≤ k, RF (xi1 · · · xir m
) = RF (xir RF (xi1 . . . xir−1 m
)), if this latter quantity is defined. Otherwise we say that e (m) is undefined. RF

Now by induction, m
− P (m
) ∈ Sλ− and

e RF (m)

e e e = RF (xi0 m
) =def RF (xi0 RF (m
))

=

induction

e RF (xi0 P (m
)) ∈ B.

e (m) = xi0 (m
− P (m
)) m − RF ` ´ e + xi0 P (m
) − RF (xi0 P (m
)) ∈ S. e (m) is the projection of m on B along S. ✷ Thus RF Suppose now that we are given a reducing family of degree λ instead of a rewriting family. Then we can further extend the above theorem with the help of the following lemma.

e only Remark that the above process allows us to define RF on monomials, and we extend it implicitly, by linearity. Remark also that this extension process is not defined in a unique way. Indeed, two different decompositions of a monoe mial m may lead to two different values of RF (m). However the following theorem shows that this extension process becomes canonical as soon as we check some commutativity conditions.

Lemma 3.3. Let F be a reducing family of degree λ for a set B connected to 1, and suppose that ∀f ∈ F, Λ(γ(f )) = Λ(f ). With the notation of theorem 3.2, the set E of monomials m such that for all decomposition of m as a product of e e variables, m = xi0 · · · xik , then RF (xi0 · · · RF (xik )) is defined, contains the set of monomials of degree less or equal to λ. Proof. Let m ∈ Mλ be a monomial of degree less or equal to λ, then m can be written as m = xi1 . . . xid with d = |m|. Let us prove by induction on k ≤ d, that pk = RF (xik RF (· · · RF (xi1 )) · · · ) is defined and that degΛ (pk ) ≤ degΛ (xik · · · xi1 ). As F is a reducing family of degree λ, for m
∈ supp(xik+1 pk ) ∩∂B, we have a rewriting rule for m
. The hypothesis that ∀f ∈ F, Λ(γ(f )) = Λ(f ) implies that m
rewrites in terms of monomials of degree bounded by degΛ (xik+1 · · · xi1 ). This proves pk+1 = RF (xik+1 pk ) is defined and degΛ (pk+1 ) ≤ degΛ (xik+1 · · · xi1 ). e e Finally, we deduce that RF (xi1 · · · RF (xid )) is defined, for

Theorem 3.2. Assume that B is connected to 1. Let F be a rewriting family, and let E be the set of monomials m such that: for all decomposition of m as a product of variables, m = xi0 · · · xik , then RF (xi0 · · · RF (xik )) is defined. Suppose that for all m ∈ E and all indexes i, j ∈ [1, n] such xi xj m ∈ E, we have: e e e e (xi RF (xj m)) = RF (xj RF (xi m)). RF e Then RF coincides with the linear projection P of E on B along the vector space spanned by the polynomials S = {xα f, α ∈ Nn , f ∈ F and xα γ(f ) ∈ E}.

255

any decomposition m = xi1 · · · xid ∈ Mλ so that m ∈ E. This ends the proof. ✷

Remark 3.7. In this proof, we have shown that if the Cpolynomials up to the degree λ reduce to 0, then the multiplication operators Mi,λ commute.

Theorem 3.4. Let F be a reducing family of degree λ for a set B connected to 1. If we have:

Finally, this leads to a new proof of theorem 3.1 of [16]:

• ∀f ∈ F, Λ(γ(f )) = Λ(f ).

Theorem 3.8. Let F be a reducing family of all degrees λ ∈ Λ for a set B of monomials, connected to 1, let RF be the corresponding reduction from B +  onto B, and let Mi : B → B such that ∀ b ∈ B, Mi (b) = RF (xi b). Then,

• Mj,λ ◦ Mi,λ− = Mi,λ ◦ Mj,λ− , for 1 ≤ i, j ≤ n, e then, we can extend RF to a linear projection RF from K[x]λ onto Bλ of kernel Fλ .

Proof. As F is a reducing family of degree λ, by lemma 3.3, we have E ⊃ Mλ . Let us prove that for all m ∈ Mλ−− and all pair of ine e such that RF (xi dexes (i, j), there exists a way to define RF e e e RF (xj m)) = RF (xj RF (xi m)). e e (m) is defined and supp(RF (m)) ⊂ As Mλ−− ⊂ Mλ ⊂ E, RF B. e e (xi m) = RF (xi RF (m)) and, similarly, We define RF e e RF (xj m) = RF (xj RF (m)). With this definition we have: e e (xi RF (xj m)) = RF

e Mi,λ (Mj,λ− (RF (m)))

= =

e Mj,λ (Mi,λ− (RF (m))) e e RF (xj RF (xi m)).

which proves the commutation property. We end the proof by applying theorem 3.2. ✷

Mj ◦ Mi = Mi ◦ Mj , for 1 ≤ i, j ≤ n iff there exist a unique projection R which extends uniquely e from K[x] onto B such that to a linear projection RF e e ker(RF ) = (F ) and (RF )|B+  = RF . Proof. Under these hypotheses, by theorem 3.4, for any e λ ∈ Λ, (RF )|B+ λ extends uniquely to a projection RF λ e from K[x]λ onto Bλ , such that ker(RFλ ) = Fλ . Since

for any λ, λ ∈ Λ such that λ < λ , we have (Bλ
)λ = Bλ , and Fλ ⊂ (Fλ
 )λ . We also have Fλ = (Fλ
 )λ so that e e (RF )|K[x]λ = RF . This defines a unique linear operaλ λ
e e e e on K[x] such that R|K[x] = RF and ker(RF ) = tor R λ λ P F F = (F ). It proves the direct implication. The λ λ∈Λ converse implication is immediate. ✷

4. THE ALGORITHM

Corrolary 3.5. With the hypothesis of theorem 3.4, we have K[x]λ = Bλ ⊕ Fλ .

The algorithm, we describe now, consists in computing a suitable set B for a basis of the quotient ring A = K[x]/I and reducing rules, in order to project onto B along I. It can be interpreted as the check that the constructed B is a basis of A. The method proceeds incrementally, until a fixed point is reached. Here are some notations, used in the description of the algorithm: • k will be the loop index at which we will consider the families of polynomials. • Pk will be the set of polynomials, from which a reducing family will be constructed. • Mk = γ(Pk ) will be the set of monomials, used as the leading terms of the family Pk . • (Pk |Mk ) will be the matrix of coefficients of the monomials of Mk in the polynomials of Pk , written row by row. • X:=PseudoSolve(A,B) is the multiplication by a pseudo inverse (eg. Moore-Penrose pseudoinverse) so that A X = B when B is in the image of A. • The function SelectM inDegγ applied to a polynomial set S is a function that defined recursively as follows:

Let us give here an even more effective way to check that we have a projection from Fλ (vector space spanned by the monomial multiples of the fi of degree λ) onto Bλ (element of degree λ of the vector space spanned by B) starting from a reducing family of degree λ, without computing explicitly the multiplication operators. Theorem 3.6. Let λ ∈ Λ. Let F be a reducing family of degree λ, for B. Assume that ∀ f ∈ F, Λ(γ(f )) = Λ(f ) and let RF be the induced reduction from B + λ onto Bλ . Then ∀f, f
∈ Fλ such that C(f, f
) ∈ B + λ , RF (C(f, f
)) = 0 e from K[x]λ onto iff RF extends uniquely as a projection RF e ) = Fλ . Bλ such that ker(RF

Proof. By theorem 3.4, we have to show that this condition is equivalent to the commutation of the operators Mi,λ
, λ
< λ on the monomials of Bλ−− . For any m ∈ Bλ−− and any i1 = i2 such that xi1 m ∈ ∂B, xi2 m ∈ ∂B, there exists f, f
∈ Fλ−  such that γ(f ) = xi1 m, γ(f
) = xi2 m. Thus, we have RF (xi1 m) = γ(f ) − f , RF (xi2 m) = γ(f
)−f
and C(f, f
) = xi2 f −xi1 f
∈ B+ λ. Consequently,

• Set d = min(Λ(g), g ∈ S) • Construct the list S
= {f ∈ S, Λ(f ) = d} • for f ∈ S
do

Mi2 ,λ (Mi1 ,λ− (m)) − Mi1 ,λ (Mi2 ,λ− (m))

– apply γ on f to get a monomial m.

= Mi1 ,λ (γ(f ) − f ) − Mi2 ,λ (γ(f
) − f ) = RF (xi1 γ(f ) − xi1 f ) − RF (xi2 γ(f
) − xi2 f
)

– Perform linear combinations between the elements of S
to eliminate m of the support of the other elements of S
and get S

.

= RF (xi2 f
− xi1 f ) = RF (C(f
, f )).

– if S

contains polynomials of degree less than d, it returns selectM inDegγ applied to S



which is zero by hypothesis. A similar proof applies if xi1 m ∈ B or xi2 m ∈ B. e Conversely, since ker(RF ) = Fλ and C(f, f
) ∈ Fλ ∩
e , we have that R (C(f , f )) = RF (C(f
, f )) = 0, which B+ F λ proves the equivalence and theorem 3.6. ✷

• if no decreasing of the degree of the members of S
occurs, return S
together with the monomial set γ(S
) constructed during the f or loop.

256

• By Increment B, we mean adding to B the monomials corresponding to the zero columns in the matrix (Pk |Mk ). • By monomials at Hamming distance 1 of B, we mean the monomials that we obtain from the monomials of B by writing them as a product of variables and changing exactly one index in this writing. • By neighbors of the monomials added to B, we mean the polynomials whose leading term is at Hamming distance 1 of the set added to B by Increment . Notice that solving a system of the form (Pk |Mk )X = Pk is equivalent (when (Pk |Mk ) is invertible) to computing a vector of polynomials, with one “leading monomial” in Mk and all the others outside. We also need the following subroutines:

8) Ck+1 := Reduce(Ck+1 , ∪i≤k+1 Pj , B) and set poolpol = poolpol ∪ Ck+1 . 9) r = #Mk+1 − Rank((Pk+1 |Mk+1 )), if r = 0 increment B and add to Pk+1 the neighbor of the monomials B is incremented with. 10) poolpol = poolpol ∪ Pk+1 . 11) For all i ∈ [1, k + 1], Pi = {the reduced polynomials in Pi }. S 12) poolpol = Reduce(poolpol, i=1..k+1 Pi , B); k = k + 1. while poolpol = ∅; Output: {Pj , j = 0 . . . k} a reducing family for all k ∈ N

Algorithm 4.1. Remainder. Let p be a polynomial, and F a family of valid rewriting rules with respect to B. Let r := rem(p, F, B) be a polynomial such that supp(r)∩γ(F ) = ∅ and p − r ∈ F .

For k ∈ N, we will denote by B k the set B computed at loop k of the algorithm 4.3, and by F k = ∪ki=0 Pi . In this section, we will denote by Rk the operator such that, given p ∈ K[x], Rk (p) returns Reduce(p, F k , B k ).

Algorithm 4.2. Reduction. Reduce(P, F, B) reduces the set of polynomials P by F a family of valid rewriting rules with respect to B. It works like this: For all p ∈ P , for all m ∈ supp(p), if m ∈ / B do the following:

Definition 4.4. We say that a polynomial f ∈ K[x] is a reducible at loop k, if Rk (f ) ∈ B k . The polynomial f is reduced at loop k, if f ∈ B k . Let us also mention now some useful facts: (r1) During the algorithm 4.3, at loop k, ∀f ∈ F k , γ(f ) ∈ ∂B k , since the polynomials in Pi are reduced (step 12). (r2) In the algorithm 4.3 (step 9 and definition of the function increment), B is constructed as

1) r0 := 1; 2) decompose m as a product of variables m = xi1 . . . xid ; 3) for j from 1 to d = |m| do

B = ∪i ∪f ∈Fi {(mi )\(γ(f ))},

a) rj = xij ∗ rj−1 ,

(1)

where mi ∈ M . (r3) At step 9, the monomials added to B, are by construction, in Mk+1 . (r4) For all monomial m ∈ B, there exists a monomial m
also in ∂B such that m
|m and m
is the leading monomial of a polynomial of F k . This comes from the fact that B is of the form (1). We prove now that the previous algorithm stops and produces a correct result, i.e. a reducing family for all degrees. To do so, we need the following lemmas and proposition.

b) rj = rem(rj , F, B), end for 4) substitute m by rd in p; Notice here that the definition of Reduce is not canonical but depends on some particular choice of a writing of the monomials as products of variables. Let us describe now the main algorithm:

Lemma 4.5. If m is a monomial, and r = Rk (m) for some index k ∈ N, then ∀m
∈ supp(r), Λ(m
) ≤ Λ(m).

Algorithm 4.3. Normal form. Input: F = f1 , . . . , fs generating an ideal I of dimension 0, and γ a choice function refining a reducing graduation Λ. Initialization: P0 = SelectM inDegγ (f1 , . . . , fs ), poolpol = {f1 . . . fs }\P0 , B = (γ(fi ))c , k = 1. Core Loop: do

Proof. This comes from the definition of Rk (see algorithm 4.2): given a monomial m, it computes a list of variables whose product equals m, it adds to a variable r, which initially equals to 1, linear combination of polynomials belonging to F k , and multiply the resulting polynomial by one variable of the list, and so on until the list is exhausted. Since Λ is a graduation, we have with the notation of algorithm 4.2, Λ(rj ) ≤ Λ(xi1 · · · xij ) for all loop j of the reduction ✷ algorithm, so that ∀m
∈ supp(rj ), Λ(m
) ≤ Λ(m).

1) Contk = {p, polynomials in poolpol s.t. supp(p) ⊂ B}. 2) For p ∈ Contk , remove from B the monomial ideal generated by γ(p). 3) Pk = { the valid rewriting rules in poolpol }\Pk−1 .

Lemma 4.6. For all d ∈ N, there exist finitely many Pk containing polynomials whose leading monomials are of size less than d.

4) Compute Ck+1 = {C(f, f
) such that γ(C(f, f
)) = xi0 γ(f ) = xi1 γ(f
), f, f
∈ Pk }, poolpol = poolpol ∪ Ck+1 .

Proof. If a set Pk contains a polynomial whose leading monomial is of size 0, then the algorithm stops as we know that this ideal I is in fact K[x]: any non constant monomial is strictly greater than 1 for any choice function, so having 1 as leading monomial means that the polynomial itself is 1. Next, remark that Pk contains a polynomial whose leading

+

5) Compute Pk+1 = ∂Pk ∩ B . 6) Mk+1 = ∂γ(Pk ) ∩ B + . 7) Pk+1 := PseudoSolve ((Pk+1 |Mk+1 ), Pk+1 ).

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the sets Λi are simultaneously reducible, i.e. f is reducible at loop k and ∀m ∈ supp(Rk (f )), k ≥ l|m| . ✷

monomial is of size d, if and only if we have excluded from B a monomial of size d. Finally remark that the monomials added to B at loop k, are in Mk+1 ⊂ ∂Mk . The lemma is true for d = 0. Assume that it is true for d − 1, and let k0 be the last index such that Pk0 contains a polynomial whose leading monomial is of size d − 1. Thus, after loop k0 no monomial of size d can be added to B. Hence, since the Pk are auto-reduced there will be at most as many Pk , k ≥ k0 containing polynomial with leading monomials of size d, as there are monomials of size d, due to the second remark. We conclude by induction. ✷

Corrolary 4.9. Let d ∈ N, there exists an index kd such that: • For all m of size at most d, supp(Rkd (m)) ⊂ B kd . • and for all m of size at most d, all m
∈ supp(Rkd (m)) of size d
, k ≥ ld
.

Definition 4.7. Given a size d, we will call ld the lowest integer such that after loop ld no Pk , k ≥ d contain polynomials whose leading monomials is of size less than d.

Remember now that for any k ∈ N, the definition of Rk (m) relies on some particular choices for the writing of m as a product of variables. As there is only finitely many monomials of size less than d and as for a given monomial, there exists only finitely many ways to write it as a product of variables. Thus, we can define the following quantity:

Notice that at step 12, some polynomials f ∈ poolpol may not be reducible (see definition 4.4), depending on the shape of B and supp(f ). However, we have the property:

Definition 4.10. For any d ∈ N, we define kd∗ as the maximum of the index kd for all the possible definitions of Rk (m).

Proposition 4.8. Given f ∈ K[x] there exists k such that f is reducible by applying Rk and such that ∀m ∈ supp(Rk (f )), k ≥ l|m| .

Lemma 4.11. Let m be a monomial, and fi be one of the generators of I then there exists a loop index k of the algorithm such that mfi reduces to 0.

Proof. We split supp(f ) into l sets Λ0 ,. . . ,Λl such that

Proof. From the above lemma there exists a loop k of the algorithm such that mfi is reducible at loop k. Let us prove now that when mfi is reducible it reduces to 0. It is true that fi reduces to 0. Suppose that for all m
of size d − 1 m
fi reduces 0 at a loop k1 , and let xj be a variable. By definition of Rk1 (xj m
fi ) = Rk1 (xj Rk1 (m
fi )) = Rk1 (xj 0) = 0. Hence this induction on the size of m shows that if mfi is reducible then it reduces to 0. This ends the proof. ✷

Λi = {m ∈ supp(f )\ ∪i−1 j=0 Λj , Λ(m) = min(Λ(m
), m
∈ supp(f )\ ∪i−1 j=0 Λj )}. We show that there exists a certain loop k at which all the monomials of all the sets Λi are simultaneously reducible. We consider here the set S0 = Λ0 , and define k0 = l|f | . For all m ∈ Λ0 , consider the computation of Rk0 (m) = Reduce(m, F k0 , B k0 ) described in definition 4.2. We denote by r1 , . . . , rd the sequence of remainders which are computed at loops (b). • Either, for all i = 1 . . . d, we have supp(ri ) ⊂ B k1 , where we define k1 = max(l|rj | j=1..d , k0 ). We have that rd ∈ B k1 and that k1 ≥ l|rd | , therefore implying that m is reducible at loop k1 and that for all k ≥ k1 , m is also reducible at loop k with the property that Rk (m) = rd . • Or, there exists m ∈ Λ0 and i0 < d such that supp(ri ) ⊂ maxj≤i (l|r | ,k0 ) j , but not supp(ri0 +1 ). Let k1 = max(l|rj | j = B 1..i0 , k0 ). Then for m
∈ supp(ri0 )\B k1 , we have, by lemma 4.5:

Lemma 4.12. For d ∈ N, at loop ld , F ld is reducing family with respect to B ld , for the set of monomials of size ≤ d. Proof. We need to prove that ∀m ∈ ∂B ld of size ≤ d, there exists one polynomial p ∈ F ld such that m = γ(p). By remark (r4), as m ∈ ∂B ld , there exists p1 ∈ F ld such that m = xi1 · · · xil γ(p1 ) and ∀l
≤ l, xi1 · · · xil
γ(p1 ) ∈ ∂B ld . Let k be the smallest integer for which there exists such a p1 ∈ F k . By lemma 4.11, xi1 p1 will be reducible at loop k1 ≥ k, but as xi1 p1 is computed from p1 at step 5, there exists a polynomial p2 ∈ F k such that γ(p2 ) = xi1 γ(p1 ). With the same arguments, for j ≤ l, there exist kj ∈ N and a polynomial pj ∈ F kj such that γ(pj ) = xi1 · · · xij γ(p). Hence there exists a polynomial p ∈ F kl such that γ(p) = m. However as γ(p) is of size d, we have by definition of ld , kl ≤ ld and p ∈ F ld . This proves that F ld is a reducing family with respect to ld ✷ B , for the monomials of size ≤ d.

Λ(m
) ≤ Λ(xi1 · · · xi0 ) < Λ(xi1 · · · xid ) = Λ(m), since Λ is a reducing graduation (see definition 2.2). This / (m). implies that m
∈ Now let S1 = S0 ∪{m
∈ supp(ri0 )\B k1 , Λ(m
) minimal}. The above construction of m
implies that S0 ⊂ S1 and, that S0 = S1 . We iterate this construction, replacing S0 with S1 , S2 with S1 , . . . . Since K[x] is Noetherian, the sequence of Si is finite. Consequently, we have proved that in both cases, there exists an index k0
such that all the monomials of Λ0 are reducible after loop k0
. Let red(Λ0 ) be the set of all these reductions, so that k0
satisfies the property k0
≥ lmax(|r|, r∈red(Λ0 )) . Consider the next sets Λi . By the same process, there exists an index ki such that all the monomials of Λi are reducible after loop ki . Let red(Λi ) be the set of all these reductions, so that k1 also satisfies the property ki ≥ lmax(|r|, r∈red(Λi )) . Hence, for k ≥ max(ki ) we have that all the monomials of

Proposition 4.13. Assume that the ideal I generated by the elements f1 , . . . , fm is zero-dimensional. Then the algorithm 4.3 stops and yields a basis B of A = R/I and a normal form R onto B, modulo I. Proof. Let us first prove that B is connected to 1 at each loop of the algorithm. To check this, remark that if B is connected to 1, it remains so when we increment it (step 9). Indeed a monomial added to B is in B + . Moreover if B is connected to 1, and if we remove from B some monomial ideals in the step 2, B remains so. Finally, as B is connected to 1 at the beginning of the algorithm, it remains so all along

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tion associated to the Degree Reverse Lexicographical order, dlex to the degree lexicographical order, M ac to Macaulay’s choice function(see 2.8), minsz to the choice function over the rational that minimize the memory needed in the reduction loop, and mix to the choice function that return either the result of minsz, or of grevlex applied to its input.

the computation, accordingly to the above statement. We show now that the algorithm stops : we prove that at some loop of index k, we have computed a representation of the quotient algebra for the monomials of size d. Let∗ d1 be the maximal size of a monomial appearing in Rkd (m) for all m of size at most d. Remark that step 8 and lemma 4.7 guaranty that all the C-polynomials whose C-leading term is of size less than d1 will simultaneously reduce to 0 for all the possible definitions of Rk at k ≤ kd∗1 (see definition 4.10). Hence, for any definition of Rk1 , any monomial m of size less than d − 2, and any variables xi , xj Rk1 (xj Rk1 (xi m)) = Rk1 (xi Rk1 (xj m)). Hence, by theorem 3.2, we deduce the following points: • We can extend the projection of ∂B k1 (restricted to monomials of size less than d) on B k1 along F k1  to the projection of the vector space spanned by the monomials of size less than d on B k1 along F k1 . • This extension coincide with Rk1 on their common set of definition. Now as I is zero dimensional, R/I is a K-vector space of finite dimension d0 . Suppose the algorithm does not stop before the loop ld0 +1 . Then in B ld0 +1 there is at least d0 + 1 monomials of size less than d0 + 1 since B ld0 +1 is connected to 1. Hence as dim(R/I) = d0 there is a linear dependence relation between these monomials in R/I. In other words there is a nonzero polynomial p = Σi=1..s qi fi ∈ I whose support is in B ld0 +1 and does not involve monomials of size greater than d0 + 1. Let d be the greatest size of the monomials appearing in supp(qi fi ). The above paragraph shows that there exists an index k such that Rk coincide with the projection from the vector space spanned by the monomials of size less or equal to d to B k along F k . Hence as all of the qi fi reduce to 0 by Rk and as Rk coincide with the projection on the vector space spanned by the monomials of size less than d, p will eventually reduce to 0. This means that there will be a polynomial of one of the Pl , l ≥ ld0 +1 whose leading monomial is one of the monomials of p, this l +1 contradicts the fact that supp(p) ⊂ Bdd00+1 . Therefore, the algorithm cannot go beyond loop ld0 +1 . Remark that at the loop k0 where the algorithm stops F k0 is a reducing family of all degree λ ∈ Λ for the set B k0 satisfying the property that ∀p ∈ F k0 , γ(p) = max{γ(m), m ∈ supp(p)} as γ refines the reducing graduation Λ. By step 8 (the C-polynomials reduce to 0), remark 3.7, and theorem ∗ 3.4, we extend Rk0 to the projection RF for all λ ∈ Λ, thus λ ∗ ∗ F∗ to the projection R of K[x] onto B along I : R|B . + = R ✷

5.

5.1 Generic equations The method we propose here is an extension of the Gr¨ obner bases computations. As such it can compute Gr¨ obner bases. The implementation we have is not as optimised as the Gr¨ obner bases ones that are being worked on for more or less ten years. An important work, mostly on linear algebra, remains to be done on our program. Hence we want to show that the method we propose here is competitive, and that it does not loose the good practical efficiency of Gr¨ obner bases computations. As the arithmetic used in our programs for doing exact computation is the rational arithmetic of gmp, which is much slower than integer computations used in the other softwares we will restrict ourself to the use of modular arithmetic. The family of examples we have chosen is the Katsura(n)2 equations. These equations are projective complete intersection with no zero at infinity. Using the Macaulay strategy, we know a priori what monomials will be leading monomials for the whole computation; so in this case we can guarantee that no test to 0 return erroneous result even using floating point arithmetic. We compare first our program to one of the best implementations available, Magma’s implementation of F4 algorithm [7]. n 7 9 10 11 12

Synaps grevlex 0.22s 3M 8.44s 5M 56.84s 13M 387.97s 45M 3072.08s 157M

Magma grevlex 0.05s 3M 1.670s 7M 13.50s 23M 96.76s 70M 1560.76s 240M

Let us mention that Gb, one of the reference implementation of Buchberger’s algorithm, spends 659s on Katsura(10). Numerically we observe that choosing the mac function also results in a better conditioning of the computations. More precisely on Katsura(6), and using a ceil of 10−10 we have: choice function grevlex dlex mac grevlex dlex mac grevlex dlex mac

EXPERIMENTATIONS

number of bits 128 128 128 80 80 80 64 64 64

time 1.98s 2.62s 1.64s 1.35s 3.98s 0.95s

0.9s

max(||fi ||∞ ) 10−28 10−24 10−30 10−20 10−15 10−19 − − 10−11

For the 64 bits computation the results computed for the grevlex and dlex orders are erroneous due to roundoff errors. The time given is the time spent in the computation of the multiplication matrices. Afterward, we used either LAPACK to perform the eigenvector computations or Maple when we needed extended precision. Because of the different nature of these tools, we do not report on the solving part timing. Finally we show here the amount of memory needed to perform the computations over Q, using GMP mpq.

The algorithm described in the previous section is implemented in the library Synaps1 (see Solve(L, newmac())). It corresponds to about 50 000 lines of C++-code. It involves a direct sparse matrix solver. The numerical approximation of the roots are obtained by eigenvalues computation, using the library Lapack (the routine dgegv) and the strategy described in [3]. The computations are performed on an AMD-Athlon 2400+ with 256M B of main memory. We show the results obtained with our implementation in the case where the graduation that we use for K[x] is the usual one. In the sequel, grevlex will refer to the choice func1

Synaps mac 0.19s 3M 6.17s 5M 32.39s 14M 252.05s 50M 1935.25s 191M

time size 2

http://www-sop.inria.fr/galaad/software/synaps/

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mac 4.22s 4.2M

minsz 30.21s 6.1M

grevlex 6.54s 4.4M

http://www-sop.inria.fr/galaad/data/

mix 7.83s 4.9M

5.2 Parallel robot

[6] I.Z. Emiris and B. Mourrain. Matrices in Elimination Theory. J. of Symbolic Computation, 28(1&2):3–44, 1999. [7] J.C. Faug`ere. A new efficient algorithm for computing Gr¨ obner Basis (F4). J. of Pure and Applied Algebra, 139:61–88, 1999. [8] J.C. Faug`ere, P. Gianni, D. Lazard, and T. Mora. Efficient computation of zero-dimensional groebner bases by change of ordering. J.S.C, 16(4):329–344, 1993. [9] Marc Giusti, Gr´egoire Lecerf, and Bruno Salvy. A Gr¨ obner free alternative for polynomial system solving. Journal of Complexity, 17(1):154–211, 2001. [10] A. Kehrein, M. Kreuzer, and L. Robbiano. An algebraist’s view on border bases. In A. Dickenstein and I. Emiris, editors, Solving Polynomial Equations: Foundations, Algorithms, and Applications., volume 14 of Algorithms and Computation in Mathematics, pages 169–202. Springer, 2005. [11] D. Lazard. Stewart platforms and gr¨ obner bases. In ARK’92, Proceedings of Advance in Robot Kinematik, Ferrare, Italia, September 1992. [12] F.S. Macaulay. The Algebraic Theory of Modular Systems. Cambridge Univ. Press, 1916. [13] H.M. M¨ oller and T. Sauer. H-bases for polynomial interpolation and system solving. multivariate polynomial interpolation. Advances Comput. Math., 12(4):335–362, 2000. [14] B. Mourrain. The 40 generic positions of a parallel robot. In M. Bronstein, editor, Proc. Intern. Symp. on Symbolic and Algebraic Computation, ACM press, pages 173–182, Kiev (Ukraine), July 1993. [15] B. Mourrain. Computing isolated polynomial roots by matrix methods. J. of Symbolic Computation, Special Issue on Symbolic-Numeric Algebra for Polynomials, 26(6):715–738, Dec. 1998. [16] B. Mourrain. A new criterion for normal form algorithms. In M. Fossorier, H. Imai, Shu Lin, and A. Poli, editors, Proc. AAECC, volume 1719 of LNCS, pages 430–443. Springer, Berlin, 1999. [17] B. Mourrain and Ph. Tr´ebuchet. Solving projective complete intersection faster. In C. Traverso, editor, Proc. Intern. Symp. on Symbolic and Algebraic Computation, pages 231–238. New-York, ACM Press., 2000. [18] B. Mourrain and Ph. Tr´ebuchet. Algebraic methods for numerical solving. In Proc. of the 3rd International Workshop on Symbolic and Numeric Algorithms for Scientific Computing’01 (Timisoara, Romania), pages 42–57, 2002. [19] B. Mourrain and Ph. Tr´ebuchet. Generalised normal forms and polynomial system solving. Technical Report 5471, INRIA Sophia-Antipolis, 2005. [20] F. Rouillier. Solving zero-dimensional polynomial systems throuhg Rational Univariate Representation. App. Alg. in Eng. Com. Comp., 9(5):433–461, 1999. [21] Hans J. Stetter. Numerical polynomial algebra. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2004.

Let us consider the famous direct kinematic problem of the parallel robot3 [14]. First we use floating point numbers to check to numerical requirements of the computations for different orders. For testing a number to be 0, we will use a leveling (here 10−8 is enough) and we will check afterward that the choices performed are the same as those done using modular arithmetic. This is equivalent to the use an hybrid arithmetic. choice function grevlex dlex mac

number of bits 128 128 128

max(||fi ||∞ ) 0.3 ∗ 10−24 0.3 ∗ 10−23 0.1 ∗ 10−24

time 2.07s 4.27s 2.22s

Here we see that choosing the right choice function can increase (but not so much in this case) the numerical accuracy of the roots. Hereafter we use the parameterization of [11] for solving, it involves more variables, gives better timings but less correct digits on the final result. number of bits 250 500

time 1.32s 2.23s

max(||fi ||∞ ) 10−63 10−140

Finally we performed tests using rational arithmetic. time size

mac 315s 17M

minsz 229.08s 14M

grevlex 201.65s 16M

mix 257.50 13M

In fact, it is not so surprising to see that the choice function γ has a big impact in terms of the computational time and of the memory required. We also mention here that over-constraining the system can result in a dramatic decrease of the computation time. Indeed expressing more constraints on the rotation than necessary gives additional quadratic equations that simplifies a lot the computations (see [19]).

6.

REFERENCES

[1] W. Auzinger and H. J. Stetter. An elimination algorithm for the computation of all zeros of a system of multivariate polynomial equations. In Proc. Intern. Conf. on Numerical Math., volume 86 of Int. Series of Numerical Math, pages 12–30. Birkh¨ auser Verlag, 1988. [2] L. Bus´e, M. Elkadi, and B. Mourrain. Using projection operators in computer aided geometric design. In Topics in Algebraic Geometry and Geometric Modeling,, pages 321–342. Contemporary Mathematics, 2003. [3] R.M. Corless, P.M. Gianni, and B.M. Trager. A reordered Schur factorization method for zero-dimensional polynomial systems with multiple roots. In W.W. K¨ uchlin, editor, Proc. ISSAC, pages 133–140, 1997. [4] D. Cox, J. Little, and D. O’Shea. Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra. Undergraduate Texts in Mathematics. Springer Verlag, New York, 1992. [5] D. Eisenbud. Commutative Algebra with a view toward Algebraic Geometry, volume 150 of Graduate Texts in Math. Berlin, Springer-Verlag, 1994. 3

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260

Domains and Expressions: An Interface Between Two Approaches to Computer Algebra Cosmin E. Oancea

Stephen M. Watt

Computer Science Department The University of Western Ontario London Ontario, Canada N6A 5B7

Computer Science Department The University of Western Ontario London Ontario, Canada N6A 5B7

[email protected]

[email protected]

ABSTRACT

enough for system developers, and otherwise it needs fixing. This has led to systems that either use modified scripting languages for their libraries (e.g. Maple), or that use modified library-building languages for their user interface (e.g. Axiom). A variant of this approach has been to build much of the mathematical support in a lower-level system implementation language, such as Lisp (e.g. with Macsyma) or C (e.g. with Mathematica). The result is that large parts of the current computer algebra systems are written in languages poorly adapted to the purpose, resulting in systems that are less flexible, less efficient and less reliable than we might wish. This paper examines the structure required for a different approach: to write libraries in a language well-adapted to large-scale computer algebra programming, together with an environment aimed at ease of use by the general end-user. It is not difficult to see that the style of programming for top-level problem solving and for libraries is quite different. For interactive problem solving, or for one-off scripts, it is important to be able to write commands quickly and succinctly. In this context, manipulation of some sort of general expression provides flexibility. On the other hand, to program large-scale computer algebra libraries, there are advantages to a language that allows efficient compilation, secure interfaces, and flexible code re-use. However, to achieve efficiency, safety and composibility requires more declarative structure. In this context, it is more natural to work with objects in precisely defined algebraic domains. Since libraries are used many times more than top-level scripts, programmers are more willing to provide this structure. Extensions to computer algebra systems are not alwasys calls to larger software components; they may equally well be collections of very fast light-weight routines. We therefore look beyond the solutions offered by loosely coupled computer algebra systems, e.g. OpenMath[7] or the software bus[8]. We choose Aldor [10] as a suitable library-building language, Maple [4] as a suitable interactive environment, and we require that Aldor libraries to be tightly coupled to Maple. That is, Aldor libraries will receive and directly operate on Maple objects in the same address space. Our solution consists of two parts: The first part allows the low-level run-time systems of Maple and Aldor to work together. It allows Aldor functions to call Maple functions and vice versa, and provides a protocol whereby the garbage collectors of the two systems can cooperate when structures

This paper describes a method to use compiled, strongly typed Aldor domains in the interpreted, expression-oriented Maple environment. This represents a non-traditional approach to structuring computer algebra software: using an efficient, compiled language, designed for writing large complex mathematical libraries, together with a top-level system based on user-interface priorities and ease of scripting. We examine what is required to use Aldor libraries to extend Maple in an effective and natural way. Since the computational models of Maple and Aldor differ significantly, new run-time code must implement a non-trivial semantic correspondence. Our solution allows Aldor functions to run tightly coupled to the Maple environment, able to directly and efficiently manipulate Maple data objects. We call the overall system Alma.

Categories and Subject Descriptors I.1.3 [Symbolic and Algebraic Manipulation]: Languages and Systems; D.2.12 [Software Engineering]: Interoperability; D.2.2 [Software Engineering]: Modules and interfaces, Software libraries; D.2.13 [Software Engineering]: Reusable software, Reusable libraries; D.2.11 [Software Engineering]: Languages—interconnection

General Terms Languages, Design, Performance

1.

INTRODUCTION

One of the positions held over the past two decades of mainstream computer algebra system design has been that there should be one over-arching language that serves both the end user and library developer. The idea has been if the language is good enough for end-users, it should be good

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. ISSAC’05, July 24–27, 2005, Beijing, China. Copyright 2005 ACM 1-59593-095-7/05/0007 ...$5.00.

261

work, together with an example of user-Alma interaction. Section 5 describes the Maple mapping, together with our type-checking mechanism. Section 6 describes the key ideas used in the Aldor and C mappings. Section 7 shows the implementation side of the example started in Section 2. Finally, Section 8 presents some conclusions.

span the two system heaps. As any low-level foreign function interface, it holds the user responsible for correct usage. This work has been reported elsewhere [9]. The second part of our solution, reported here, implements a high-level correspondence between Maple and Aldor concepts. The aim has been to bridge the semantic differences between the two environments, to allow Aldor domains to appear to the user as Maple modules, and Maple modules to appear as Aldor domains. While our semantic correspondence works both ways, in practice we are primarily interested in using Aldor libraries in the Maple environment. We use a tool to generate Aldor, C and Maple code that wraps the Aldor library exports, as well as supporting run-time support code to do dispatch and caching. The resulting package, which we call Alma, allows standard Aldor libraries to be used in a standard Maple environment [4]. More precisely, Alma can be seen as a software component architecture to achieve connectivity among two computer algebra systems. It gracefully handles usererrors (type-checking), supports reflective features to describe components’ types and functionalities, provides a useroriented interface (Maple “look and feel”), and employs highlevel optimizations. Thus, our approach is more challenging and quite different than previous work on low-level foreign function interfaces, and consequently the internal architecture of the proposed framework is more complex. We present two validations of this architecture: First, we describe the mappings of the Aldor language features to Maple, and the Alma type-checking process (Section 5). Second, we present a comprehensive example, in which approximatively 1160 Aldor exports have been made available to the Maple user (Sections 2 and 7). Earlier results leading to this approach have been reported in [3] [6].

2. EXAMPLE This section presents an example where a Maple user employs the functionality of the Aldor BasicMath library to solve a mathematical problem in a way not supported natively in Maple. The BasicMath library was developed at NAG by Moreno Maza and others as part of the FRISCO project, and provides Aldor with a set of types and algorithms for computer algebra. It is an extensive library, comprising about 103700 lines of Aldor code. read "mtestgcd-wrap.mpl": # Construct polys af1 := MapleToAldorPoly(x*y^2 - 4*y + 5*x): af2 := MapleToAldorPoly(6*x*y - y^2 + 5): am := MapleToAldorPoly(x^2 + 1): # Form triangular set and gcd by Aldor package. trset := TriPack:-empty(): rchain:= TriPack:-regularChain(am, trset): ggcd := TriPack:-regularGcd(af1, af2, rchain): ggcd := genstep(ggcd): ggcd := TriPack:-reducedForm(ggcd, rchain): # Get the GCD as a Maple expression. AldorToMaplePoly(ggcd); y - x

We see the following as contributions of this work: • Aldor has been found to offer efficiencies comparable to hand-coded C++ [1]. Our approach therefore allows extension libraries to operate with efficiencies comparable to Maple kernel routines. • These extensions are written in a high-level language, well-adapted for mathematical software. It allows the programmer to ignore lower-level details and have natural integration of dynamic components into the Maple environment. • Aldor is designed for mathematical “programming in the large” and provides linguistic support for such concepts as generic algorithms, algebraic interface specification and enforcement, dynamic instantiation, etc. Our approach allows the Maple system to benefit from these features. Alternatives, such as C++, do not provide this. • Authors of large Aldor libraries often wish to make their functionality available through a main-stream computer algebra system. Two examples are Bronstein’s library for differential operators, Sumit [2], and Moreno Maza’s library for triangular sets, Triade [5]. The current work makes this relatively easy.

Figure 1: A Maple session computing a GCD in (R[x]/Sat(mx ))[z, y] using the Alma framework The Maple session presented in Figure 1 shows the Alma interface. The example computes the greatest common divisor of two polynomials in (R[x]/Sat(x2 + 1))[z, y] by invoking the Aldor BasicMath library and using its support for regular chains. The session uses the file mtestgcd-wrap.mpl to act as a wrapper between the Alma system and the user. The implementation of this file is explained in Section 7, after we have described the necessary concepts. The example first creates the Alma objects corresponding to the given Maple polynomials. The regular chain containing the polynomial m is constructed, and the greatest common divisor of f 1 and f 2 with respect to the regular chain is computed. Finally, the reduced form of ggcd is computed, and it is converted to a Maple polynomial. The functions empty, regularChain and regularGcd have the interfaces exactly as exported by the Aldor library. TriPack is the instantiation of an automatically generated Maple module wrapper corresponding to the Aldor package RegularTriangularSet.

The remainder of this paper is organized as follows: We start with an example in Section 2: we show a Maple session computing the polynomial GCD over a tower of algebraic extensions using Aldor’s BasicMath library. Section 3 briefly introduces the aspects of the Maple and Aldor programming environments needed to understand Alma. Section 4 presents a high level architectural view of the Alma frame-

3. ASPECTS OF MAPLE AND ALDOR This section briefly presents some of the aspects of Aldor and Maple systems that we used in our architectural design. Maple uses a dynamically typed language that supports first class functions. Typically, functions use dynamic type tests to implement polymorphism, and name overloading is

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not supported. Modern versions of Maple have adopted the concept of modules to organize packages and libraries. A module is a first-class Maple object and provides a collection of name bindings. Some of these bindings are accessible to Maple code outside the module, after the module has been constructed; these are the exports of the module [4]. Figure 2 shows a Maple module and its use.

-- File Example.as: import from SingleInteger; define Module(R:Ring) : Category == AbelianGroup with { *: (R, %) -> %; ++ Scalar multiplication coerce: R -> %; coerce: String -> %; } ++ Polynomial domain over ring R Polynomial(R: Ring) : Module(R) == add { (r: R) * (x: %) : % == ...; coerce(r: R) : % == ...; coerce(s: String) : % == ...; ... }

makeZp := proc( p ) module() export plus; plus := (a,b) -> a + b mod p; end module: end proc: z5 := makeZp(5); # create the module z5:-plus(2,4); # add 2 and 4 mod 5.

Figure 3: An Aldor Category/Domain Example

Figure 2: A Maple Module and Its Use As they are first class objects, modules can be returned by functions. A module’s exported functions can reference environment variables visible at the moment of their creation (i.e. it is a closure). In Figure 2 the module returned by the makeZp function references makeZp’s parameter p. It exports the plus operation whose functionality is to add numbers modulo p.

4. ALMA DESIGN This presents an overview of Alma’s design. The main ideas that guided the design are summarized below: • Alma should automatically generate any needed (Maple, C, and Aldor) stubs, and keep the system’s internals hidden from the user. • Alma should provide a dynamic (interactive) type-checking mechanism that gracefully handles user needs, and errors. • Alma should allow Maple to interact with Aldor components in an efficient manner, introducing only a minimal overhead cost. • Alma should extend the Maple language only as needed, by providing mappings for foreign programming language concepts such as overloading, domains, etc. • Alma should be simple to use, rendering a Maple “look and feel” to Aldor code.

Aldor is a strongly typed functional programming language with a higher order type system. The type system has two levels: domains and categories. Each value belongs to some unique type, known as its domain. Domains are in principle run-time values, but they belong to type categories which can be determined statically. Categories specify properties of domains, such as which operations they export, and are used to specify interfaces and inheritance hierarchies. The main difference between the two-level domain/category model and the single-level subclass/class model of objectoriented programming is that a domain is an element of a category, whereas a subclass is a subset of a class. This difference eliminates a number of deep problems in the definition of functions with multiple related arguments. Dependent products and mapping types are fully supported in Aldor. Generic programming is achieved through explicit parametric polymorphism, using functions which take types as parameters and which operate on values of those types, eg: f(R:Ring,a:R,b:R):R == a*b - b*a. In Aldor, within a domain-valued expression, the name % refers to the domain being computed, is fixed-pointed, and can be used as a type name. An example of an Aldor program is presented in Figure 3. It defines a parametrized category Module(R), representing a simplified version of the mathematical category of R-Modules. Module(R) declares as exports a scalar multiplication and two conversion operations. Polynomial has the dependent mapping type: (R: Ring)-> Module(R), taking one parameter R, which is a domain satisfying the Ring category, and produces a type beloning to the category of R-modules. Static analysis can use the fact that R provides all the operations required by Ring, thus allowing static resolution of names and separate compilation of parameterized modules. Names can be overloaded, and are resolved based on their static type. The first line in the Aldor code in Figure 3 makes the exports of the SingleInteger domain available throughout the file.

4.1 Rationale of the Design Figure 4 introduces the main components of the Alma architecture. The module that does the stub code generation is located inside the Aldor compiler. It receives as input an Aldor program, and generates the usual compiled binary representation of it, together with Aldor, C, and Maple stubs for the program’s exports. Among these there may be exports that have their definition in some Aldor library. The Maple stub becomes the interface between the user and the Alma system. It uses the functionality of the type checking module, in order to ensure a correct call to the Aldor library. Otherwise, if no type-checking is performed, an incorrect call on the user’s part would most likely produce a low-level fatal error. The type-checking module is designed to provide useful feedback to the user in the case of an erroneous invocation. For example it will list the allowed export types for a given export name. Once the program has reached a mature phase, one may want to eliminate the type-checking overhead. If a fast implementation is desired, the Alma code generation module is able to produce code in which no type checking is performed. The type checking module is implemented mostly in Maple, but it also uses Aldor run-time system enhancements (the “has” operation that tests if a given domain satisfies a given category). Our architecture allows Maple to share a single address

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Aldor Program

4.2 Example of Correspondence

Maple Environment

We present a simple interaction between the Maple-user and the Alma framework. Assume the user wants to use the functionality of the Aldor code in Figure 3, and the stubs have already been generated by calling the Aldor compiler with the appropriate options on the Example.as file.

Maple Program ALDOR COMPILER

Maple Stub

ALMA module

Aldor

Aldor

C

Binary

Stub

Stub

ALMA

Caching

JITC

Type Checking

Maple

>read("MapleExampleStub.mpl"): # line 1 >with(Example); # line 2 module() export Polynomial, ... end module >Polynomial("help"); # line 3 Domain Type: Polynomial(R: Ring) : Module(R) Exports:*:(R,%)->%;coerce:(R)->%;coerce:(String)->%; Comment: Polynomial domain over ring R >Polynomial("help", "*"); # line 4 Functional Type: *: (R, %) -> %; Comment: Scalar multiplication >SI_dom := SingleInteger:-Info:-asForeign; # line 5 ["d", 1856856, module() export ...] >int_obj := Alma:-AldorInt(5); # line 6 ["o", 5, module() export ... ] >poly_si_dom := Polynomial(SI_dom); # line 7 ["d", 1848300, module export ...] >poly_obj := poly_si_dom:-coerce(int_obj); # line 8 module() export ... end module >wrong_obj:=Polynomial(SI_dom):-coerce(SI_dom);#line 9 no function with this signature! candidates: coerce:(SingleInteger)->Polynomial(SingleInteger) coerce:(String) -> Polynomial(SingleInteger)

Library

Aldor Library

Aldor Run Time System

A+

M+

Maple Run Time System

BINARIES

Figure 4: High-Level Architecture Overview: white arrows mean “generates,” normal arrows mean “uses,” dashed boxes are user source code, light boxes are generated code.

space with optimized Aldor components from a library. However, the cost of calling an Aldor function can be somewhat expensive: The initial use of the Aldor stub may require a number of expensive runtime operations, such as domain instantiation, that cannot be statically optimized by the Aldor compiler. (These operations may involve parameters that are known only at runtime.) Thus, for performance reasons, the Maple stub uses a cache module (implemented based on Maple’s remember option) to store previously computed domain/category types. Aldor-closure objects corresponding to functional Aldor exports are also cached, so that they can be invoked directly, thus by-passing the Aldor stub. Alma supports function-level just in time re-compilation of the C and Aldor stubs (JITC module). More precisely, if an export is found to be “hot” and depends on type-parameters known only at run-time, Alma will build, and recompile a specialized C and Aldor stub corresponding to that export. Since most type-parameters are now instantiated, the Aldor compiler will find better opportunities for aggressive optimizations (like inlining), thus improving the application’s performance. In order to successfully complete a foreign Aldor invocation, the Maple stub calls the C stub that forwards the request to the Aldor stub, and this invokes the correct Aldor export on valid Aldor parameters, returning a value to the C stub. The C stub creates foreign Maple objects and returns them to the Maple stub. The functionality of the Aldor and Maple run-time system enhancements modules (A+ and M+ in the figure) is to synchronize Maple’s and Aldor’s garbage collectors (see [9]). Alma’s objects expose rich reflective features that can be queried by the user. This allows one to find the functionality of the corresponding Aldor component, its type, etc. Alma’s foreign objects can also be manipulated in the same way as any ordinary Maple objects: They may be used in Maple operations (such as map, apply), while Alma’s internal invocation mechanism is completely transparent to the user.

Figure 5: User-Alma interaction. Lines starting “>” are user input; the others are Maple output The first line in Figure 5 imports the Maple stub into Maple’s environment. On line 3, the user asks for information about the Polynomial domain. Alma answers by providing the type information, exports, and the comments associated with the Polynomial domain (see Figure 3). Similarly, on line 4, the user asks about the * export of the Polynomial domain. All Aldor domains/categories are translated into Maple modules, or functions producing modules, if parameterized. They export an Info module that encapsulates the type’s reflective features. The asForeign export of the Info module stores a Maple foreign object corresponding to the Aldor domain it represents. At present, our implementation represents a foreign Maple object as a list that contains a classification identifier (“d” means domain, “c” means category, “f” means function, “o” means object, etc.), a pointer to the Aldor object (for primitive types this will be the value), a Maple structure representing the Aldor type, and some additional information used to synchronize the garbage collectors This is illustrated by Alma’s response to the user command at lines 5, 6, 7. Line 5 creates a foreign domain type object corresponding to the Aldor SingleInteger domain. Line 6 creates an object of type SingleInteger which in fact is just a primitive integer value, as one can see in the Maple representation of the int obj. Next, on line 7, another domain-type object is created, corresponding to the Polynomial(SingleInteger) Aldor type. If the user would like to verify first that the SingleInteger domain satisfies the Ring category, he can look in the SingleInteger:-Info:-supertypes export. Note that the interaction with our framework is quite intuitive, as our mapping closely follows Aldor’s specification structure and semantics. Types are run-time values both in Aldor and in our mappings: the user has to construct them first in order to use their exports. Types are also first class val-

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ues, therefore they are constructed and used in the same way a regular object is used. Finally, on line 8, the coerce function is called, and as result, a foreign Maple object of Polynomial(SingleInteger) Aldor type is returned. The last line in our example (line 9) shows how our framework reacts to an erroneous input: The type checking module detects that the parameter to the coerce export is neither of type SingleInteger nor of type String, so the incorrect Aldor library invocation is aborted. In addition, feedback is provided to the user with respect to the valid type signatures of the coerce function. Also note that while Maple does not support overloading, our mapping behaves as though it does. To the user it seems as though one can call two functions with the same name and with different parameter types, as they appear in the Aldor specification.

5.

1 ‘Polynomial‘ := proc() option remember; ## type cache 2 local ret,tmp_fct,b,ret_param,ALMA_getObject,args4; 3 if(args[1]="help") then ... return; fi; 4 args4 := args; 5 if nargs=1 then 6 b := true; 7 if b then b:=type(args[1],TC(Ring()));fi; 8 if b then ## TYPE: Module(R:Ring) 9 ret := module() 10 export ‘*‘, Info, fcts; 11 Info := module() ##metadata:reflective+profiling 12 export GenExports,GenInfo,hash,self,asForeign, 13 type,asForeign,supertypes,printExports, 14 domArgs,domArgsOpt,optimizeOn,profile; 15 GenInfo:=["Polynomial",[["Ring"]], 16 ["Apply","Module",[args4[1]]]]; 17 GenExports:=[["*",[args4[1],"%"],["%"]], ...]; 18 domArgs:=args4; domArgsOpt:=[]; 19 optimizeOn:=[0]; profile:=[[0]]; 20 end module; 21 fcts := module() 22 export ‘*‘, ‘*clos‘, ‘coerce‘, ‘coerceclos‘; 23 local ‘*cstubname‘;‘*cstubname‘:="starFrMyPolyT"; 24 ‘*clos‘:=proc(arg)option remember;#closure cache 25 local tmp_fct, ret; 26 tmp_fct:=define_external(convert(‘*cstubname‘, 27 symbol),’MAPLE’, ’LIB="./libctestJIT.so"): 28 ret:=tmp_fct(Alma_map(lst->lst[2], [op(arg)]), 29 ["f",[args4[1,3],Info:-self],[Info:-self]]); 30 return ret; 31 end proc; 32 ‘*‘ := proc() 33 local ret, cached_clos, b, ret_param; 34 if nargs=2 then 35 b := true; 36 if b then b:=type(args[1],TC(args4[1,3]));fi; 37 if b then b:=type(args[2],TC(Info:-self));fi; 38 if b then 39 if (Info:-optimizeOn[1]=1) then 40 cached_clos:=‘*clos‘(domArgsOpt); 41 else cached_clos:=‘*clos‘(domArgs); fi; 42 Info:-profile[1,1]:=Info:-profile[1,1]+1; 43 if(Info:-profile[1,1]=Alma:-JITtreshold) then 44 ‘*cstubname‘:="starFrMyPolySpec"; 45 Info:-optimizeOn[1]:=1; 45 OptimizeAldor(Info:-self, 1); fi; 46 ret:=callAldorClosure(cached_clos, 47 map(lst->lst[2],[args]),[Info:-self]); 48 return ret; 49 fi; fi; 50 print("Context: Polynomial(R:Ring);"); 51 print("Candidates: *(R,%)->(%)"); 52 error "No function with this signature"; 53 end proc; ... end module; 54 ‘*‘:=fcts:-‘*‘; 55 end module; 56 ret:-Info:-self:=ret; 57 Alma_getObject:=proc() local tmp_fct, ret1; 58 tmp_fct:=define_external(’cPolynomialOfT’, 59 ’MAPLE’, ’LIB’="./libctestJIT.so"); 60 ret1:=tmp_fct(map(lst->lst[2],[op(args4)]), 61 [ret:-Info:-self]); return ret1; 62 end proc; 63 ret:-Info:-asForeign:=‘Alma_getObject‘(); 64 ret:-Info:-type:=Module(args[1]); 65 ret:-Info:-supertypes:=[Type]; return ret; 66 fi; fi; 67 print("Context:"); 68 print("Candidate:Polynomial:(R:Ring())->Module(R)"); 69 error "No function with this signature"; 70end proc;

THE MAPLE STUB

We now turn our attention to the internals of the system, starting with the generated Maple stubs. The Maple mapping addresses the issues that arise from matching the Aldor’s strongly typed system with Maple’s dynamically typed system. In particular, one of the challenges is in matching the compile-time parametric polymorphism of Aldor’s dependent types with the dynamic polymorphism of Maple’s module-producing functions. For a rich connectivity between Maple and Aldor to exist, Aldor’s features, such as run-time domain types, overloading, dependent types and mapping types, need to be mapped to Maple. The key for the translation of these features is to create, via the Maple stub, dynamic types corresponding to the hierarchy of available Aldor types, and to design a dynamic type-checking mechanism for the foreign Maple objects. Alma’s type-checking phase is greatly simplified, compared to the static Aldor type-checking, as it happens at the application’s run-time where most parameters have completely instantiated types.

5.1 Mapping Rules The code in Figure 6, is an extract of the Maple stub corresponding to the Aldor Polynomial domain defined in Figure 3. We use this to help present the high-level concepts ideas used to interface Maple with Aldor. For space reasons, we have excluded the code for the coerce exports, the “help” option, or some of the exports of the Info module. An Aldor domain-producing function (e.g., Polynomial) is translated into a Maple function which at run-time yields a module. In addition it encapsulates the necessary information for type-checking its parameters and exports. This is done lines 7, 36, and 37 in Figure 6; type/TC is the Alma type-checker that ensures the consistency of the mapped Maple code with the Aldor type system. Aldor’s nested domains are mapped into nested Maple modules. The rest of the Aldor domain exports are mapped to Maple module exports. Name overloading in our mapping is achieved by concatenating the different implementations for the same name and using a single function in which dynamic type tests identify the right code to be executed. Modules corresponding to Aldor’s domains and categories export an Info module containing metadata (reflective features and profiling information) associated with that type. These standardized exports of the Info module are computed at the domain/category-type module creation time. Our mapping exploits Maple’s support for closures. Each

Figure 6: Part of MapleExampleStub.mpl

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of the generated functions that produces a type will set a variable with a unique name to point to its parameters list, thus guaranteeing access to its parameters from a function declared in a nested scope. Line 36 and 37 in Figure 6 typecheck the *: (r:R, x:%)->% export of the Polynomial: (R:Ring)->Module(R) parameterized domain. Notice that here R is a type variable, as it is given as a parameter to the Polynomial domain, and is used as a type in its implementation. R can be accessed by means of the args4 variable in the Polynomial’s function outer scope. The type(args[1],TC(args4[1,3])) call invokes the Alma typechecker (type/TC) to verify if r is of type R. This uses the representation knowledge that the third element in Aldor’s domain foreign object representation is the Maple module object that maps the corresponding Aldor domain). Both r and R are known only at run-time, and are accessible through the closure’s environment. Lines 32-53 in Figure 6 show the implementation of the * export of the Polynomial domain. Lines 34-37 verify that the number and type of the parameters are consistent with the Aldor definition. If the optimizeOn entry associated with the * export is set, then this export has already been type specialized and re-compiled (see Section 6). In this case, the *clos function is invoked on the non-inlined parameters (domArgsOpt, line 40). Otherwise it is invoked on all domain parameters (domArgs, line 41). The *clos function returns an Alma closure-object corresponding to the Aldor * export. This is invoked with valid parameters on line 46 by means of the callAldorClosure Alma’s systemfunction. Its first parameter is the Alma closure object, the second is a list of Aldor valid arguments, while the third is the Alma type of the result. If the profiled information corresponding to the * export shows that it is advantageous to JIT recompile the C/Aldor stub (line 43), the Alma systemfunction OptimizeAldor is called (line 45). In order to return the Alma closure-object corresponding to the Aldor * export, the *clos function requires access to the C stub via the Maple’s define external function. The call to define external links in an externally defined function, and produces a Maple procedure that acts as an interface to this external function [4]; the tmp fct, computed on line 26 of Figure 6, is such an interface. The first parameter of the tmp fct is a list of Aldor parameters on which the Aldor stub is to be invoked, while the second argument (["f",[args4[1,3],Info:-self],[Info:-self]]) is the Alma type of the closure object (* receives two parameters: one of type R – where R is a type-parameter of the Polynomial domain, and another one of type %, yielding an object of type %). We note that, the type/closure cache of the Alma system is easily implemented with Maple’s option remember (lines 1 and 24).

(domain-type object), a (object of type SingleInteger), and fun (function), which have all been defined in Figure 7. For objects that do not correspond to domain/category Aldor types, the third element of Alma’s foreign object layout (rows 3 and 4, column 2) is their Alma type. The layout of the domain/category-type objects does not include their types, but rather themselves (see rows 1 and 2 in Figure 8). For example MyCat(SI) is the Alma module-type associated with the Aldor MyCat(SingleInteger) category. This is because their Alma type is readily accessible by means of their reflective features (the Info:-type export). -- any Aldor domain satisfies Type -- any Aldor category type satisfies Category define MyCat(T:Type):Category == with; --Type: MyDom(T:Type):MyCat(T) == add; --Type: fun(A:Type, o:A, obj:MyDom(A)): A == o;--Type: SI == SingleInteger; a: SI := 3::SI; --Type:

Category Domain Function Object

Figure 7: Aldor Specification Row 4 in Figure 8 shows the Alma type corresponding to the fun function. It is composed from a classification identifier “f”, a list containing the Alma types of its parameters and another list containing the Alma types of its returns. A list whose first argument is the “l” tag (link) indicates that the parameter’s type is itself passed as a parameter to this function, the remaining list’s arguments giving the index in the current type where the type-parameter was introduced. The "a" tag stands for “apply the second argument of the set to the rest of the set’s arguments.” It is used only if the type expression involves a type-parameter that has not yet been computed (thus the type of the “apply” cannot be computed yet in this case).

5.3 Type Checking Let us now consider Alma’s type-checking mechanism. In Aldor, every value is a member of a unique domain that determines the interpretation of its data. For the current version of the language, only the domain of all domains, and the domain of all functions produce non-trivial subtype lattices [10][11]. This means that user-developed domains cannot create subtypes, the only non-trivial sub-typing lattices for our type system are the lattices of categories and functions; a non-function object is of a unique type, and cannot satisfy any other type. To type-check that a foreign Maple object o is of Alma type d (i.e. a Maple module corresponding to an Aldor domain type), the Aldor type of o (found through the foreign object layout), and the Aldor object representation of d are compared, either directly or by hash codees. To verify that an Alma domain-object belongs to a certain category type, the Aldor run-time system is invoked (“has” operation) via Alma’s Aldor stub.

5.2 Foreign Object Layout Aldor Expr. MyCat(SI) MyDom(SI) a fun

We now briefly describe the Alma foreign object layout. Where necessary we shall provide details on Aldor’s type system semantics. In Aldor, types and functions are first class values. Therefore, besides “regular” objects, we have to define proper formats for foreign Alma type/closure objects, and to design proper Maple types for them. Figure 8 shows the foreign object layout for the Aldor expressions: MyCat(SI) (category-type object), MyDom(SI)

Associated Alma Foreign Object Layout ["c", ptr to obj, MyCat(SI)] ["d", ptr to obj, MyDom(SI)] ["o", 3, SI] ["f",ptr to clos,f tp] where l1:=["l",1], f tp:=["f",[Type,l1,["a",MyDom,[l1]]],[l1]];

Figure 8: Foreign Object Layout. The Aldor expressions in the first column are defined in Figure 7

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astarFrPolynomialT(T:Ring) : (Ring,Polynomial(T)) -> Polynomial(T) == { _*$Polynomial(T); -- (***) }

To test that a foreign Maple functional object S1->T1 is of a functional type S2->T2, one has to verify that S2 is a subtype of S1, and T1 is a subtype of T2. When testing this, a run-time unification algorithm is used, which computes and works with the fix-point representation of a type. Otherwise for mutually recursive types the algorithm will never end.

6.

-- Code generated by the JIT re-compilation module -SI == SingleInteger; astarFrPolynomialSpec() : (SI,Polynomial(SI)) -> Polynomial(SI) == { _*$Polynomial(SI); -- this can be aggresively optimized -}

THE C AND ALDOR STUBS

The role of the Aldor and C stub is to re-direct the user’s call to the Aldor library. These are not necessarily accessible to the user, and do not resemble the structure of the mapped Aldor specification. The C and Aldor mappings, if not used inside our framework form un-safe code, as they assume that the type-checking has already been performed at the Maple stub level. The C stub is the glue between the Maple and Aldor stubs, as both languages expose a basic interoperability layer with C. The C stub for the Aldor export *:(R, %)->% is presented in Figure 9. When invoked, the C stub (starFrPolynomialT) identifies the Aldor objects to be passed as parameters to the Aldor stub (list args), and calls the Aldor stub (represented by astarFrPolynomialT) on these arguments (casted to void pointers). The resulting Aldor object (ret) is combined with its Alma type (ret type, also received as a parameter by the C stub) to form an Alma foreign closureobject that is returned to the Maple stub. This is accomplished through the use of the Alma system function makeForeignObject. The created closure-object, when called (via callAldorClosure Alma system-function), uses the Aldor-C interoperability layer for executing a closure call (CCall). The C stub generated by the JIT recompilation module looks very much like the standard one. The only difference is that it invokes a different Aldor function (astarFrPolynomialSpec) which takes fewer parameters. In our case it takes no parameters as the SingleInteger parameter of the Polynomial has been already inlined in the * Aldor stub export generated by the JIT re-compilation module.

Figure 10: Aldor Stub Mapping ric polymorphism of the Aldor specification/library to the Maple user who can now instantiate Aldor types at application’s run-time and call their exports. A domain functional export is represented as a function that takes as parameters all the parameters of the domains in which it is nested (starting with the uppermost one), and that returns the desired closure to the C stub. All the other exports shall return an object (for example domain/category types). As can be seen, the Aldor stub is quite simple. We employ the type inference mechanism to do the difficult work of identifying which of the possible overloaded * functions we return. The Aldor compiler will identify more opportunities to aggressively optimize the astarFrPolynomialSpec Aldor stub export (generated by JIT re-compilation) – for example it can inline all the SI operations (+, -, *) that appear in the * export of the Polynomial(SI) domain.

7. EXAMPLE IMPLEMENTATION We now show the details a library author must be aware of to use Alma. This completes the example of Section 2, which showed only the end-user’s point of view. -- File testgcd.as: #include "basicmath" N == NonNegativeInteger; R == Integer; lv: List Symbol == [+"z",+"y",+"x"]; V == OrderedVariableList(lv); Q == Fraction(Integer); P == SparseMultivariatePolynomial(Q, V); gcdPack==GcdOverTowersOfAlgebraicExtensionsPackage(lv); T == RegularTriangularSet(Q, lv); VT == ValueWithRegularChain(P, T); BB == Boolean; SI == SingleInteger;

/************ C stub for *$Polynomial(T) ************/ extern FiClos astarFrPolynomialT(void* D); ALGEB starFrPolynomialT(MKernelVector kv, ALGEB args){ ALGEB list_args, ret_type, result; FiClos ret; list_args = (ALGEB)args[1]; ret_type = (ALGEB)args[2]; ret = astarFrPolynomialT((void*)MapleToInteger32(kv, MapleListSelect(kv,list_args,1))); result = makeForeignObject(kv,"f",ret,ret_type); return result; }

Figure 11: Aldor specification used as input to the Alma framework Figure 11 shows the Aldor specification that must be provided as input to the Alma compiler to make available part of BasicMath’s exports to the Maple environment. The compilation generates Maple, C, and Aldor stubs that each have about 1160 exports. It is, in our opinion, easy to see why a naive, non-automatic integration of this library in the Maple environment is not a practical solution: It requires a good deal of effort, not to mention the maintenance cost. If the exports of the library are changed, the Maple mapping must be altered as well. Figure 12 presents the hand-written wrapping code that will create the necessary Alma types and functions and will ease the use of the Alma system. This is not, strictly speaking, necessary and could be done by the end user. However it is likely that it needs to be created only once, and may be

/****** C stub for *$Polynomial(SingleInteger) ******/ /** Code generated by the JIT re-compilation module **/ extern FiClos astarFrPolynomialSpec(); ALGEB starFrPolynomialSpec(MKernelVector kv,ALGEB args) { ALGEB ret_type, result; FiClos ret; ret_type = (ALGEB)args[1]; ret = astarFrPolynomialSpec(); result = makeForeignObject(kv,"f",ret,ret_type); return result; }

Figure 9: C Stub Mapping Figure 10 illustrates the main ideas employed in the Aldor stub generation. The Aldor stub exposes the paramet-

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used in different programs. We underscore that the Maple and Aldor generated stubs are generic and can be instantiated over various types. The Alma user may also work with SparseMultivariatePolynomial(R,V), not only with the P defined in Figure 11.

Maple. New programs can work naturally with both Maple and Aldor native objects, while the Aldor compiler enforces mathematical interface requirements and generates efficient code. Most importantly, we have re-examined one of the most basic assumptions of modern computer algebra system design: that algebra code should be written either in the toplevel user language or in the low-level systems implementation language. We believe that we have demonstrated that top-level problem solving and library development can successfully use different mathematical programming languages.

# Import from generated Aldor file. read "mtestgcd.mpl": with(testgcd): # Problem-independent abbreviations. STR := String: CHAR := Character: S := Symbol: # Wrapper for this package. TriPack := RegularTriangularSet(Q:-Info:-asForeign,lv): GenP := Generator(P:-Info:-asForeign): GenCHAR := Generator(CHAR:-Info:-asForeign): GenVT := Generator(VT:-Info:-asForeign):

9. REFERENCES [1] L. Bernardin, B. Char, and E. Kaltofen. Symbolic computation in java: An appraisement. In Proc. ISSAC 1999, pages 237–244. ACM, 1999. [2] M. Bronstein. Sum-it: A strongly-typed embeddable computer algebra library. In Proceedings of DISCO’96, Karlsruhe. Springer LNCS 1128, 1996. [3] Y. Chicha, M. Lloyd, C. Oancea, and S. M. Watt. Parametric polymorphism for computer algebra software components. In Proc. 6th International Symposium on Symbolic and Numeric Algorithms for Scientific Comput., pages 119–130. Mirton Publishing House, 2004. [4] Maple User Manual. Maplesoft, a division of Waterloo Maple Inc., 2005. [5] M. Moreno Maza. Technical report TR 4/99, on triangular decompositions of algebraic varieties. Technical report, NAG Ltd, Oxford, UK, 1999. [6] C. Oancea and S. M. Watt. A Framework for Using Aldor Libraries with Maple. In Actas de los Encuentros de Algebra Computacional y Aplicaciones, pages 219–224, 2004. [7] Special issue on OpenMath. ACM SIGSAM Bulletin, 34(2), June 2000. [8] J. Purtilo. Applications of a software interconnection system in mathematical problem solving environments. In Symposium on Symbolic and Algebraic Manipulation (SYMSAC 86), pages 16–23. ACM, 1986. [9] S. M. Watt. A study in the integration of computer algebra systems: Memory management in a Maple-Aldor environment. In Proc. International Congress of Mathematical Software, pages 405–411, 2002. [10] S. M. Watt. Aldor. In J. Grabmeier, E. Kaltofen, and V. Weispfenning, editors, Handbook of Computer Algebra, pages 154–160, 2003. [11] S. M. Watt, P. A. Broadbery, S. S. Dooley, P. Iglio, S. C. Morrison, J. M. Steinbach, and R. S. Sutor. AXIOM Library Compiler User Guide. Numerical Algorithms Group (ISBN 1-85206-106-5), 1994.

MapleToAldorPoly := almaPolyToAldor(P,SI,N,R,Q,S,CHAR,STR): AldorToMaplePoly := almaPolyToMaple(P,GenP,Q,N,R,S,STR,GenCHAR): # Utility function. genstep := proc(ggcd0) local ggcd, vgcd, str; ggcd := GenVT:-‘step!‘(ggcd0): vgcd := GenVT:-value(ggcd): str := AlmaNewString(SI,CHAR,STR)("val"): VT:-apply(vgcd, str): end:

Figure 12: Maple wrapper used in Figure 1 The code in Figure 12 constructs various Alma types and constants corresponding to BasicMath types/constants which will be needed in the computations already presented in Figure 1. Note that Alma has also generated Maple exports corresponding to the Aldor constants lv, N, R, Q, T, ... (Figure 11), and these can now be directly manipulated in the Maple file. The utility function (genstep) receives as parameter a generator object containing (gcdi , toweri ) pairs, obtained by calling the TriPack:-regularGcd function, and returns gcd1 . The functions almaPolyToAldor and almaPolyToMaple are Alma system components that return to the user Maple to Aldor and Aldor to Maple polynomial conversion closures.

8.

CONCLUSIONS

We have described an approach to using efficient, externally defined, high-level mathematical libraries within Maple. These can extend Maple in an effective and natural way. Our implementation allows Aldor domains to appear as Maple modules, and allows Aldor programs unfettered direct access to Maple objects. This allows very efficient interaction between the two environments. At this point Alma is most useful in two settings: The first setting is to allow kernel-like efficiency in core mathematical extensions of Maple. The difference between Alma and using C code via Maple’s foreign function interface is that it is possible to work at a high mathematical conceptual level and not worry about details such as garbage collection. The second setting is to allow complex Aldor packages to be used naturally from Maple. These packages typically have their own internal representation for the mathematical objects they manipulate. We forsee a third setting where Alma will be used: as an alternative for writing new libraries for

268

Symbolic-Numeric Completion of Differential Systems by Homotopy Continuation Greg Reid∗

Jan Verschelde†

Allan Wittkopf‡

Wenyuan Wu§

ABSTRACT

1.

Two ideas are combined to construct a hybrid symbolicnumeric differential-elimination method for identifying and including missing constraints arising in differential systems. First we exploit the fact that a system once differentiated becomes linear in its highest derivatives. Then we apply diagonal homotopies to incrementally process new constraints, one at a time. The method is illustrated on several examples, combining symbolic differential elimination (using rifsimp) with numerical homotopy continuation (using phc). Categories and Subject Descriptors: G.1.8 General Terms: Algorithms, Design Keywords: Component of solutions, diagonal homotopy, dae (Differential Algebraic Equation), differential elimination, hidden constraint, homotopy continuation, numerical algebraic geometry, numerical jet geometry, path following, polynomial system, witness set.

Over and under-determined systems of ode and pde arise in applications such as constrained multibody mechanics and control systems (e.g. differential-algebraic equations (dae) arise in constrained Lagrangian mechanics [20]). Much progress has been made in exact differential elimination methods, theory and algorithms for nonlinear systems of pde. For example see Boulier et al. [3], Chen and Gao [5], Hubert [9], Mansfield [12], Seiler [20], Reid, Rust et al. [18], Wu [30]. Such methods enable the identification of all the hidden constraints for a system of pde and the automatic statement of an existence and uniqueness theorem for its solutions. They give a geometrical view of its solution space [17, 20] and enable the determination of its symmetry properties. They enable the computation of initial data and associated formal power series solutions in the neighborhood of a point. Algorithmic membership tests (specifically in the radical of a differential ideal) can be given [3, 9]. They can ease the difficulty of numerical solution of dae systems [26]. This paper is a sequel to [14] and [7] in which we develop theory and methods for using numerical homotopy continuation methods in the differential elimination process to identify missing constraints for systems of differential equations. In [14] such methods were first introduced by combining the Cartan-Kuranishni approach with homotopy methods. During the application of that approach all equations are differentiated up to the current highest derivative order, resulting in potentially large numbers of pde. These pde are treated as polynomial equations in jet space, and their large number implies that the number of continuation paths that must be tracked can be impractically large. In this paper we process the leading linear pde using the rifsimp algorithm [29] and the leading nonlinear pde using PHCpack [27], applying diagonal homotopies [23]. The correctness of our results can be certified if the constraints are free from multiplicities and the numerical representations are well conditioned. This paper is organized as follows. In §2 we introduce some basic material for symbolic differential elimination, and in §3 we give a short overview on recent developments in homotopy methods. In §4 we present our method and briefly outline some optimizations in §5. Examples are given in §6 and concluding remarks in §7.

∗ Ontario Research Centre for Computer Algebra, c/o Department of Computer Science, University of Western Ontario, London, Ontario, N6A 5B7, Canada. Email: [email protected]. URL: http://www.orcca.on.ca/˜reid. Support from NSERC is gratefully acknowledged. † Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 851 South Morgan (M/C 249), Chicago, IL 60607-7045, U.S.A. Email: [email protected] or [email protected]. URL: http://www.math.uic.edu/˜jan. This material is based upon work supported by the National Science Foundation under Grant No. 0134611 and Grant No. 0410036. ‡ Centre for Experimental and Computational Mathematics, Simon Fraser University, Burnaby, British Columbia, Canada. Email: [email protected]. URL: http://www.cecm.sfu.ca/˜wittkopf. Support from NSERC is gratefully acknowledged. § Ontario Research Centre for Computer Algebra, c/o Department of Applied Mathematics, University of Western Ontario, London, Ontario, N6A 5B7, Canada. Email: [email protected]. Support from NSERC is gratefully acknowledged.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. ISSAC’05, July 24–27, 2005, Beijing, China. Copyright 2005 ACM 1-59593-095-7/05/0007 ...$5.00.

2.

INTRODUCTION

SYMBOLIC DIFFERENTIAL ELIMINATION

Consider a polynomially nonlinear system of pde R = (R1 , . . . , Rl ) = 0 with independent variables x = (x1 , . . . , xn )

269

and dependent variables u = (u1 , . . . , um ) over C with coefficients from some computable extension of Q. As in [3, 18, 20] solutions and derivatives are replaced by formal (jet) variables, allowing manipulation of equations without first assuming that solutions exist [13]. In particular, denoting the p-th order jet variables corresponding to derivatives as u, the jet variety of a qth order system in J q = Cnq is

pde which are linear in their highest derivatives with respect to the ranking, and its complement the leading nonlinear pde. During its execution these two sets are continually updated until they satisfy certain theoretical conditions for termination [18, 29]. One condition is that the integrability conditions, after reduction with respect to the leading nonlinear pde, should lie in the algebraic ideal generated by the leading nonlinear pde. Also the once-differentiated set of the leading nonlinear pde, after reduction with respect to the leading linear pde, should lie in that ideal. Violation of these conditions gives new equations which are appended to the system, and the process above is repeated. In existing implementations [29] the membership is symbolically tested using Gr¨ obner Bases and Triangular Set methods.

p

V (R) := {(x, u, u, . . . , u) ∈ J q : R(x, u, u, ..., u) = 0}. (1) 1


Here nq = n + m

q

n+q q

1



q

is the number of independent

variables, dependent variables and derivatives of order less than or equal to q. We restrict to the subset of the variables of J q that actually appear in the given system.

3.

Example 2.1. Throughout this article we use the following running example, first introduced in [16], see also [7]: ∂ 2 u(x, y) ∂ 2 u(x, y) − = 0, 2 ∂y ∂x∂y




∂u(x, y) ∂x

r

+

∂u(x, y) − u(x, y) = 0. ∂x (2)

For the case r = 2, this is a differential polynomial system R = (uyy − uxy , u2x + ux − u) = 0 in the jet space of second order J 2 = C8 and has jet variety V (R) = {(x, y, u, ux , uy , uxx , uxy , uyy ) ∈ J 2 : uyy − uxy = 0, u2x + ux − u = 0}. The rifsimp algorithm [18, 29] takes on input a ranking of partial derivatives. A ranking of derivatives [18] is a total ordering on the set of all jet variables (excluding independent variables) that is invariant under differentiation and satisfies a positivity condition. Definition 2.2 (Highest Derivative). The highest derivative of a pde ψ is the largest derivative appearing in the pde in the ranking. A pde is classified as leading linear or leading nonlinear with respect to a ranking according as whether it is linear or not in its highest derivative.

Our tool to numerically solve polynomial systems is homotopy continuation. Homotopy methods define families of systems, embedding a system to be solved in a homotopy, connecting it to a start system whose solutions are known. Continuation methods are then applied to track the paths defined by the homotopy, leading to the solutions. By random choices of constants in the homotopy one can prove that, except for an algebraic set of bad choices of constants, singularities and diverging paths can only occur at the end of the paths, when the system to be solved has singular solutions or fewer solutions than the generic root count. In [24], a new field “Numerical Algebraic Geometry” was described which led to the development of homotopies to describe all irreducible components (all meaning: for all dimensions) of the solution set of a polynomial system. We briefly list key ingredients of this developing field, see also [25]: 1) Witness Sets: are the key data in a numerical irreducible decomposition. A witness set for a k-dimensional solution component consists of k random hyperplanes and all isolated solutions in the intersection of the component with those hyperplanes. The degree of the solution component equals the number of witness points. Witness sets are equivalent to lifting fibers in a geometric resolution [10]. 2) Cascade of Homotopies: Candidate witness points are computed efficiently using a cascade of homotopies, peeling off the hyperplanes in going from high to lower dimensional solution components. This idea of cutting with hyperplanes to determine the dimensions of solution components appeared in Giusti and Heintz [6]. 3) Factorization. Using monodromy loops, certified by linear traces, a pure dimensional solution set is factored into irreducibles. See e.g. [4] for the related approximate multivariate polynomial factorization. 4) Diagonal Homotopies. We intersect two solution components: A and B. Let A be defined by polynomial equations f (u) = 0, cut by hyperplanes K(u) = 0, and B similarly defined by g(v) = 0 and L(v) = 0. Consider the homotopy

Example 2.3. Consider the ranking of partial derivatives: u ≺ ux ≺ uy ≺ uxx ≺ uxy ≺ · · ·

NUMERICAL ALGEBRAIC GEOMETRY

(3)

Then (2) has hd(uyy −uxy ) = uyy and hd(u2x +ux −u) = ux . Hence uyy − uxy is leading linear and u2x + ux − u is leading nonlinear with respect to the ranking. Definition 2.4 (Formal Total Derivative). The  ∂ ∂ + Σm formal total derivative is Dxj = ∂x =1 uxj ∂u
+ · · · j Given a list of equations N = 0, D(N ) is the list of first order total derivatives of all equations of N with respect to all independent variables (i.e. D(N ) := [Dxj Nk ]) and forms a single prolongation of N . For Example 2.1 with N = u2x + ux − u = 0 and formal total derivatives Dx and Dy we have: D(N ) = [2ux uxx + uxx − ux , 2ux uxy + uxy − uy ]. Implementations of symbolic differential elimination algorithms have devoted much attention to decreasing the order of such prolongations, while still determining all the obstructions to the existence of formal power series solutions (e.g. see [20, 29]). As input the rifsimp algorithm takes a polynomially nonlinear system of pde, together with a ranking of derivatives [18]. It partitions the system into two disjoint sets:



  f (u) = 0  g(v) = 0   H(u, v, t) =  (1 − t) +  K(u) = 0  L(v) = 0

 f (u) = 0 g(v) = 0  t. (4) u−v =0  M (u) = 0

For t = 0, we start at the product of the two witness sets A × B. At t = 1, we are at the diagonal

270

u − v = 0. The above homotopy is just a cartoon description of diagonal homotopies, see [23]. Intersecting components is done symbolically in the geometric resolution, see [10].

• L is a set of leading linear pde in solved form with respect to its highest derivatives in the ranking ≺, where hd(L) are all distinct, and no member of hd(L) is a derivative of any other member;

These methods have been implemented in PHCpack [27], see [22] for a description of some of its added capabilities. An interface to PHCpack within Maple is described in [11].

• N is a set of leading nonlinear pde with no dependence on hd(L) or derivatives of hd(L); • P is a set of inequations (the pivots) with no dependence on on hd(L) or derivatives of hd(L);

4. SYMBOLIC-NUMERIC COMPLETION ALGORITHM

• the integrability conditions of the leading linear pde after reduction wrt L are in N ;

Pseudo-code is given for the main algorithm HybridRif and its subroutines.

• D(N ) after reduction wrt L is in N . In addition, an existence and uniqueness theorem is provided for its output [18]. We work with a modified version of the rifsimp algorithm, called riflin, which works with only the leading linear part so no membership tests are performed and D(N ) is not computed. In contrast to rifsimp only a subset of all constraints are determined and an existence uniqueness result can not be stated.

4.1 Using Witness Sets A basic step of our method is to detect whether a new constraint lowers the dimension of some component of the current system. As we append constraints, the general membership test of [22] simplifies to substituting the witness points of the current constraints into the presumed new constraint. Algorithm 4.1. W new = MembershipTest(W, p, ) Input: Witness set W ; a polynomial p; a tolerance . Output: W new = {w ∈ W : |p(w)| > }.

4.3

For correctness of our methods we need to test whether the constraints are free from singularities. Numerically this is done by setting thresholds on the condition numbers of the witness points. While ill-conditioned points are not necessarily points with with multiplicity > 1 (extreme values of coefficients can lead to a bad scaling), for our current homotopy methods there is no difference in practice.

Algorithm 4.6. [N, P, L, f ail] = HybridRif(S, ≺, ) Input : S, a polynomially nonlinear system of differential equations and inequations over Q and a ranking ≺; a tolerance . Output: [N, P, L, f ail], where N: leading nonlinear part; P: pivots (inequations); L: leading linear part; fail: true if witness sets are not reduced.

Algorithm 4.2. boolean = IsReduced(W, ) Input: Witness set W ;  tolerance on inverse condition number. Output: true if all points in W have good condition; false otherwise.

[N (0) , P (0) , L(0) ] := riflin(S, ≺) W (0) := WitnessSet(N (0) , ) Repeat from j = 1 Q(j) := N (j−1) ∪ P (j−1) ∪ L(j−1) ∪ D(N (j−1) ) [N (j) , P (j) , L(j) ] := riflin(Q(j) , ≺) if hd(L(j−1) ) = hd(L(j) ) then [N (j−1) , P (j−1) , L(j−1) ] := [N (j) , P (j) , L(j) ]; else W (j) := W (j−1) ; Repeat for p in N (j) \N (j−1) W new := MembershipTest(W (j) , p, ); if W new = ∅ then W (j) := WitnessSet(p, ); W (j) := Intersect(W new , W (j) ); end if; end loop; if W (j) = W (j−1) then f ail :=not IsReduced(W (j−1) , ); return [N (j−1) , P (j−1) , L(j−1) , f ail]; end if; j := j + 1; end if; end loop.

A witness set for a hypersurface (defined by one multivariate polynomial) is obtained by cutting the hypersurface by a random line and applying a univariate root finder. Algorithm 4.3. W = WitnessSet(p, ) A polynomial p ; a tolerance . Output: Witness set W for p. Diagonal homotopies [23] are used to compute a numerical representation of the intersection of two solution components given by witness sets. Algorithm 4.4. W = Intersect(W1 , W2 ) Input: Witness sets W1 , W2 representing A, B resp. Output: Witness W representing A ∩ B.

4.2 Specification of rifsimp & riflin In brief the rifsimp algorithm described in [18, 29] has the following input and output characteristics: Algorithm 4.5. [N, L, P ] = rifsimp(S, ≺) Input: Output:

The main algorithm

The symbolic algebraic equation manipulation in rifsimp is replaced with the numerical diagonal homotopy method.

S, a system of polynomially nonlinear pde and inequations over Q, and a ranking ≺. [N,L,P], where

In practice HybridRif will abort reporting failure as soon as a witness set shows intolerably high condition numbers.

271

In general however this does not necessarily imply S ⊇ C. For example consider S = {(x, u) : (x − 3)(u − 1) = 0} and C = {(x, u) : (u − 2) = 0} then with x ˆ = 3, Sxˆ ⊇ Cxˆ , but S ⊇ C. But note that components of form such as x − 3 = 0 are not fibred over X and are not of interest for pde, since they imply that the ‘independent variables’ are instead dependent on each other. Such non-fibred components are avoided, with high probability, by setting x = x ˆ. Assume C has only one component fibred over X and Sxˆ ⊇ Cxˆ . If S ⊇ C, then C ∩ S is a proper algebraic subset of C [21] which means dim C > dim (C ∩ S) and since C is fibred over X, Cxˆ is not empty. Therefore dim Cxˆ > dim (C ∩ S ∩ xˆ ) = dim (Cxˆ ∩ Sxˆ ), contradicting Sxˆ ⊇ Cxˆ , so S must contain the whole component of C. This technique can often dramatically decrease the B´ezout bound of the system and number of paths for the witness set by homotopies in MembershipTest. Note that the degree d of a pde, when the independent variables are fixed to constants, is invariant under prolongation. Hence the B´ezout degree of the prolongation of a single pde, is dN where N is the number of pde in the prolongation.

4.4 Termination Conditions Algorithm IsReduced is implemented using estimates for the inverse condition number of the Jacobian matrix at the witness points. A standard Noetherian argument, which is a minor variation of that in [18], shows that the linear part L(j) must eventually stabilize. Further it is easily shown that hd(L(j) ) = hd(L(j−1) ) ⇒ P (j) = P (j−1) . (j−1)

(5)

(j)

, N , ) = ∅ used The condition MembershipTest(W to terminate HybridRif corresponds to the symbolic test involving the difference of two varieties: V (N (j) )\V (P (j−1) ) ⊇ V (N (j−1) )\V (P (j−1) ).

(6)

As HybridRif will fail when it encounters singularities or ill-conditioned representations, its termination is not absolute as is the case when the symbolic conditions (5) and (6) are applied. However, when HybridRif terminates without failure, the final witness set can be certified as follows: every witness point is an approximate zero in the sense of [2].

5. OPTIMIZATIONS

6.

It will be advantageous, but not theoretically necessary, to remove redundant equations. A polynomial is redundant if after its removal the geometry of the solution set has not changed. By repeated calls to Algorithm 4.1, we can implement the following.

EXAMPLES

6.1

Illustrative Example

The simple illustrative system (2) with r = 2 has S := [uyy − uxy = 0, u2x + ux − u = 0]

new

= Shrink(N, W, ) Algorithm 5.1. N Input: N a set of polynomials; W witness set representing N −1 (0);  is tolerance for Algorithm 4.1. Output: N new cuts out same solution set as N .

(7)

on entry into HybridRif. We assume the ranking is given by (3). Since the independent variables x, y do not appear explicitly they are not used in dimension counts. At the first iteration, applying riflin to S yields the single leading linear pde in the solved form L(0) , and a single leading nonlinear pde N (0) :

The number of paths followed by homotopy methods is perhaps the most important aspect of their computational cost. In the case of dense polynomial square systems, this number is given by the B´ezout degree. As a system is prolonged (differentiated) the number of equations can grow dramatically and the product of the degrees of these equations (their B´ezout degree) can grow even more explosively. Thus methods for decreasing this number are a priority in the development of homotopy methods for pde systems. It is also advantageous to fix the value of the independent variables to random numbers: x = x ˆ where x belongs to the space of independent variables X. Extended graphs of solutions of pde belong to components which are fibred over X. This condition is ensured for each component of V (N ) ⊆ J q that is fibred over X. Let πX denote the projection onto X, that is πX (x, v) = x where v = (u, ..., u). Thus at

N (0) u2x + ux − u = 0

P (0) ∅

L(0) uyy = uxy

(8)

We first calculate D(N (0) ) = [2ux uxx + uxx − ux , 2ux uxy + uxy − uy ] then Q(1) = L(0) ∪ P (0) ∪ N (0) ∪ D(N (0) )

(9)

and apply riflin to Q(1) to obtain N (1) u2x + ux − u = 0 u2y − uy ux = 0 uy ux − u2y = 0

q

P (1) L(1) (2ux + 1) = 0 uyy = uxy = uxx =

uy 2ux +1 uy 2ux +1 ux 2ux +1

(10)

We remove the obvious duplicate equation in N (1) by a simple implementation of Algorithm 5.1 although this is not necessary for the correctness and termination of HybridRif. Next we check whether the leading linear part is stable or not. Since hd(L(1) ) = [uyy , uxy , uxx ] = hd(L(0) ) = [uyy ] we return to the beginning of the major loop. We first compute

neighborhoods O(ˆ x, vˆ) of regular points (ˆ x, vˆ) ∈ V (N ) we x, vˆ) ∩ V (N )) = dim X. Equivalently we have dim (πX O(ˆ have dim πX T(ˆx,ˆv) V (N ) = dim X = n where T(ˆx,ˆv) V (N ) is the tangent space to V (N ) at (ˆ x, vˆ). Suppose we are given a system with variety C and a hypersurface S. We can already test S ⊇ C, by use of the algorithm MembershipTest and substitution of generic points, but the number of continuation paths can be impractically ˆ} = large. Instead we set x = x ˆ, Cxˆ = {(x, v) ∈ C : x = x ˆ} = xˆ ∩ S where xˆ ∩ C and Sxˆ = {(x, v) ∈ S : x = x x, v)} is a linear space. By application of Memberxˆ = {(ˆ shipTest and substitution of generic points we can determine if Sxˆ ⊇ Cxˆ by following far fewer continuation paths.

D(N (1) ) = [2ux uxx + uxx − ux , 2ux uxy + uxy − uy , −uxy (ux − uy ) − uy (uxx − uxy ), −uyy (ux − uy ) − uy (uxy − uyy )] (11) then Q(2) = L(1) ∪ P (1) ∪ N (1) ∪ D(N (1) ). Next riflin is applied to Q(2) which after removing redundant equations

272

gives: 2

(2)

(2)

N P L u2x + ux − u = 0 (2ux + 1) = 0 uyy = uy ux − u2y = 0 uxy = uxx =

3

3

3

uy 2ux +1 uy 2ux +1 ux 2ux +1

X

2

2 Ux Xu

3

+12U

(12) 3

2

3

2 Xx Xu 2

2

+ X Ux Xuu = 0, 3

3

3

3

2

3

3 Xx 3

3

3

2

2

(15)

= 0, 2

3

2xX Ux Xxu + X Ux Xu − xX Uxx Xu + 2xX Ux Xx Xu − X Uu Xx +xX

2

2 Uu Xx

3

3

− 2xX Uxu Xx + xX Uu Xxx + 12xU

2

2 Xx Xu

=0

augmented with the condition that the Jacobian of the transformation does not vanish: Xx Uu − Xu Ux = 0. Application of the HybridRif Algorithm with the ranking graded first by total order of derivative, then with ∂u ≺ ∂x and finally lexicographically with U ≺ X, i.e.: U ≺ X ≺ U u ≺ Xu ≺ U x ≺ Xx ≺ · · ·

(16)

gives the leading linear system Xxx

=

Uxx

=

Uxu

=

Uuu

=

6 X 3 − 5 x2 Xx Uu X − x3 Uu Xx 2 , Xu = 0 5x3 XUu 3 2 3 2 2 3 2 3 20 x U Uu Xx − 20 u X Uu Xx − 5 x Uu X Ux Xx

2 3 2 5 3 3 /(5x Uu Xx X ) +4 x Ux X Uu Xx + 6 Ux X 2 x3 Uu Xx 2 − 5 x2 Xx Uu X + 3 X 3 5x3 XXx 0

(17)

together with the condition Uu = 0, Xx = 0. The constraint leading nonlinear equations found by HybridRif are: 3

3

2

x Uu Xx − X = 0, 2

3

2

3

4

6

2

3

2

4

−200 x uXx X Uu + 200 x Xx Uu U − 27 x Uu X Xx + 36 X 2

4

2

3

2

6

7

4

−25 x Uu X Xx + 16 x Uu XXx = 0,

(13)

3

6

2

2

6

4

3

3

2

5

−200 x Xx Uu Ux − 16 x Uu Xx + 12 x Uu X Xx + 90 xX Uu 3

2

2

2

3

2

2

4

2

2

6

5

−1200 x X Uu Xx U + 680 x uXx X Uu + 85 x Uu X Xx

where R = [φ1 = uyy − uxy = 0, φ2 = u2x + ux − u = 0] is the input system above. In [7] the following dimensions are computed using homotopy continuation:

5

6

2

2

2

3

+720 uX Uu − 171 X − 200 x uXx X Uu = 0, 432 uX

10

4

2

7

7

4

4

2

4

+ 792 x X Xx Uu + 204 x X Xx Uu − 60 x X Xx Uu 3

2

7

9

5

2

2

4

2

6

−3600 x X Xx U − 200 x X Xx Uu Ux + 200 x uX Xx Uu 2

8

6

6

4

4

2

5

5

3

2

5

3

6

4

2

4

−1200 x uX Xx Uu − 2400 x X Xx Ux Uu − 2400 x X Xx Uu U

dim(D2 R) = 1

dim π(DR) = 1 dim π(D2 R) = 1 dim π 2 (D2 R) = 1

2

3

+4u X Ux Xu + 4x U

φ1 = 0, φ2 = 0, Dx (φ1 ) = 0, Dy (φ1 ) = 0, Dx (φ2 ) = 0, Dy (φ2 ) = 0, Dxx (φ2 ) = 0, Dxy (φ2 ) = 0, Dyy (φ2 ) = 0

3

x X Ux Xxx − 4u X Uu Xx − x X Uxx Xx + x X Ux Xx

Here hd(L(2) ) = hd(L(1) ), so the membership test is applied to N (2) to test V (N (2) )\V (P (2) ) ⊇ V (N (1) )\V (P (1) ). First we compute the witness set of each polynomial in N (1) in (u, ux , uy )-space by WitnessSet. There are two paths to be followed for each polynomial. Then the witness set W (1) for N (1) is computed by Intersect, yielding four witness points resulting from tracking 4 paths. During the application of MembershipTest points in W (1) are evaluated in the system N (2) . Since IsReduced (W (1) , ) = true and MembershipTest(W (1) , N (2) , ) = ∅, the termination conditions are met and the algorithm returns [N (1) , P (1) , L(1) ]. For this example it can be checked that the outputs of HybridRif and the fully symbolic algorithm rifsimp are the same. In §6.2 an example is given where the outputs of HybridRif and rifsimp differ. Comparison with a Numerical Geometrical Completion Method: Here we compare HybridRif with a numerical geometrical completion method [1, 7, 14] which is a variation of the symbolic Cartan-Kuranishi method [13, 20]. In [7] the first application of the interpolation-free method of [14] is given to the example system above. The method when applied to an input system R involves computing dim π  Dk R where π : J q → J q−1 is the usual projection until the criteria of projected involution [1] are satisfied. The output of the method of [7] consists of

dim(DR) = 1

2

3

+ 2X Uu Xxu − X Xx Uuu − 2X Xu Uxu + 2X Uu Xx Xu

3

dim(R) = 2

2

4U Xu − X Xu Uuu + X Uu Xuu + X Uu Xu = 0,

(2)

3

2

7

9

6

2

8

2

+1632 x uX Xx Uu + 6336 x uX Xx Uu − 800 x Xx Uu U 6

2

4

3

3

3

−1400 x uX Xx Uu + 800 x u Xx Uu X − 990 x X Xx +54 xX

(14)

10

=0

2

(18)

Application of the initial data algorithm [29] to the leading linear equations (17) above yields the following initial data

and show π(DR) is an involutive system. In the computations, the worst B´ezout number that appears is 64 which is much bigger than 4, the number of continuation paths that had to be followed in the application of HybridRif above.

X(x0 , u0 ) = X 0 , U (x0 , u0 ) = U 0 , Xx (x0 , u0 ) = Xx0 , Ux (x0 , u0 ) = Ux0 , Uu (x0 , u0 ) = Uu0 .

6.2 System for Discrete Symmetries of uxx = x1 ux + x43 u2

Reference [15] solves the problem of determining the full diffeomorphism pseudogroup of point transformations (x, u) → (ˆ x, u ˆ) of the form x ˆ = X(x, u), u ˆ = U (x, u), leaving invariant the ode uxx = x1 ux + x43 u2 . Requiring that these transformations leave the ode invariant leads [15] to a system of nonlinear pde for the unknown functions X, U :

273

(19)

Then the existence and uniqueness theorem [18] implies that formal power solutions to the system exist at points where the constants X 0 , U 0 , Xx0 , Ux0 , Uu0 satisfy the constraint nonlinear equations (18). In this example N (1) consists of the first 3 nonlinear equations in (18) with degrees 6,13,12 respectively, and the corresponding linear part (L(1) ) becomes stable. Next the witness set is constructed for N (1) . Next N (2) is obtained with all 4 nonlinear equations of (18) with degrees 6, 13, 12 and 19 respectively. Application of MembershipTest shows that

d

the fourth equation is geometrically new so the witness set of its intersection is computed using Intersect. Because of the high total degree in this example, we use techniques to decrease the number the continuation paths followed by phc. The first technique is to specialize the independent variables to random fixed values as discussed in §5. In particular the degrees of the uncovered constraints (18) decrease dramatically from 6, 13, 12, 19 to 3, 7, 6, 10. A second key to success, was to use mixed volumes instead of B´ezout Bounds. In particular in the application of diagonal homotopies, this decreased the number of paths needing to be followed for N (1) from 126(= 3 · 7 · 6) to 3 and the number of paths for N (2) from 1260(= 3 · 7 · 6 · 10) to 4. Application of diagonal homotopies showed the existence of 1 dimensional components for the constraint nonlinear system (3 dimensional if we include x, u in the dimension count). This agrees with the explicit computations in [15]. Denote by Glie the Lie subgroup of symmetries in a connected component of the identity of the full symmetry group G of the ode. Our dimensional computation correctly reveals the dimension of Glie as 1 as determined by a more conventional linearized calculation in [15]. The degree determined by our calculations is 4 and corresponds to the cardinality of G/Glie which is in agreement with [15] (indeed there it is shown that the factor group is isomorphic to Z4 ). Further calculations using phc on the full constraint nonlinear system reveals that there are 4 degree one, one dimensional components (fixing x, u to constants) whose equations can be interpolated if desired. These computations are again in agreement with the explicit ones in [15]. Interestingly high degree singular components of natural geometric origin violating the invertibility condition Xx Uu − Xu Ux = 0 arose in our calculations and initially caused some numerical difficulties. Such components were excluded by inclusion of the invertibility condition. For the system above this is equivalent to Xx Uu = 0, since Xu = 0. Consequently we also have X = 0 and U = 0. Comparison with a Numerical Geometrical Completion Method: We compare HybridRif with a numerical geometrical completion method [1, 7, 14] which is a variation of the symbolic Cartan-Kuranishi method [13, 20]. The method when applied to an input system R (18) involves computing dim π  Dk R until the criteria of projected involution [1] are satisfied. The system R has B´ezout number 12288 which is reduced to 1875 after substituting random values for the independent variables. The prolongation of DR which has 18 equations with B´ezout number 50096498540544. After specializing the independent variables it reduces to 177978515625 which was still too high. Comparison with the rifsimp symbolic algorithm: Application of rifsimp with the ranking (16) yielded the leading linear system: Ux,x

=

ram:Z ram:C

=

d ram:Z ram:C

=

Uu

=

5 2.5 20.7

6 8.3 62.2

7 16.5

8 128.7

inc

inc

9 inc inc

2 1.5 1.5

5 1.8 1.8

8 2.0 2.0

11 2.1 2.0

14 2.2 2.2

17 2.3 2.3

20 2.5 2.6

with the leading nonlinear equations 32u3 XU − 64u2 x2 Ux U − 32xu2 U 2 − 8u2 x2 Ux X +8xu2 XU − 32u2 x3 Ux2 − 2x3 uUx X − 16x3 uUx U +x2 uXU − 12ux4 Ux2 − 8x2 uU 2 − x3 U 2 − x5 Ux2 = 0, (uX − xU )(xX + 4uX + 4xU ) = 0

(21)

and the inequations X + 8U = 0, X = 0, U = 0. Unlike the example of §6.1, this differs from the result obtained by HybridRif. This discrepancy is resolved by noting that both systems define the the same locus in jet space. Finally we note that Hydon [8] gives an elegant and efficient method which exploits the knowledge of the Lie group Glie to considerably ease computation of the full group G.

6.3

Random first order ODE R(ux , u) = 0

In this section we apply our symbolic-numeric approach to a class of random ode R(ux , u) = 0 for a single dependent variable u. The efficiency of this approach is compared with that of using the symbolic rifsimp algorithm. Differentiation of R(ux , u) = 0 gives Rux uxx + Ru ux = 0. The following cases are easily obtained: Case 1: uxx = − RRuuux , Rux = 0, R(ux , u) = 0 x Case 2: S2 = {R = 0, Ru = 0, Rux = 0}, ux = 0 Case 3: S3 = {R(0, u) = 0, Rux (0, u) = 0}, ux = 0

(22)

For random differential polynomials R, system S2 in (22) consists of two random polynomials in one variable and system S3 in (22) consists of three random polynomials in two variables. For random R systems S2 and S3 will be inconsistent with high probability and Case 1 will be the only consistent case. A full analysis of all the singular cases for such ode is given in the classic work of Hubert [9]. Two subclasses of random ode with degrees d from 1 to 20 were considered. One subclass had random coefficients consisting of integers between −99 and 99, and the other random subclass had random coefficients consisting of complex numbers of the form (a + bi)/(max{|a|, |b|} + 1) where a, b are random integers between −10 and 10. The computations were carried out using Maple 9, and phc (release 2.3 beta) on a 1.5 GHZ Pentium M, with 512 MB of RAM, running under Windows XP. As shown in Table 1 the RAM was exhausted at relatively low degree d = 9,

(U 2 (4U + X)(8u + x))/ (u(32uU 2 + 4xU 2 + 8x2 Ux U + 32uxUx U

Xu

4 1.4 11.2

Table 2: phc memory consumption (MB) applied to a class of random polynomial ODE R(ux , u) = 0 with integer coefficients, and a class with complex rational coefficients. Here d = degree(R).

(−16x2 uU 2 − 4x3 U 2 − 128xu2 U 2 +

+8uXU + xXU + 4xuXUx + x2 Ux X)) 0 X + 8U 8u + x

3 0.6 6.8

Table 1: rifsimp memory consumption (MB) applied to a class of random polynomial ODE R(ux , u) = 0 with integer coefficients, and a class with complex rational coefficients. Here d = degree(R). RAM=inc indicates the memory of machine was exhausted.

−384u2 x2 Ux U − 16x4 Ux U − 160x3 uUx U + +4x2 uXU + 16xu2 XU + 128u3 XU − 3x4 XUx + −80u2 x2 Ux X − 32x3 uUx X)(x3 (4u + x)2 (X + 8U )) Xx

2 0.24 3.40

(20)

274

means that generally fewer constraints need to be stored than would be required to represent the ideal. In addition, the maintenance of the constraints in their introduced form helps to preserve sparsity, and reduce equation and coefficient growth typical of Gr¨ obner methods. It also allows flexibility in using alternative and sparse methods to control expression swell. Such methods include encoding the constraints by straight line programs, or using memory management based on ordered storage strategies [29] or directed acyclic graph structures as used by Lecerf in his implementation of the algorithms in [10]. The methods were applied to a number of examples starting with an easy illustrative example in §6.1. Secondly a system for discrete symmetries of moderate difficulty for symbolic methods was considered. Although the output was implicit, it illustrated that useful features of the symmetries could be extracted by the new hybrid methods (such as the number of discrete symmetries, and the degree of the components of the group). On this example, HybridRif was compared with a geometrical approach based on a numerical version of the Cartan-Kuranishi algorithm. It demonstrated that far fewer continuation paths were needed by HybridRif than the numerical geometrical method developed in earlier work. Finally in §6.3 we considered a class of random first order ode. On systems which are denser and of higher degree, numerical methods have an advantage while symbolic methods can perform better on lower degree, highly structured sparse systems. We caution that the sample size is too small to make emphatic statements. Certainly it indicates that there is scope to improve rifsimp’s algebraic processing by using alternative symbolic and numeric algorithms. This paper belongs to a series initiated in [28], continued in [14] and [7], aimed at developing “Numerical Jet Geometry”, as a subfield of “Numerical Algebraic Geometry”. Ultimately, this development will lead to methods enabling the practical processing of approximate input systems.

10

5 log[2](t)

0

-5

1

1.5

2

2.5

3

3.5

4

log[2](d)

Figure 1: Time-Degree Statistics for a Random polynomial ODE R(ux , u) = 0 plotted on a log − log scale. d = degree(R) and t is the time to apply the differential elimination process using: rifsimp (comp. coeffs. ♦); rifsimp (integ. coeffs. ); phc (comp. coeffs. +); phc (integ. coeffs. o). and this was dramatically worse for complex coefficients where exhaustion occurred at d = 7. As seen in Table 2 RAM usage by phc was dramatically lower and more stable than that of rifsimp. While changing from random integer to complex coefficients barely affected the RAM consumed by phc, it dramatically increased RAM usage by rifsimp. Degree-time statistics for rifsimp and phc are shown in Figure 1. The positive concavity of the two curves for rifsimp indicates its complexity is more than polynomial. The approximately linear curves for phc in Figure 1 on the loglog scale is typical for a polynomial-time method. However the worst case complexity of HybridRif is at least exponential, considering its application to systems of linear homogeneous pde in a single dependent variable. In that case its output is isomorphic to a Gr¨ obner Basis. Groundbreaking work on reducing the complexity to polynomial time for ode was done by Sedglovacic [19]. The memory usage statistics show the discrepancy between rifsimp and phc growing with increasing degree, and when changing from integer to random complex coefficients. The symbolic differential elimination program Rosenfeld Groebner had similar memory and time behavior to rifsimp on the random class of ode.

8.

REFERENCES [1] J. Bonasia, F. Lemaire, G. Reid, R. Scott, and L. Zhi. Determination of Approximate Symmetries of Differential Equations. Centre de Recherches Math´ematiques, CRM Proceedings and Lecture Notes. Vol 39, pages 233–250, 2004. [2] L. Blum, F. Cucker, M. Shub, and S. Smale. Complexity and Real Computation. Springer–Verlag, 1998. [3] F. Boulier, D. Lazard, F. Ollivier, and M. Petitot. Representation for the radical of a finitely generated differential ideal. Proc. ISSAC 1995. ACM Press. 158–166, 1995. [4] G. Ch`eze and A. Galligo. Four Lectures on Polynomial Absolute Factorization. In A. Dickenstein and I.Z. Emiris (Eds.), Solving Polynomial Equations: Foundations, Algorithms, and Applications. Volume 14 of Algorithms and Computation in Mathematics 14, Springer-Verlag, pages 339–392, 2005. [5] Y. Chen and X.-S. Gao. Involutive Bases of Algebraic Partial Differential Equation Systems. Science in China (A), 33(2), page 97–113, 2003. [6] M. Giusti and J. Heintz. La d´etermination de la dimension et des points isol´ees d’une vari´et´e

7. DISCUSSION Our method applies to intrinsic (exact) systems of polynomially nonlinear pde and relies on splitting the system into a leading linear subsystem and its complement. Welldeveloped (linear) symbolic methods are applied to the leading linear part of the system. The success of this strategy enables the shrinking of the number of genuinely nonlinear equations that are dealt with by the numerical continuation methods. The use of diagonal homotopies allows handling the constraints incrementally, exploiting the structure of the leading nonlinear systems, and leads to a further decrease in the number of paths to be followed. Note that one could – at least in theory – replace the use of witness sets and diagonal homotopies in HybridRif by lifting fibers and using geometric resolutions [10]. In contrast to Gr¨ obner methods, the fact that only geometrically new constraints are used

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[19] A. Sedoglavic. A probabilistic algorithm to test local algebraic observability in polynomial time. J. Symbolic Computation 33(5): 735–755, 2002. [20] W.M. Seiler. Involution - The formal theory of differential equations and its applications in computer algebra and numerical analysis. Habilitation Thesis, Univ. of Mannheim, 2002. [21] A.J. Sommese and J. Verschelde. Numerical homotopies to compute generic points on positive dimensional algebraic sets. Journal of Complexity 16(3):572-602, 2000. [22] A.J. Sommese, J. Verschelde, and C.W. Wampler. Numerical irreducible decomposition using PHCpack. In M. Joswig and N. Takayama, editors, Algebra, Geometry, and Software Systems, pages 109–130. Springer–Verlag, 2003. [23] A.J. Sommese, J. Verschelde, and C.W. Wampler. Homotopies for intersecting solution components of polynomial systems. SIAM J. Numer. Anal. 42(4):1552–1571, 2004. [24] A.J. Sommese and C.W. Wampler. Numerical algebraic geometry. In The Mathematics of Numerical Analysis, Volume 32 of Lectures in Applied Mathematics, edited by J. Renegar, M. Shub, and S. Smale, 749–763, 1996. Proceedings of the AMS-SIAM Summer Seminar in Applied Mathematics, Park City, Utah, July 17-August 11, 1995, Park City, Utah. [25] A.J. Sommese and C.W. Wampler. The Numerical solution of systems of polynomials arising in engineering and science. World Scientific Press, Singapore, 2005. [26] J. Tuomela and T. Arponen. On the numerical solution of involutive ordinary differential systems. IMA J. Numer. Anal. 20: 561–599, 2000. [27] J. Verschelde. Algorithm 795: PHCpack: A general-purpose solver for polynomial systems by homotopy continuation. ACM Transactions on Mathematical Software 25(2): 251–276, 1999. Software available at http://www.math.uic.edu/~jan. [28] A. Wittkopf and G.J. Reid. Fast differential elimination in C: The CDiffElim environment. Computer Physics Communications, 139: 192–217, 2001. [29] A. Wittkopf. Algorithms and Implementations for Differential Elimination. Ph.D. Thesis, Simon Fraser University, 2004. [30] W.-T. Wu. On the foundations of algebraic differential geometry. Mathematics-Mechanization Research Preprint No. 3, pages 1–26, 1989.

276

Algorithms for Symbolic/Numeric Control of Affine ∗ Dynamical Systems Jean-Guillaume Dumas

Aude Rondepierre

Universite´ Joseph Fourier, Laboratoire de Modelisation ´ et Calcul, 50 av. des Mathematiques. ´ B.P. 53 X, 38041 Grenoble, France.

Universite´ Joseph Fourier, Laboratoire de Modelisation ´ et Calcul, 50 av. des Mathematiques. ´ B.P. 53 X, 38041 Grenoble, France.

[email protected]

[email protected]

ABSTRACT

theory lacks generic algorithms, specially when singular subarcs appear[14, 19, 2]. Furthermore, in “real-life”, optimal control problems are fully nonlinear. Therefore most of the algorithms presented here have been developed towards their application to the hybrid control of nonlinear dynamical systems: in [20], we propose a piecewise affine approximation by way of a hybrid automaton. In each cell, the local optimal control problem is affine and techniques developed here will be applied. In this paper we consider a linear dynamical system:  ˙ X(t) = AX(t) + Bu(t) (1) X(0) = X0

We consider a general linear dynamical system and want to control its behavior. The goal is to reach a given target by minimizing a cost function. We provide a new generic algorithm with together exact, symbolic and numerical modules. In particular new efficient methods computing a block Kalman canonical exact decomposition and the optimal solutions are presented. We also propose a new numerical algorithm under-approximating the controllable domain in view of its analytical resolution in the context of singular sub-arcs. Categories and Subject Descriptors: I.1.2 [Symbolic and algebraic manipulation]: Algorithms, J.1.7 [Computer Applications]: Command and control.

where ∀t ≥ 0, X(t) ∈ Rn and u(t) ∈ Um = {s1 , . . . , sp } ⊂ Rm is the control. We want to control the system (1) from an initial state X0 to a target Xf = 0 at an unspecified time tf , in such a way that the functional: J(X0 , u(.)) = R +∞ l(X(t), u(t))dt is minimized. 0 Here, we provide a full implementation analyzing linear optimal control problems as general as possible. Our algorithm is divided in four steps: (1) Canonical transformation (see §2). (2) Approximation of the controllable set (see §3.2). (3) Computation of optimal solutions (see §4). (4) Inverse transformation (see §2.2).

General Terms: Algorithms. Keywords: Affine Optimal Control Problems, Canonical Transformation, Controllability.

1.

INTRODUCTION

Aerospace engineering, automatics and other industries provide a lot of optimization problems, which can be described by optimal control formulations: change of satellites orbits, flight planning, motion coordination [7] ([16] for more applications in aerospace industry). Optimal control has so become a more and more challenging domain and its theory has been extensively developed for many years. Nevertheless, the problem of synthesis of optimal feedback is not solved, even for linear systems. In some specific cases like time-optimal control problems, adequate solutions have been found [18, §3],[2, 17, 16]. Also, control

Each step can be done in several different ways and some salient features of our presentation are: • a new and more efficient implementation by block matrices of the exact computation of Kalman decomposition. • symbolic computation of the boundaries of domains, where the optimal control is constant (see e.g. algorithms 5 and 6).

∗Work partially supported by the Region Rhˆ one-Alpes (Calcel project).

• a new numerical method to compute an under-approximation of the controllable domain. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. ISSAC’05, July 24–27, 2005, Beijing, China. Copyright 2005 ACM 1-59593-095-7/05/0007 ...$5.00.

• an efficient implementation of the optimal solution computation, for a very large class of cost functions using subroutines previously defined (see algorithm 7). Our approach enables the high dimensions treatment, even when compared to numerical softwares. Indeed, numerical methods developed from the Hamilton-Jacobi-Bellman

277

2.1.1 Simplification to rk(B) = m

(HJB) theory are known to suffer from the dimension: they generally require to generate a grid over a bounded region of the state space. If the state dimension is n and the number of discretization points per dimension is 50 (which is the minimum acceptable: 100 could still be a bit sparse), one has to consider 50n grid points. Despite the development of efficient techniques for the choice of the discretization points like adaptative mesh, computations grow exponentially in the state dimension. Consequently dimension 4 or 5 cannot be exceeded, while e.g. aerospace [16] requires treatments of dimensions 6 or 7. By the use of Hybrid Computation [4] combining numerical analysis and computer algebra, we are now able to deal with high dimensions (see [20, Part II]): first the mesh is made on the fly to reduce the complexity. Then, at the vertices of the mesh, only a linear interpolation of our complex system is performed. In each cell, the system is linear and one need to develop methods as symbolic as possible: basically, an analytical approach must allow to improve the hybrid approximation. The paper is organized as follows: in section 2, we will develop explicit algorithms to compute exactly the canonical transformation of any linear optimal control problem and then the exact inverse transformation. In section 3, we provide a numerical controllability analysis and then in section 4, the analytical computation of optimal solutions of the canonical problem.

2.

Lemma 1. Let us consider the linear system (1). There ˜ ∈ Rn×rk(B) and a linear mapping exists a full rank matrix B rk(B)×m ˙ ˜ Φ∈R such that: X(t) = AX(t) + BΦu(t). Proof. b = rk(B). If b < m, then there exists a column ˜ 0 ] where B ˜ ∈ Rn×b permutation P ∈ Rm×m s.t.: BP = [B|B ˜ and rk(B) = b. Moreover, the column vectors of B0 are ˜ i.e.: ∃Λ ∈ Rb×m−b , B0 = linearly dependent of those of B, ˜ ˜ [Ib |Λ]P −1 and Φ = [Ib |Λ]P −1 . BΛ. Hence: B = B In the following, we will denote by F ullRank(B) the algo˜ Φ) from a matrix B as in the lemma. rithm computing (b, B,

2.1.2 Block Kalman Canonical Form Now we want to decompose the state space of our linear system into a controllable part and an uncontrollable one. The classical method is to introduce the linear subspace W (A, B) = span(B, AB, . . . , An−1 B) and then prove that W is the first subspace of Rn satisfying both: (i) Im(B) ⊂ W (ii) W is A-invariant. The method is then to decompose ¯ : one has to compute a basis the state space Rn into W ⊕ W of the subspace W (A, B) and to complete it for the whole state space. The matrix T of theorem 1 would be the change matrix from the canonical basis to the computed basis. In this paper we propose a new approach via block matrix computation developed in collaboration with C. Pernet: we use the so-called LQU of a x × y matrix of – » P decomposition U1 U2 is x × y, U1 is an upper rank r, where U = 0 0 triangular r × r invertible matrix, L is x × x, lower blocktriangular, and P and Q are permutation matrices [10].

CANONICAL TRANSFORMATION

Linear control systems have been widely analyzed. In [13, 12], Kalman considers constant linear optimal control problems without constraints on the control. In this context, we have two well-known results: the first one is a controllability criterion (see [13, 3] for more details), and the second is the following decomposition theorem:

Algorithm 1 BlockKalmanForm Require: A n × n matrix, B n × m matrix. Ensure: r, T, A1 , A2 , A3 , B1 as in theorem 1. 1: K = [B|AB| . . . |An−1 B]; 2: (L, Q, U1 , U2 , P, r) = LQU P (K T ); 3: if r = n then 4: Return (n, In , A, ∅, ∅, B). 5: end if – » Ir T , lower triangular. 6: Form δ = [Ir |0]Q LQ 0 – » Ir . 7: Form d = [Ir+1..nm |0]QT LQ 0 T 8: G = [Ir |0]QT K» . – Ir T T 9: C1 = G(A P U1−1 δ −1 ) 0 » – Ir U1−1 δ −1 ) 10: C2 = [0|In−r ]P (AT P T 0 » – −U1−1 U2 11: C3 = [0|In−r ]P AT P T In−r » – Ir 12: Q1 = [Im |0]Q {Q1 is m × r} dδ −1 –T » G 13: Return (r, , C1T , C2T , C3T , QT1 ). [0|In−r ]P

Theorem 1 ([12] Kalman Canonical Structure). Let A and B be real matrices having respective sizes n × n and n × m. There exists an invertible n × n matrix T such that: – – » » A1 A2 B1 T −1 AT = T −1 B = 0 A3 0 where r = rk([B AB . . . An−1 B]) = rk([B1 A1 B1 . . . An−1 B1 ]) 1 A1 is a r dimensional square matrix and B1 a r × m matrix. There exist many numerical algorithms computing the Kalman canonical form of full rank linear dynamical systems. Next, we consider rank deficient systems, for which exact computation of the rank is needed. Therefore we propose a new explicit and exact algorithm for the Kalman decomposition. Our approach is to use block versions of the linear algebra algorithms as in [5] in order to improve the locality of the computations and treat larger problems faster. Indeed, we are then able to compute exactly the rank of the system and use the LQUP decompostion of [6] (nowadays quite as fast as numerical routines) to perform the decomposition.

2.1 Block Canonical decomposition We consider the general linear system (1). Our decomposition is divided into two steps: we first reduce the system to one with a full rank mapping of the control and second apply a LQUP decomposition to the Kalman matrix.

Theorem 2. Algorithm 1 is correct and its arithmetic complexity is O(nω m)1 . 1 where ω is the exponent of matrix multiplication (3 for the classical algorithm and 2.3755 for Coppersmith-Winograd’s)

278

2.2.2 State Space

Proof. [20] It has three parts and is actually another, constructive, proof of Kalman’s theorem: 1. First, use the generalization of the companion matrix decomposition to prove that GAT = C1 G. 2. Second, use the latter to show that T −1 AT is block triangular. 3. Show that T −1 B has generic rank profile. 4. Now for the complexity: building the Kalman matrix is n matrix multiplications n × n by n × m, each requiring [6, Lemma O(nω−1 m) operations. Following ´ 4.1], the LQUP ` decomposition requires O nω−1 (mn + n) operations. Those two costs dominate the remaining operations: two triangular inversions O(r ω ), some permutations and column selections, and small matrix multiplications (GAT is O(rnω−1 ) and dδ −1 is O(nmr ω−1 ) where r ≤ n.).

By construction, the change matrix T is non singular. Therefore, a trajectory Y (.) from an initial point Y0 corresponds to a trajectory X(.) = T Y (.) from the initial point X0 = T Y0 . Every trajectory is necessarily related to a control, the table 1 displays the correspondence between each trajectory. Initial Problem (1) (X(.), u(.)) (T Y (.), Ψ˜ u)

→ ←

Canonical Problem (3) (T −1 X(.), Φu) (Y (.), u ˜(.))

Table 1: Matching trajectories Proof. The key point here is that a trajectory (X(.), u(.)) in the X-space is a solution of R tthe system (1): X(t) = eAt X0 + eAt 0 e−Aw Bu(w)dw −1 T −1 X(t) = e(T AT )t T −1 X0 Rt −1 −1 + e(T AT )t 0 e−(T AT )w T −1 Bu(w)dw] −1 T −1 X(t) = e(T AT )t T −1 X0 Rt −1 −1 ˜ + e(T AT )t e−(T AT )w T −1 BΦu(w)dw]

Our implementation and constructive proof of the Kalman decomposition are based on LQUP factorization and block matrix computation. The better locality induced by this block version enables the use of very fast Basic Linear Algebra Subroutines, even with symbolic computations [6]. Therefore the computation time is improved. Moreover if we first apply the algorithm FullRank of paragraph 2.1.1, the system (1) can be replaced by another linear one: » – » – A1 A2 B1 Y˙ (t) = Y (t) + u ˜(t) (2) 0 A3 0  Y (t) = T −1 X(t) via possibly two variable changes: u ˜(t) = Φu(t) Next, we use these decomposition in order to define a canonical optimal control problem, simpler to solve.

0

Then (T −1 X(.), Φu(.)) is a solution of (2), i.e. a trajectory in the Y -space.

2.2.3 Cost Function Let X0 be a controllable point. The value function related to the initial control problem (1) is defined by: V (X0 ) = R +∞ inf u(.) 0 l(X(t), u(t))dt. We want to define a new value R +∞ ˜ function V˜ (Y0 ) = inf u(.) l(Y (t), u ˜(t))dt such that the ˜ 0 two related optimal control problems are equivalent. First, the idea is to define a new cost function ˜ l, such that the value function is invariant by canonical transforl(Y, u ˜) → mation (i.e.: V (X0 ) = V˜ (T −1 X0 )). In this case, ˜ l(T Y, Ψ˜ u) and the new optimal control problem becomes: R +∞ ˜ 0, u “Minimize J(Y ˜(.)) = 0 ˜ l(Y (t), u ˜(t))dt with respect to the control u ˜(.) under the dynamic (3) (2) and the constraints: ∀t ≥ 0, u ˜(t) ∈ Conv{˜ s1 , . . . , s˜p
}”.

2.2 Inverse transformation In this section, the focus is on the explicit construction of a new linear optimal control problem under the dynamic (2). A new cost function and new state and control spaces have to be constructed and initial solutions have to be recovered.

2.2.1 Control Space In this paragraph we focus on the construction of a new control space for the linear system (2). By assumptions (see section 1), the control u(.) satisfies: ∀t ≥ 0, u(t) ∈ Um = Conv(s1 , . . . , sp ). Moreover the image of a polyhedron in finite dimension by a linear mapping is polyhedral. So the new control polyhedron is: ΦUm = Conv(Φs1 , . . . , Φsp ). Note that, if rk(B) = m, then Φ = Im , so that no control change is needed. When rk(B) < m, the main difficulty to build our optimal control problem is that there is not any invertible relation between u and u ˜ ; consequently to switch from one control problem to the other, we will first need to define the pseudo-inverse of the control change matrix: s˜1 , . . . , s˜p
are the vertices of ΦUm . We introduce the Moorem×b ) [21] of the matrix Φ = Penrose pseudo-inverse –Ψ ∈ R » I b −1 [Ib |Λ]P : Ψ = P defined by: ∀i ∈ {1, . . . , p }, Ψ˜ si = 0 sk , where k = min{j ∈ {1, . . . , p} / Φsj = s˜i }. By linearity, Ψ is also well defined on the whole polyhedron ΦUm ,
P
˜= indeed: ∀˜ u ∈ ΦUm , ∃(αi )i=1...p
∈ [0, 1]p , pi=1 αi = 1, u Pp
Pp
˜i . Hence Ψ˜ u = i=1 αi Ψ˜ si and the proposition 1 i=1 αi s is proven:

We then have to verify that optimal solutions of this new problem correspond to optimal solutions of (1): ˜∗ (.)) be an optimal solution Proposition 2. Let (Y ∗ (.), u u∗ (.)) is an optimal solution of the of (3). Then (T Y ∗ (.), Ψ˜ initial problem (1) and V (T Y0 ) = V˜ (Y0 ). (The proof is by inspection of J(X0 , Ψ˜ u∗ ).)

2.2.4 Algorithms To conclude the section, we describe two algorithms: SimplifySystem and InverseTransformation. From one given optimal control problem, SimplifySystem allows to define the canonical optimal control problem (see §2.1) ; once this problem is solved, InverseTransformation exactly computes the related optimal solutions of the initial problem (1) by the use of proposition 2. In the following algorithms, the pseudo-inverse Ψ of Φ is given e.g. by [21]. In this section we achieved the transformation of any linear optimal control problem into a canonical one. Moreover we have proved that optimal solutions of the canonical problem give optimal solutions of our initial problem. We have also proposed exact computation algorithms for switching

˜ Proposition 1. (i) ΦΨ = Ib , (ii) ∀u ∈ Um , Bu = BΦu, ˜u ˜ = BΨ˜ u. (iii) ∀˜ u ∈ ΦUm , B

279

Nevertheless we do not need to compute eNt . Indeed, we can recursively show (by starting from n − r to k + 1) that: ([0|In−r−k ]Q∗ )Y2 (0) = 0 Hence: ∀t ≥ 0, ([0|In−r−k ] Q∗ )Y2 (t) = 0. So under the variable change: Y˜2 = (Q∗ Y2 )1..k , the system (2) then becomes: ( ˜2 Y˜2 (t) + B1 u(t) Y˙ 1 (t) = A1 Y1 (t) + A ˙ ˜ ˜3 Y˜2 (t) Y2 (t) = A

Algorithm 2 SimplifySystem Require: A, B, Um = [s1 , . . . , sp ], l. ˜, ˜ l. (Data for the new opEnsure: r, T, Φ, Ψ, A1 , A2 , A3 , U timal control problem: r, the state change matrix, the control change, the dynamic, the control space and the cost function). {Definition of the new control space:} ˜ M ):=FullRank(B); 1: (b, B, 2: Ψ := P seudoInverse(Φ); ˜ := ConvexHull(Φs1 , . . . , Φsp ); 3: U {Definition of the new optimal control problem:} ˜ 4: (r, T, A1 , A2 , A3 , B1 ) := BlockKalmanF orm(A, B); {Definition of the new cost function:} 5: ˜ l := (Y, u ˜) → l(T Y, Ψ˜ u) ˜, ˜ 6: Return (r, T, Φ, Ψ, A1 , A2 , A3 , U l).

– – » Ik ˜3 = (D +N ) Ik is stable. and A 0 0 We have shown that the analysis of the uncontrollable part of the system (2) leads to define a subspace of the state space, namely {(Y1 , Y˜2 , 0) ∈ Rr × Rk × Rn−r−k }. In this subspace, Y˜2 (.) trajectories converge towards 0. From now on, we therefore restrict our analysis to a system (2) where the matrix A3 is stable. »

where: A˜2 = A2 Q

Algorithm 3 InverseTransformation Require: T, Ψ, Y ∗ , u ˜∗ . ∗ 1: Return (T Y , Ψ˜ u∗ , Um , ΦUm ).

3.2 Under-Approximation of the Controllable Domain Now, we assume w.l.o.g that the points si defining the control boundaries are such that: si ∈ / Convj=i (sj ). Therefore, each point si is a vertex of the polytope Um and we have (see §2): rk(B) = m, rk([B|AB| . . . |An−1 B]) = n. We want to find the set of controllable points of our system. By time reversal, we come down to the computation of the attainable set from the target point 0. In [1], for safety verification, the idea is to compute a conservative over-approximation of the attainable set. They can thus certify that the system can not escape from an admissible set of states. On the contrary, we need a guaranty that Y0 is controllable. Therefore we instead compute an underapproximation of this set. Let us start by defining the controllable set C in our context: R C = {Y ∈ Rn /∃T ≥ 0, ∃u : [0, T ] → Um , Y = 0T e−Aτ Bu(τ )dτ }. ˙ Indeed, any solution of a linear system R t Y (t) = AY (t)+Bu(t) has the form: Y (t) = eAt Y (0) + 0 eA(t−s) Bu(s)ds.

to one problem to the other. Now, we can work on the canonical problem.

3.

CONTROLLABLE DOMAIN

In this section, we consider the canonical optimal control problem previously defined and raise the question of its controllability: how to compute the set of initial points Y0 for which the control problem (2) with the constraints Y (0) = Y0 ; Y (tf ) = 0 and ∀t ≥ 0, u(t) ∈ Um = {s1 , . . . , sp } ⊂ Rm ) admits a solution. Let us state: ∀t ≥ 0, Y (t) = (Y1 (t), Y2 (t)) where: Y1 (t) ∈ Rr and Y2 (t) ∈ Rn−r . Thus the state space splits clearly up into an uncontrollable part (Y˙ 2 = A3 Y2 ) and a controllable one (Y˙ 1 = A1 Y1 + A2 Y2 + B1 u). We study the controllability question in the two configurations.

3.1 Stabilization of the uncontrollable part

Proposition 3. The controllable domain C is a convex subset of the state space.

Let us consider the uncontrollable part: Y˙ 2 (t) = A3 Y2 (t)

(4)

The main idea of the proof (given in [20]) is to define (by convexity and at maximal time) a new control from that of some controllable points within C. Now we can introduce our under-approximation of the domain by time-reversal of the control polytope:

Clearly, 0 ∈ R is an equilibrium point of (4). Thus the target 0 is reachable from everywhere if 0 is a stable focus of (4). In other words, the matrix A3 has to be stable (all its eigenvalues have negative real parts). In the following, we prove that the non-stability of A3 involves constraints on Y2 (0), so that we can easily come down to the case of a stable matrix A3 : we apply the Schur decomposition to A3 and choose to sort its eigenvalues in such a way that: ∀i = 1 . . . k, Re(αi ) < 0 and ∀i = k + 1 . . . (n − r), Re(αi ) ≥ 0. Then there exists a unitary Q ∈ Cn×n such that: Q∗ A3 Q = D + N where D = diag(α1 , . . . , αn−r ) and N ∈ C(n−r)×(n−r) is strictly upper triangular. Moreover (4) is easily solvable: ∀t ≥ 0, Y2 (t) = eA3 t Y2 (0). Hence: n−r

Corollary 1. Let Yi (.) be the trajectory from 0 by time reversal according to u = si . If C(t) = Conv1..k (Yi (t)), then Z t e−Aτ Bu(τ )dτ C(t) ⊂ C and ∀Y ∈ C(t), ∃ a control u, Y = 0

Any point in C(t) is said controllable at least in time t and C(t) is an under-approximation of the controllable set in time t. This gives us an algorithm to build our under-approximation in time T . Nevertheless for a given time T , the quality of the approximation could be very poor (see example 1, figure 1-(a)). To refine it, we choose to discretize the time interval [0, T ] in N subintervals. The under-approximation in time T is the convex hull of under-approximations in time j ∗h for

∗

Q∗ Y2 (t) = eQ A3 Qt Q∗ Y2 (0) eNt Q∗ Y2 (0) = e2(D+N)t Q∗ Y2 (0) = eDt3 α1 t 4 4 e 5 Q∗ Y2 (0) 4 = 4 αn−r t e

280

P: “Minimize H with respect to the control variable u ∈ Um under the constraints: ∂H (Y (t), u(t), λ(t)) (5) Y˙ (t) = ∂λ ˙ T = − ∂H (Y (t), u(t), λ(t)) (6) λ(t) ∂Y

j = 1..N −1 (where h = T /N ) and the quality is significantly improved (see example 1, figure 1-(b)). We have thus defined the following algorithm, UnderApproximation, computing a set of controllable points. Algorithm 4 UnderApproximation Require: A, B, U, T, h (where U = Conv{s1 , . . . , sp }). Ensure: an under-approximation with a step h = T /N of the controllable domain in time T. 1: ApproxVertices:=[0]; 2: for all time step j (from 1 to N) do 3: for all vertex si do 4: Yi (.) = trajectory from 0 with u = si ; 5: ApproxVertices:=ApproxVertices ∪{Yi (jh)}; 6: end for 7: end for 8: Return ConvexHull(ApproxVertices);

(2D Under-Approximations). » – » – Let us con1 0 1 1 sider the system: Y˙ = Y + u with u ∈ 0 2 0 2 Conv([0, 0]T , [1, 0]T , [0, 1]T ). The following figures show in dashes under-approximations of the controllable set (represented in plain line) for three refinements. Example 1

-0,8

-0,6

-0,4

-0,2

4.1 Singular control Let us consider the optimization problem P. By definition, P is a linear program. It thus admits solutions which may occur on the boundary of the polyhedral set Um . Now, any solution (Y, u, λ) of the Hamiltonian system (5) is said to be extremal and distinguish regular and singular solutions:

-0,2

Definition 1. An extremal (Y (t), u(t), λ(t)) is called regular on an interval [t0 , t1 ], if there exists k s.t., for almost all t ∈ [t0 , t1 ],

-0,4

[l1 (Y (t)) + λT (t)B]sk < min{[l1 (Y (t)) + λT (t)B]si ; i = k}

x -1

and H(Y  (t), λ (t), u (t)) = 0 along the optimal trajectory.” Our algorithm is divided in two main steps: first, the controllable set is partitioned (see 4.1) in domains, inside which the optimal control is constant. In practice, we propose symbolic algorithms computing the boundaries of these cells (see §4.2). The second step requires to compute an optimal trajectory from an entry point to the target within each cell. In this section, the cost function l is assumed linear in the control: l(Y, u) = l0 (Y ) + l1 (Y )u. The case were this function is nonlinear is actually simple: indeed, the Hamiltonian optimization problem could be solved by classical tools,since not Hu = ∂H = 0 is then solvable in the control variable u. ∂u

0 0

y

Therefore, for any regular extremal (Y (t), u(t), λ(t)), the optimal control is given by the relation:

-0,6

-0,8

u(t) = si if [l1 (Y (t)) + λT (t)B]si < min{[l1 (Y (t)) + λT (t)B]sj }

-1

j=i

(a)

(b)

Consequently one can define a partition of the controllable set (see definition 2) as follows:

Figure 1: Under-approximations in time T = 5 (a) without refinement (N = 1) (b) by discretizing (N = 5 in dash-dots - N = 30 in dashes, nearly superposed)

4.

Definition 2. An optimal trajectory Y (.) belongs to the domain Γi on a time interval [t0 , t1 ] if the condition: ∀t ∈ [t0 , t1 ], ∀j ∈ {1, ..., m}−{i}, [l1 (Y (y))+λT (t)B](si −sj ) < 0 holds. Thus at any point of the domain Γi , the optimal control is u(.) = si and the related field vector is AY + Bsi .

OPTIMAL SOLUTIONS

Now, we introduce the switching function Si,j , that describes transitions between the domains Γi and Γj :

In this section, we present some theoretical results and algorithms for solving linear optimal control problems. The algorithm is as general and symbolic as possible to design optimal controllers. Recall that we want to control a linear system: Y˙ (t) = AY (t) + Bu(t), A ∈ Rn×n , B ∈ Rn×m from a controllable initial state Y0 to a final state Yf = 0 at an unspecified time tf using the admissible control functions u ∈ Um = Conv1..p (sp ) ⊂ Rm in such a way that: R +∞ J(Y0 , u(.)) = 0 l(Y (t), u(t))dt is minimized. According to the decomposition algorithm developed in section 2, we also assume: rk(B) = m and rk([B|AB| . . . |An−1 B]) = n. To solve this canonical system, we now introduce the Hamiltonian function: H(Y, u, λ) = l(Y, u) + λT AY + λT Bu. The pseudo-Hamiltonian formulation of the optimal control problem and the Pontryagin Minimum principle provide us the following optimization problem [18, §1], [2, §2],[17, §4]:

Definition 3 (Switching function). Si,j (t) = Hu (Y (t), u(t), λ(t))(si − sj ) = [l1 (Y (t)) + λT (t)B](si − sj ) Then, the single zeros of Si,j give us the switching time between the domains Γi and Γj . However it may also be possible to find time intervals where the switching function is identically equal to zero. This typically corresponds to the appearance of singular arcs in each face of the polyhedral control set. Thus singular trajectories are: Definition 4. [22] A trajectory Y (.) is called ij-singular on a time interval [t0 , t1 ] if the condition “Si,j (t) = 0 and ∀k = i, j, Sj,k (t) < 0” holds for almost all t ∈ [t0 , t1 ].

281

Algorithm 5 ij-singular boundary Require: i and j, indices of the considered Γ domains. Require: H(Y, v, λ). Ensure: ϕ, where ϕ(X) = 0 defines the ij-boundary Ensure: u the ij-singular optimal control. Ensure: λ the optimal Pontryagin parameter. 1: Compute the smallest integer K such that: 2K ∂ ( d Hv ) = 0 (where Hv = ∂H ) ∂v dt2K ∂v 2: if the Legendre-Clebsh (LC) condition [14, 19]: d2K ∂ (−1)K ∂v ( dt 2K Hv ) ≥ 0 is not satisfied then 3: Return “no singular solution”. 4: end if “ i ” d } {(S) is 5: Solve (S) {H = 0, Hv = 0, dt i Hv = 0 i=1..2K linear in v and λ, hence we obtain the exact singular values of v and λ in relation with Y . The remaining relation gives the equation (ϕ(Y ) = 0) of the boundary.} 6: Return (ϕ(Y ), sj + (si − sj )v(Y ), λ(Y )).

Just note that definition could be naturally extended to Isingular trajectories (I ⊂ {1, . . . , p}). According to definitions 2 and 4, we show that ij-singular trajectories geometrically correspond to the boundary between Γi and Γj . On this singular boundary, the optimal control is said to be singular and satisfies: Proposition 4. Let us consider an ij-singular trajectory Y (.) on a time interval [t0 , t1 ]. Then: ∀t ∈ [t0 , t1 ], u(t) ∈ [si , sj ]. Likewise, on an I-singular trajectory, u(t) ∈ Convk∈I (sk ).

4.2 Boundaries computation At this point of our analysis, we have partitioned our state space in domains delimited by: - singular boundaries (see e.g. [17, fully optimal problem]). - mixed and non singular boundaries (see [11, ex. 1]).

While the related boundary is bounded, the whole boundary between Γi and Γj is necessarily also made of a regular part. The next paragraph is devoted to its computation.

- non singular boundaries (see [18, time-optimal problems]). In our linear control problem, the Hamiltonian has the form: H0 (Y, λ)+H1 (Y, λ)u. Now, let us consider the boundary between domains Γi and Γj . Then, in the whole paragraph, we use H with the form: H(Y, v, λ) = H0 (Y, λ) + H1 (Y, λ)(sj + (si − sj )v) where v ∈ [0, 1] (since u ∈ [si , sj ] with proposition 4) to show how to symbolically compute the considered boundary, when it exists.

4.2.3 Mixed boundaries In this paragraph we assume that we have already computed the singular boundary between two domains Γi and Γj and check the existence condition of these boundary. So we have its equation: ϕ(Y ) = 0 under the constraint 0 ≤ v(Y ) ≤ 1, the singular control u∗ and the related λ∗ . We now want to compute the related regular boundary (see algorithm 6).

4.2.1 Switch rules In this paragraph we briefly describe a method to compute the allowable “switching directions” [11] in the state space. d From examination of the sign of dt Hv (Y (t), v, λ(t)) at switching points (i.e. Hv (Y (t), λ(t)) = 0 and H(Y (t), v, λ(t)) = 0), it is possible to determine whether switchings from u = si to u = sj are allowed in a given region of the state space.

Algorithm 6 MixedBoundary Require: i and j, indices of the considered Γ domains. Require: ϕ equation of the ij-boundary. Require: λ optimal Pontryagin parameter on the ijboundary. Ensure: a parameterization of the non singular boundary between Γi and Γj . 1: Parameterize the singular boundary (by the implicit functions theorem) :ψ(ξ) (i.e. such that ϕ(ψ(ξ)) = 0). 2: for s ∈ {si , sj } do 3: Compute the trajectory Y [ψ(ξ), s] from ψ(ξ) by time reversal with u = s: Rt Y [ψ(ξ), s](t) = eAt ψ(ξ) + eAt 0 e−Aw Bsdw 4: Solve the Euler-Lagrange equations (6) with the initial condition λ[ψ(ξ), s](0) = λ∗ (ψ(ξ)). 5: Compute the first time t(ξ) < 0 for which the switching condition between si and sj holds, i.e.: Si,j (t) = 0 (see definition 3). No solution t(ξ) invalidates the singular boundary so that the boundary between Γi and Γj is necessarily regular. 6: end for 7: Return the switching curve equations(if they exist): Y [ψ(ξ), s](t(ξ)) = 0, s ∈ {si , sj }

4.2.2 Singular boundaries In this paragraph we present a symbolic algorithm computing singular boundaries when they exist. This algorithm is essentially based on the Pontryagin maximum principle [18] and classical results in the theory of singular extrema (see [14, 19, 2] for more details). We show on table 2 some performances of this algorithm in high dimension where Um is a random simplex in Rm and n = m. Note that we still have to check that the n cpu (s) n cpu (s)

2 0.16 9 4.21

3 0.22 10 7.03

4 0.35 11 10.53

5 0.56 12 19.06

6 0.91 13 31.38

7 1.51 14 53.85

8 2.43 15 94.18

Table 2: Symbolic singular boundaries timings so-computed boundary really exist in the controllable domain and that the switching conditions are satisfied: ∀k ∈ / {i, j}, Si,k < 0. However, we show next that these conditions are not always sufficient to determine if a computed singular boundary is valid or not. Such cases appear when the computed singular control explicitly depends of the state Y .

Consider the system [11, Example 1]: 8 » – » – 0 1 1 < ˙ X(t) = X(t) + u(t) 0 0 −1 : X(0) = X0

282

(7)

where X ∈ R2 , |u| < 1 and the performance index to be RT minimized is: J(X, u(.)) = 0 21 x1 (t)2 dt. (Note that (7) already is under its canonical form since rk([B|AB]) = 2). [11] provides the full analysis of the singular controls and the graph of the allowable switching regions [11, fig. 5]. There, the singular boundary is defined by φ(x1 , x2 ) = x1 + 2x2 = 0 and −1 ≤ x2 ≤ 1 and the switching function is S(t) = 2(λ1 (t) − λ2 (t)). Due to the constraint −1 ≤ x2 ≤ 1, the singular boundary does not allow us to draw a partition of the state space. However, we are able to complete [11]’s results by the computation of the whole boundary between the controls u = 1 and u = −1: as we have a valid singular boundary, we can now apply our M ixedBoundary algorithm as shown on figure 2: 1. Parametrize the singular boundary: ψ(ξ) = (−2ξ, ξ). 2/3. For u=1, compute the trajectory from ψ(ξ) according to u = 1 by time reversal: Y [ψ(ξ), 1](t) = [− 12 t2 + ξt + t − 2ξ, −t + ξ]. 4. Solve the Euler-Lagrange equations S(0) = 0 5. Compute the first timept(ξ) such that S(t(ξ)) = 0 holds: t(ξ) = 2ξ − 2 ξ 2 − 3ξ + 3 6. Repeat steps 3., 4. and 5. with u = −1. 7. Return the parametrized switching curve equation: p p 8 (2ξ ξ 2 − 3ξ + 3 − 2ξ 2 + 6ξ − 6 − 2 ξ 2 − 3ξ + 3, > p > < −ξ + 2 ξ 2 − 3ξ + 3, p ξ ∈ [−1, 1]) p 2 > 6ξ + 6 + 2 ξ 2 + 3ξ + 3, (2ξ ξ + 3ξ + 3 + 2ξ 2 + > p : 2 −ξ − 2 ξ + 3ξ + 3, ξ ∈ [−1, 1])

5

piecewise constant with values in {si , sj }). Let us distinguish two possible configurations: - For all (i, j) ∈ {1, . . . , m}2 , i = j, the boundary between Γi and Γj is non singular. In this case, one method is to compute all the switching functions Si,j (see definition 3). After that we can study the zeros of Si,j and deduce the transition time τ between two of them. By time reversal, we start from the origin and build the switching curve. This method is well described in [17]. - There exists k ∈ {1, . . . , m} − {i, j}, such that the boundary between Γj and Γk exists and is singular. In this case we come down to the same technique than for mixed boundaries: the idea is to take a parameterization of the singular boundary between Γj and Γk . We consider by time reversal the trajectory from a point of this boundary according to u = s ∈ {sj , sk } and compute the first negative time for which the switching condition: Sj,k (t) = 0 holds. This latter algorithm, N onSingBoundary is based on the following proposition [20]: Proposition 5. Let Y (.) be an optimal trajectory from an initial point Y0 and u(.) the associated optimal control. We assume that there exists a time T > 0, such that: ∃@ > 0, Y(.) regular over [T − @, T ] and Y (T + .) is ij-singular. Then: ∀t ∈ [T − @, T ], u(t) ∈ {si , sj }. In conclusion, we have proposed three algorithms to compute boundaries between the domains Γi . We can now define a general one Boundary that compute the boundary between two given domains Γi and Γj : (ϕ, ω, u) := Boundary(i, j) where ϕ is the equation of the boundary, 1 if ϕ(Y ) = 0 is singular and ω is defined by: ω(Y ) = 0 otherwise ∗ u is the related optimal control. We therefore have all the necessary subroutines to solve our problem.

From u=+1 to u=−1

4

From u=−1 to u=+1 3

4.3 Linear Optimal Control

u=1

2

In this section, we detail the general algorithm for solving any linear optimal control problem. The principle is as follows: after a virtual partition of the state space in Γi domains, one follows the trajectory within each cell of the partition. Indeed, in each cell and in every boundaries, the control is known thanks to the algorithms of section 4.2. When the trajectory reaches a boundary, there is a switching of cell and a control change. This goes on till the target is reached. Once the canonical problem is solved with algorithm 7, we just have to apply the inverse transformation 3 to come down to optimal solutions of our initial control problem.

y 1

From u=−1 to u=+1

A

0

-1

x

From u=+1 to u=−1

B

i

From u=+1 to u=−1

-2 -15

-10

-5

0 x

Switch rules axes

5

10

From u=−1 to u=+1

Exact regular part

Approximation of the regular part

Singular boundary

Figure 2: MixedBoundary in Gibson’s problem

5. CONCLUSION We have presented an algorithm for solving general linear optimal control problems. First we propose an explicit method to transform any problem into a canonical one by the way of a block Kalman decomposition. We have also developed generic algorithms solving the canonical problem even when complex boundaries occur. Yet, two important new features of our algorithms are that we give a full generic implementation and that they are mainly symbolic. Also,all the algorithms presented here have been implemented in Maple and work in high dimensions2 .

Remark 1. In some cases the equation Si,j (t) = 0 is not solved by Maple, so that we cannot express the time t(ξ) in step 5. Nevertheless, by discretizing the given singular boundary, we can compute a discretization of the regular part of the searched boundary. Then, by interpolation, we find again the whole boundary.

4.2.4 Non Singular Boundaries In this paragraph, we consider the case where there is no singular or mixed boundary between the two domains Γi and Γj . The optimal control is then called bang-bang (i.e.

2 The maplets are available online at: www-lmc.imag.fr/ lmc-mosaic/Jean-Guillaume.Dumas/SHOC

283

Algorithm 7 LinearOptimalControl Require: A, B, Y0 , l(Y, u) = l0 (Y ) + l1 (Y )u and {s1 , . . . , sm }. Ensure: Optimal trajectory, control and value function. 1: V := 0; {Initialize switching functions} ∂ 2: Si,j = ∂u H(Y, sj + (si − sj )v, λ) = l1 (Y ) + λT B(si − sj ). {Virtual partition of the state space} 3: I = {i ∈ [|1, m|]/{λ/∀j = i, Si,j < 0} = ∅}. 4: for all i ∈ I and j ∈ I such that i < j do 5: (ϕi,j , ωi,j , ui,j ) := Boundary(i, j). ˜ j )j∈I is the induced partition of the controllable set. 6: (Γ 7: end for {Identification of the domains where u = si } 8: for all j ∈ I do S ˜ j == 9: if ∂ D k {Y s.t. ϕi,k (Y ) = 0} then ˜j . 10: Γi := Γ 11: end if 12: end for 13: k := 0; T0 := 0; {Within each cell, reach the boundary} 14: while Yk = 0 do 15: Find i s.t. Yk ∈ Γi . 16: if Yk ∈ ∂Γi then 17: Find j s.t. ϕi,j (Yk ) = 0. 18: u := ui,j (Yk ); 19: if ωi,j (Yk ) == 1 {0 is reached on this ij-singular boundary.} then 20: Tk+1 := Solution of Y [Yk , u](t) = 0; 21: break while loop; 22: end if 23: else 24: u := si ; 25: end if {Piecewise solution} 26: Compute Tk+1 = inf{t > 0; Y [Yk , u](t) ∈ ∂Γi } 27: Yk+1 := Y [Yk , u](Tk+1 ). 28: u := u for t ∈ [Tk , Tk+1 ]; 29: Y := Y [Yk , u] for t ∈ [Tk , Tk+1 ]; R T −T 30: V := V + 0 k+1 k l(Y [Yk , u](t)dt 31: end while 32: Return (Y, u , V )

trajectories from Pj that go through (or tend towards) Pj by time reversal according to u = si . Also, a rigorous proof of the convergence of our under-approximation towards the real controllable set has still to be completed.

6. ACKNOWLEDGMENTS We would like to thank Kevin Hamon for his collaboration and for the work done towards generic algorithms (see [9]).

7. REFERENCES [1] E. Asarin, T. Dang, and A. Girard. Reachability of non-linear systems using conservative approximations. In Proceedings of the 2003 Hybrid Systems: Computation and Control, pages 20–35, Apr. 2003. [2] A. Bryson and Y. Ho. Applied Optimal Control. Hemisphere, 1975. [3] D. Delchamps. State Space and Input-Output Linear Systems. Springer-Verlag, 1988. [4] J. Della Dora, A. Maignan, M. Mirica-Ruse, and S. Yovine. Hybrid computation. In ISSAC’2001, London, Ontario, July 2001. [5] J.-G. Dumas, T. Gautier, and C. Pernet. Finite field linear algebra subroutines. In ISSAC’2002, Lille, France, pages 63–74, July 2002. [6] J.-G. Dumas, P. Giorgi, and C. Pernet. FFPACK: Finite Field Linear Algebra Package. In ISSAC’2004, Santander, Spain, July 2004. [7] R. Fierro, A. K. Das, V.Kumar, and J. P. Ostrowski. Hybrid control of formations of robots. 2001. [8] G. H. Golub and C. F. Van Loan. Matrix computations third ed.. Johns Hopkins University Press, Baltimore, MD, USA, 1996. [9] K. Hamon. Contrˆ ole optimal et algorithme de calcul des trajectoires. Technical report, LMC-IMAG, 2004. [10] O. Ibarra, S. Moran, and R. Hui. A generalization of the fast lup matrix decomposition algorithm and applications. Journal of Algorithms, 3:45–56, 1982. [11] C. Johnson and J. Gibson. Singular solutions in problems of optimal control. IEEE Transactions on Automatic Control, 8:4–15, 1963. [12] R. Kalman. Canonical structure of linear dynamical systems. In Proceedings of the National Academy of Sciences, pages 596–600, 1961. [13] R. Kalman. Mathematical description of linear dynamical systems. Siam Journal on Control, 1:152–292, 1963. [14] H. Kelley, R. Kopp, and H. G. Moyer. Singular Extremals, pages 63–101. Academic Press, 1967. [15] C. Pernet. Calcul du polynˆ ome caract´eristique sur des corps finis. Master’s thesis, Universit´e Joseph Fourier, jun 2003. [16] H. Pesch. A practical guide to the solutions of real-life optimal control problems. Parametric Optimization. Control Cybernet, 23:7–60, 1994. [17] E. Pinch. Optimal Control and the Calculus of Variations. Oxford University Press, 1993. [18] L. Pontryagin, V. Boltiansky, R. Gamkrelidze, and E. Michtchenko. Th´ eorie math´ ematique des processus optimaux. Editions de Moscou, 1974. [19] H. Robbins. A generalized Legendre-Clebsch condition for the singular cases of optimal control. IBM Journal of Research and Development, 11(4):361–372, 1967. [20] A. Rondepierre and J.-G. Dumas. Hybrid optimal control of dynamical systems. Technical report IMAG-ccsd -00004191, arXiv math.OC/0502172. Fvrier 2005. [21] B. D. Saunders. Black box methods for least squares problems. In ISSAC’2001, London, Ontario, 2001. [22] M. Zelikin and V. Borisov. Optimal chattering feedback control. Journal of Mathematical Sciences, 114(3):1227–1344, 2003.

Further developments already are in progress: • Complete the whole algorithm for a cost function nonlinear in the control. In this case, the Hamiltonian optimization =0 problem could be solved by classical tools. Indeed, ∂H ∂u can now be solved in the control variable u. • The UnderApproximation and solving algorithms have been performed on linear dynamical systems under the canonical form where A2 = 0 (see §3.2 and section 4). These two algorithms have to be extended for any canonical form (see (2)). In practice, this corresponds to the appearance of a perturbation time function t → A2 eA3 t Y2 (0) in the dynamical system. The technique does not change, but practical implementations are slightly more complex. • The UnderApproximation could be refined and a study of the approximation error has still to be made. The idea is to consider cases where the dynamical system for u = si admits one (or an infinite number of) equilibrium point Pi (note that 0 is an equilibrium point when u = 0). The underapproximation can e.g. be completed by the convex hull of

284

Finding Telescopers with Minimal Depth for Indefinite Nested Sum and Product Expressions Carsten Schneider∗ Research Institute for Symbolic Computation J. Kepler University, A-4040 Linz, Austria

[email protected] in constructing a ΠΣ∗ -extension and in solving problem A, that covers besides telescoping also creative telescoping [9]. In particular, if G = K, i.e., (F, σ) is a ΠΣ∗ -field, and if K is σ-computable1 , all these steps (constructing (F, σ) and solving A) can be done completely automatically; see [4]. A: Given a ΠΣ∗ -extension (F, σ) of (G, σ) and f = (f1 , . . . , fn ) ∈ Fn . Find all g ∈ F and c = (c1 , . . . , cn ) ∈ Kn with P (1) σ(g) − g = cf , where cf = n i=1 ci fi .

ABSTRACT We provide algorithms that find, in case of existence, indefinite nested sum extensions in which a (creative) telescoping solution can be expressed with minimal nested depth. Categories and Subject Descriptors: I.1.2 [Symbolic and Algebraic Manipulation]: Algebraic algorithms General Terms: Algorithms Keywords: Difference fields, symbolic summation

1.

Within this approach it is crucial to find an appropriate (F, σ) so that the depth can be reduced by telescoping. For instance, finding (F, σ) is not so obvious, if one wants to reduce the depth of the left hand sides as follows, k n n n 3 (3) X X 1 X Hj Hj X Hj (j Hj − 1) (3) = H − , (2) n k3 j=1 j 2 j2 j5 j=1 j=1 k=1 !2 (2) (2) n k n X X X Hj Hj (2) 2 (4) = −(H + H ) + (n + 1)× n n j3 j3 j=1 j=1 k=1 !2 (2) (2) (2) (2) (4)
n n X X Hj Hj (jHj )2 − Hj + j 2 Hj × + ; j3 j5 j=1 j=1 P Pn 1 (r) 1 here Hn := n i=1 i and Hn := i=1 ir , r > 1. E.g., if one wants to find the right hand side of (2), one needs the sum P (3) 3 (3) extensions Hn and n − 1)/j 5 which do not j=1 Hj (j Hj occur on the left hand side; see Example 1.1. In short, using [1, 7] these two extensions must be adjoined manually. Subsequently, we solve this problem automatically by solving problem B.

INTRODUCTION

Indefinite nested sums and products in rational terms can be represented in ΠΣ-extensions [1]. More precisely, take a difference field (G, σ), i.e., a field G together with a field automorphism σ : G → G, and let K be its constant field, i.e., K = constσ G := {k ∈ G | σ(k) = k}. Then a ΠΣ∗ -extension (F, σ) of (G, σ), a restricted version of ΠΣ-extensions, is a difference field with constσ F = K of the following form: F = G(t1 ) . . . (te ) is a rational function field and σ : F → F is extended from G to F by the recursively defined application σ(ti ) = ai ti (product) or σ(ti ) = ti + ai (sum) with ai ∈ G(t1 ) . . . (ti−1 ) for 1 ≤ i ≤ e. It is a Σ∗ -extension (resp. Π-ext.), if for all ti we have σ(ti ) = ti + ai (resp. σ(ti ) = ai ti ); (F, σ) is a ΠΣ∗ -field over K if G = K. Note that the nested depth of these sums/products gives a measure of the complexity of expressions. For instance, the sum of the left hand side of (2) has depth four, whereas the expression on the right hand side has only depth three. In this article we try to reduce the depth of such indefinite sums by telescoping: First construct a ΠΣ∗ -field, say (F, σ), in which the summand f ∈ F can be represented, and afterwards try to find a telescoper g ∈ F for σ(g) − g = f where the depth of g is not larger thanP the depth of f . Given such a g, one gets, roughly speaking, n k=0 f (k) = g(n+1)−g(0). So far, methods have been developed in [1, 7] that assist

B: Given a ΠΣ∗ -extension (F, σ) of (G, σ), K := constσ F and f ∈ Fn . Decide if there are 0 6= c ∈ Kn and g in a ΠΣ∗ -extension (F(x1 ) . . . (xr ), σ) of (F, σ) such that (1) and2 δG (g) = δG (cf ) hold; if yes, compute such a solution. Remark: In [5] we considered the special case σ(xi ) − xi ∈ F which is too restricted to find the closed forms from above. More precisely, we introduce depth-optimal ΠΣ∗ -extensions, a special class of ΠΣ∗ -extensions, in which we can represent constructively nested sums and products and in which we can solve B; see Theorems 6 and 8. It turns out that only Σ∗ -extensions are needed to solve problem B. The resulting algorithms are implemented in the summation package Sigma [4].

∗ Supported by the SFB-grant F1305 and the grant P16613N12 of the Austrian FWF.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. ISSAC’05, July 24–27, 2005, Beijing, China. Copyright 2005 ACM 1-59593-095-7/05/0007 ...$5.00.

1 E.g., a rational function field A(x1 , . . . , xr ) over an algebraic number field A is σ-computable; for a definition see [6]. 2 δG (f ) denotes the maximal depth of the sums and products occurring in f ∈ F over the ground field G; see Sec. 2.

285

2.

PROPERTIES AND DEFINITIONS

G set δ(g) = 0. If δG is defined for (G(t1 ) . . . (ti−1 ), σ) with i > 1, we define δ(ti ) = δG (ai ) + 1, and for g ∈ G(t1 ) . . . (ti ) we define3 δ(g) = max({δG (ti ) | ti occurs in g} ∪ {0}). We define δG (f ) = maxi δG (fi ) for f = (f1 , . . . , fn ) ∈ Fn . The depth of (F, σ) over G, δG (F), is defined by δG ((0, t1 , . . . , te )). Convention: Throughout this article the depth is defined over (G, σ); we set δ := δG . We might use the depth-function without mentioning G. Then we assume that the corresponding difference fields are ΠΣ∗ -extensions of (G, σ). In all our examples we will assume that G = Q. • Let (F(t1 ) . . . (te ), σ) be a ΠΣ∗ -extension of (F, σ) with σ(ti ) = αi ti + βi . This extension is called ordered if δ(ti ) ≤ δ(ti+1 ). The extension has maximal depth d if δ(ti ) ≤ d. If there is a permutation τ : {1, . . . , e} → {1, . . . , e} with ατ (i) , βτ (i) ∈ F(tτ (1) ) . . . (tτ (i−1) ) for all 1 ≤ i ≤ e, then the generators of the ΠΣ∗ -extension (F(t1 ) . . . (te ), σ) of (F, σ) can be reordered without changing the ΠΣ∗ -nature of the extension. In short, we say that (F(t1 ) . . . (te ), σ) can be reordered to (F(tτ (1) ) . . . (tτ (e) ), σ) if there exists such a τ . On the rational function field level we identify two such fields. • Let (F, σ) and (F0 , σ 0 ) be difference fields. Then a σ-monomorphism/σ-isomorphism is a field monomorphism/isomorphism τ : F → F0 with σ 0 (τ (a)) = τ (σ(a)) for all a ∈ F. Suppose that (F, σ) is a ΠΣ∗ -extension of (H, σ). A σ-monomorphism/σ-isomorphism τ : F → F0 is called an H-monomorphism/H-isomorphism if τ (a) = a for all a ∈ H. The proofs of the following statements can be found in [8].

First we work out that the problem to represent sums in Σ∗ -extensions can be reduced to telescoping. Theorem 1. [1] Let (F(t), σ) be a difference field. (1) Then this is a Π-extension of (F, σ) iff σ(t) = a t, t 6= 0, . a ∈ F∗ and there is no n 6= 0 and g ∈ F∗ with an = σ(g) g (2) Then this is a Σ∗ -extension of (F, σ) iff σ(t) = t + a, t∈ / F, a ∈ F, and there is no g ∈ F with σ(g) − g = a. Namely, Theorem 1.2 shows that indefinite summation/telescoping and building up Σ∗ -extensions are closely related. E.g., if one fails to find a g ∈ F with σ(g) − g = f ∈ F, i.e., one cannot solve the telescoping problem in F, one can adjoin the solution t with σ(t) + t = f to F in form of the Σ∗ -extension (F(t), σ) of (F, σ). Note that with similar techniques one can represent products in Π-extensions; see [6]. Summarizing, by solving A, nested sums can be represented in Σ∗ -extensions. In Section 3 we will show that by refined telescoping we obtain also refined Σ∗ -extensions. These new aspects can be illustrated as follows. Example 1. (1) Given the left hand side of (2), say Sn , telescoping produces the ΠΣ∗ -field (Q(t1 )(t2 )(t3 )(t4 ), σ) over Q with σ(t1 ) = t1 + 1, σ(t2 ) = t2 + t11+1 and σ(tr+1 ) = tr+1 + σ( ttrr ) for r = 2, 3. Namely, there is no g ∈ Q(t1 ) with 1

σ(g) − g = t11+1 and no g ∈ Q(t1 ) . . . (tr ) with σ(g) − g = σ( ttrr ) for r = 2, 3. Here t4 represents Sn with depth 4. We 1 can improve this situation by solving problem B for F := Q(t1 )(t2 )(t3 ): We obtain the Σ∗ -extension (F(x1 )(x2 ), σ) of (1+(t1 +1)t2 )x1 1 (F, σ) with σ(x1 ) = x1 + (t1 +1) 3 , σ(x2 ) = x2 + (t1 +1)3 t3 and g := x1 t3 − x2 such that σ(g) − g = σ( t3 ); see Exp. 6.2.

Lemma 1. Let (F(x), σ) with σ(x) = αx + β be a ΠΣ∗ extension of (F, σ). (1) Let a, f ∈ F and suppose there is a solution g ∈ F(x) with σ(g) − ag = f , but no solution in F. If x is a Π-extension then f = 0 and a = σ(h) αm for some h ∗ h ∈ F, m 6= 0; if x is a Σ -extension then f 6= 0 and a = 1. (2) Let (F(x)(t), σ) be a Π-extension of (F(x), σ) with α0 := σ(t)/t ∈ F. Let (a1 , a2 ) ∈ F be homogeneous over F, a0 := (a1 α0i , a2 ) with i 6= 0 and f ∈ Fn . Then a0 is inhomogeneous over F(x) and V(a0 , f , F) = V(a0 , f , F(x)).

1

Then Sn is represented by g, which gives (2). (2) Suppose we have represented Sn with t4 ∈ Q(t1 ) . . . (t4 ) as above, and suppose that we want to simplify Sn0 given on the left hand side of (3). Then we adjoin the Σ∗ -extension (3) 1 x1 with σ(x1 ) = x1 + (t1 +1) , and 3 in order to represent Hj

Proposition 1. Let (E, σ) be a ΠΣ∗ -extension of (F, σ) with K := constσ F. Then the following holds: (1) Let a ∈ F2 be homogeneous over F and f ∈ F. If there is a g ∈ E \ F with σa g = f then there is not such a g in F. i) ∈ F. If (2)Let E = F(t1 ) . . . (te ) with σ(ti ) − ti ∈ F or σ(t ti Pe σ(g) − g = f for g ∈ E then g = i=1 ci ti + w where ci ∈ K i) and w ∈ F; moreover, ci = 0, if σ(t ∈ F. ti 0 (3) Let (F(t), σ) and (F(t ), σ) be Σ∗ -extensions of (F, σ) with g ∈ F(t0 ) \ F s.t. σ(g) − g = σ(t) − t. Then there is an F-isomorphism τ : F(t) → F(t0 ) with τ (t) = g. (4) Let τ : F → F0 be a σ-isomorphism for (F, σ), (F0 , σ). Then there is a ΠΣ∗ -extension (E0 , σ) of (F0 , σ) with a σisomorphism τ 0 : E → E0 where τ 0 (a) = τ (a) for all a ∈ F.

1 +1)t2 )x1 look for a solution of σ(g) − g = (1+(t . In this case (t1 +1)3 we compute g = x1 t3 + t4 which is reflected by the identity n n k n (3) X X X X Hj (j 3 Hj −1)) Hj Hj (3) 1 = H − , (3) n j5 j2 k3 j2

j=1

j=1

k=1

j=1

i.e., we have increased the depth by telescoping! This examples illustrates the advantages of (F, σ) with F := Q(t1 )(t2 )(t3 )(x1 )(x2 ): Sn can be represented by x1 t3 − x2 , and Sn0 is given by x2 with depth 3. Finally, we introduce further definitions and properties. • Let (F, σ) be a difference field with K = constσ F, a = (a1 , a2 ) ∈ F2 , f = (f1 , . . . , fn ) ∈ Fn and p ∈ F. We write σa p := a1 σ(p) + a2 p, σ(f ) := (σ(f1 ), . . . , σ(fn )) and f p := (f1 p, . . . , fn p). a is called homogeneous over F if a1 a2 6= 0 and σa g = 0 for some g ∈ F∗ . Let V be a subspace of F over K and suppose that a 6= 0. Then we define the solution space V(a, P f , V) as the subspace (c1 , . . . , cn , g) ∈ Kn × V | σa g = n i=1 ci fi of the vector space Kn × F over K. Note that the dimension is at most n + 1; see [1]. Summarizing, problem A is solved if one finds a basis of V(a, f , F). • Let (G(t1 ) . . . (te ), σ) be a ΠΣ∗ -extension of (G, σ) with σ(ti ) = ai ti or σ(ti ) = ti + ai . Then the depth-function over G, δG : G(t1 ) . . . (te ) → N0 , is defined as follows. For any g ∈

3.

DEPTH-OPTIMAL ΠΣ∗ -EXTENSIONS

In this section we introduce depth-optimal ΠΣ∗ -extensions and motivate its relevance to symbolic summation. Afterwards we show how the problem to represent sums in δoptimal extensions and how problem B can be reduced to problem C given below. To this end, we develop algorithms that solve C in Section 5. A Σ∗ -extension (F(s), σ) of (F, σ) with σ(s) = s + f is 3

286

g is given by g = g1 /g2 with g1 , g2 ∈ G[t1 , . . . , ti ] coprime.

called depth-optimal, in short δ-optimal, if there is no ΠΣ∗ extension (H, σ) of (F, σ) with maximal depth δ(f ) such that σ(g) − g = f holds for some g ∈ H \ F. A ΠΣ∗ -extension (F(t1 ) . . . (te ), σ) of (F, σ) is called δ-optimal if all the Σ∗ extensions are δ-optimal. First we give some examples; for a proof of Lemma 2 see [8].

(F(t1 )(x), σ). Note that (F(x)(t1 ), σ) is a ΠΣ∗ -extension of (F, σ). If t1 is a Π-extension, we are done. Otherwise, suppose that t1 is a Σ∗ -extension which is not δ-optimal. Then there is a ΠΣ∗ -extension (H, σ) of (F(x), σ) with maximal depth δ(t1 ) − 1 and g ∈ H with σ(g) − g = σ(t1 ) − t1 . Since δ(x) < δ(t1 ), (H, σ) is a ΠΣ∗ -extension of (F, σ) with maximal depth δ(t1 ) − 1. Consequently, (F(t1 ), σ) is not a δ-optimal Σ∗ -extension of (F, σ), a contradiction.

Lemma 2. A ΠΣ∗ -extension (G(t1 ) . . . (te ), σ) of (G, σ) with δ(ti ) ≤ 2, σ(t1 ) = t1 + 1 and constσ G = G is δ-optimal.

• Now we can show that a δ-optimal Σ∗ -extension (S, σ) of (F, σ) is “depth-optimal”: Given a ΠΣ∗ -extension (H, σ) of (F, σ), one can construct a ΠΣ∗ -extension (E, σ) of (S, σ) in which the elements of H can be embedded by an Fmonomorphism τ : H → E without increasing the depth. Remark. (E, σ) and τ can be computed, if one can solve A.

Example 2. Consider the ΠΣ∗ -field (Q(t1 )(t2 )(t3 )(t4 ), σ) from Exp. 1.1. t1 , t2 are δ-optimal extensions by Lemma 2. Moreover, t3 is δ-optimal by Exp. 4. t4 is not δ-optimal since we find the extension (Q(t1 )(t2 )(t3 )(x1 )(x2 ), σ) and g := x1 t3 − x2 s.t. σ(g) − g = σ(t4 ) − t4 . Later we will see that the reordered extension (Q(t1 )(t2 )(x1 )(t3 )(x2 ), σ) is δ-optimal; see Exp. 5 for x1 and Exp. 6.2 for x2 .

Theorem 3. Let (F, σ) be a ΠΣ∗ -extension of (G, σ); let (S, σ) be a Σ∗ -extension of (F, σ) which gives a δ-optimal ordered extension of (G, σ) by reordering. Then for any ΠΣ∗ ext. (H, σ) of (F, σ) with maximal depth d there is a ΠΣ∗ extension (E, σ) of (S, σ) with maximal depth d and an Fmonomorphism τ : H → E where δ(τ (a)) ≤ δ(a) for a ∈ H.

Next, we work out some important properties. • In Example 1.2 we have illustrated that in a ΠΣ∗ -extension (F, σ) of (G, σ) we might arrive at a solution g ∈ F of σ(g) − g = f with f ∈ F where δ(g) > δ(f ) + 1. This bad situation cannot happen in δ-optimal extensions; see Theorem 2.

Proof. Let (D, σ) be the δ-optimal ordered ΠΣ∗ -ext. of (F, σ) that we get by reordering the Σ∗ -extension (S, σ) of (F, σ). Moreover, let (H, σ) be a ΠΣ∗ -extension of (F, σ) with maximal depth d, i.e., H := F(t1 ) . . . (te ) with di := δ(ti ) ≤ d. Suppose that δ(ti ) ≤ δ(ti+1 ), otherwise we can reorder it without loosing any generality. We will show that there is a ΠΣ∗ -extension (E, σ) of (D, σ) with maximal depth d and an F-monomorphism τ : H → E with δ(τ (a)) ≤ δ(a) for all a ∈ H. Then reordering of (D, σ) proves the corresponding result for the extension (S, σ) of (F, σ). Besides this we will show that there is a Σ∗ -extension (A, σ) of (H, σ) and a σ-isomorphism ρ : E → A such that ρ(τ (a)) = a for all a ∈ F(t1 ) . . . (te ); this property is needed to handle the Π-extension case in the proof step (II). Induction base: If e = 0, i.e., H = F, the statement is proven by taking (E, σ) := (D, σ) with the F-monomorphism τ : F → D where τ (a) = a for all a ∈ F and by taking (A, σ) := (S, σ) with the σ-isomorphism ρ : D → A where ρ(a) = a for all a ∈ D. Induction assumption: Otherwise, suppose that 1 ≤ i < e and write H0 := F(t1 ) . . . (ti−1 ). Assume that there is a ΠΣ∗ -extension (E, σ) of (D, σ) with maximal depth di−1 and a Σ∗ -extension (A, σ) of (H0 , σ) with A := H0 (s1 ) . . . (sr ) together with an F-monomorphism τ : H0 → E with δ(τ (a)) ≤ δ(a) for all a ∈ H0 and a σ-isomorphism ρ : E → A with ρ(τ (a)) = a for all a ∈ H0 . Induction step: (I) First suppose that ti is a Σ∗ -extension. Define f := τ (σ(ti ) − ti ) ∈ E. Note that

Lemma 3. Let (E, σ) with E = F(t1 ) . . . (te ) be a δ-optimal ordered ΠΣ∗ -extension of (F, σ) where d := δ(F), δ(ti ) > d. (1) If σ(g) − g = f for g ∈ E and f ∈ F with δ(f ) < d, then δ(g) ≤ d. (2) If (F(x1 ) . . . (xr ), σ) is a Σ∗ -extension of (F, σ) with βi := σ(xi ) − xi and δ(xi ) ≤ d, then there is the Σ∗ extension (E(x1 ) . . . (xr ), σ) of (E, σ) with σ(xi ) = xi + βi . Proof. (1) Suppose we have σ(g) − g = f with g ∈ Pk E and m := δ(g) > d. By Prop. 1.2, g = i=1 ci ti + h ∗ where tk is a Σ -ext. with δ(tk ) = m and ck 6= 0. Set H := F(t1 ) . . . (tk−1 ). By Prop. 1.1 there is no g 0 ∈ H∗ with σ(g 0 ) − g 0 = f . Therefore by Thm. 1 one can construct a Σ∗ -ext. (H(s), σ) of (E, σ) with σ(s) = s + f where δ(s) = δ(f ) + 1 ≤ d < m. Note that σ(g 0 ) − (g 0 ) = σ(tk ) − tk with P g 0 = (s − k−1 i=1 ci ti )/ck ∈ H(s). Hence tk is not δ-optimal, a contradiction. Therefore δ(g) ≤ d. (2) For r = 0 we are done. Otherwise, let i ≥ 1 be minimal s.t. (E(x1 ) . . . (xi ), σ) is not a Σ∗ -ext. of (E, σ). Then there is a g ∈ E(x1 ) . . . (xi−1 ) with σ(g) − g = βi . By Lemma 3.1 it follows that δ(g) ≤ d, i.e., g ∈ F(x1 ) . . . (xi−1 ), a contradiction to Thm. 1.2. Theorem 2. Suppose that (F, σ) is a δ-optimal ordered ΠΣ∗ -extension of (G, σ) and f ∈ F∗ . If σ(g) − g = f for g ∈ F∗ then δ(f ) ≤ δ(g) ≤ δ(f ) + 1. Proof. Since δ(σ(g) − g) ≤ δ(g), δ(f ) ≤ δ(g). If δ(F) = δ(f ), δ(g) = δ(f ). Otherwise, take the δ-optimal ordered ΠΣ∗ -ext. (F, σ) of (H, σ) with F = H(t1 ) . . . (te ), δ(H) = δ(f )+1, δ(ti ) > δ(f )+1. By Lemma 3.1, δ(g) ≤ δ(f )+1.

δ(f ) ≤ δ(σ(ti ) − ti ) < di

• We show a reordering property; the general case that reordering gives again a δ-optimal extension is skipped here.

(4)

by assumption. (I.i) Suppose that there is no g ∈ E with σ(g)−g = f . Then we can construct the Σ∗ -ext. (E(y), σ) of (E, σ) with σ(y) = y+f by Thm. 1 and the F-monomorphism τ 0 : H0 (ti ) → E with τ 0 (a) = τ (a) for all a ∈ H0 and τ 0 (ti ) = y. With (4) we have δ(y) = δ(f ) + 1 ≤ di and consequently, using our induction assumption, δ(τ 0 (a)) ≤ δ(a) for all a ∈ H(ti ). Moreover, the Σ∗ -ext. (E(y), σ) of (D, σ) has maximal depth di . Furthermore, by Prop. 1.4 we can construct a Σ∗ -ext. (A(ti ), σ) of (A, σ) with the σ-isomorphism ρ0 : E(y) → A(ti ) with ρ0 (a) = ρ(a) for all a ∈ A and ρ0 (y) = ti . Hence ρ0 (τ (a)) = a for all a ∈ H0 and ρ0 (τ 0 (ti )) =

Lemma 4. Let (F(t1 ) . . . (te )(x), σ) be a ΠΣ∗ -extension of (F, σ) where (F(t1 ) . . . (te ), σ) is a δ-optimal extension of (F, σ) and δ(x) < δ(ti ). By reordering (F(x)(t1 ) . . . (te ), σ) is a δ-optimal ΠΣ∗ -extension of (F(x), σ). Proof. If e = 0 nothing has to be shown. Suppose that the lemma holds for e ≥ 0. Consider (F(t1 ) . . . (te )(x), σ) as claimed above with e > 0. Then by the induction assumption (F(t1 )(x)(t2 ) . . . (te ), σ) is a δ-optimal ΠΣ∗ -extension of

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ρ0 (y) = ti , i.e., ρ0 (τ 0 (a)) = a for all a ∈ H0 (ti ). By reordering we get a Σ∗ -ext. (A0 , σ) of (H(ti ), σ) with our isomorphism ρ0 : E(y) → A0 . This shows the induction step for this particular case. (I.ii) Suppose there is a y ∈ E with σ(y) − y = f . Since (E, σ) is a ΠΣ∗ -extension of (D, σ) with maximal depth di−1 ≤ di , we can apply Lemma 4 and obtain by reordering of (E, σ) a δ-optimal ordered ΠΣ∗ extension (G(z1 ) . . . (zl )(x1 ) . . . (xu ), σ) of (G(z1 ) . . . (zl ), σ) where δ(G(z1 ) . . . (zl )) ≤ di and δ(xj ) > di . Hence with (4) we can apply Lemma 3.1 and get δ(y) ≤ di , i.e., δ(y) ≤ δ(ti ). In particular, we get the F-monomorphism τ 0 : H0 (ti ) → E with τ 0 (a) = τ (a) for all a ∈ H0 and τ 0 (ti ) = y. Then by the previous considerations and our induction assumption it follows that δ(τ 0 (a)) ≤ δ(a) for all a ∈ H0 (ti ). What remains to show is that there is a Σ∗ -ext. (A0 , σ) of (H0 (ti ), σ) with a σ-isomorphism ρ0 : E → A0 with ρ0 (τ 0 (a)) = a for all a ∈ H0 (ti ). Define h := ρ(y) ∈ A. Then σ(h) − h = ρ(f ). Let j be minimal such that h ∈ H0 (s1 ) . . . (sj ). By Prop. 1.3, (H0 (s1 ) . . . (sj−1 )(sj ), σ) and (H0 (s1 ) . . . (sj−1 )(ti ), σ) are isomorphic with λ(a) = a for all a ∈ H0 (s1 ) . . . (sj−1 ) and λ(ti ) = h. Hence the reordered (H0 (ti )(s1 ) . . . (sj−1 ), σ), (H0 (s1 ) . . . (sj−1 )(sj ), σ) are isomorphic with λ. By Prop. 1.4 there is a Σ∗ -ext. (A0 , σ) of (H0 (ti )(s1 ) . . . (sj−1 ), σ) which is isomorphic to (A, σ) with λ : A0 → A where λ(a) = a for a ∈ H0 and λ(ti ) = h. Take the σ-isomorphism ρ0 : E → A0 with ρ0 (a) := λ−1 (ρ(a)) for all a ∈ E. Then ρ0 (τ 0 (a)) = λ−1 (ρ(τ (a))) = λ−1 (a) = a for a ∈ H0 and ρ0 (τ 0 (ti )) = λ−1 (ρ(y)) = λ−1 (h) = ti , i.e., ρ0 (τ 0 (a)) = a for a ∈ H0 (ti ). (II) Suppose that ti is a Π-ext., i.e., α := σ(ti )/ti ∈ H0 . Moreover, assume that there is a g ∈ E and an n > 0 with σ(g)/g = τ (α)n . Then there is a g 0 ∈ A with σ(g 0 )/g 0 = ρ(τ (α))n = αn . Let j be minimal s.t. g 0 ∈ H0 (s1 ) . . . (sj ). We have j ≥ 1, since otherwise ti is not a Π-ext. over H0 . Applying Lemma 1.1 shows that such a solution g 0 ∈ H0 (s1 ) . . . (sj ) does not exist, a contradiction. Therefore, there is a Π-extension (E(y), σ) of (E, σ) with σ(y) = f y where f := τ (α). Now we can follow the proof idea as in case (I.i) to complete the induction step; see [8].

Lemma 6.3 this will always happen, if δ(f ) < δ(F). Otherwise, if there is no such g and δ(F) = δ(f ), take the Σ∗ -extension (F(s), σ) of (F, σ) with σ(s) − s = f and δ(s) = δ(F) + 1. Then we can apply the following result. Theorem 5. Let (F, σ) be a ΠΣ∗ -extension of (G, σ), and let (S, σ) be a Σ∗ -extension of (F, σ) which gives a δ-optimal ordered extension of (G, σ) by reordering. Let 0 6= a ∈ F2 and f ∈ Fn . If (a, f ) is S-complete and V(a, f , S) = V(a, f , F), then (a, f ) is F-complete. Proof. Suppose (a, f ) is not F-complete, i.e., there is a ΠΣ∗ -extension (H, σ) of (F, σ) with maximal depth d := δ(F) and g ∈ H \ F, c ∈ Kn s.t. σa g = cf . By Thm. 3 there is a ΠΣ∗ -extension (E, σ) of (S, σ) with maximal depth d and an F-monomorphism τ : H → E. Hence σa τ (g) = cf . Since (a, f ) is S-complete, τ (g) ∈ S. By V(a, f , S) = V(a, f , F), τ (g) ∈ F; a contradiction. Namely, by Theorem 5 ((1, −1), (f )) is F-complete. Hence by Theorem. 4 s is δ-optimal, and thus (F(s), σ) is a δoptimal ordered ΠΣ∗ -extension of (G, σ); see Example 4. • Solving B: Suppose we have given a δ-optimal ordered ΠΣ∗ -extension (F, σ) of (G, σ) and given f ∈ Fn . Then Theorem 6 tells us how we can solve B by solving C. Theorem 6. Let (F, σ), (S, σ) be as in Theorem 5. Suppose that ((1, −1), f ) is S-complete for f ∈ Fn . Then the following holds: If there is a solution g and c ∈ Kn for problem B, there is also a g ∈ S with (1) and δ(g) = δ(cf ). Proof. Let (S, σ) be such an extension and suppose that we have a solution of B, i.e., a ΠΣ∗ -extension (H, σ) of (F, σ) with H = F(x1 ) . . . (xr ), a c ∈ Kn and a g ∈ H∗ with σ(g) − g = c f =: f and δ(g) = δ(f ). Hence δ(g) ≤ d := δ(F). Remove all xi from H where δ(xi ) > d. This gives a ΠΣ∗ extension (H0 , σ) of (S, σ) with maximal depth d where g ∈ H0 . By Thm. 3 there is a ΠΣ∗ -extension (E, σ) of (S, σ) with maximal depth d and an F-monomorphism τ : H0 → E. Thus σ(τ (g)) − τ (g) = f with τ (g) ∈ E. Since ((1, −1), f ) is S-complete, τ (g) ∈ S.

Finally, we explain how problem B and the problem to represent sums in δ-optimal extensions can be solved. Let (F, σ) be a ΠΣ∗ -extension of (G, σ) with d := δ(F), a ∈ (F∗ )2 , and 0 6= f ∈ Fn . (a, f ) is called F-complete, if for any ΠΣ∗ -extension (E, σ) of (F, σ) with maximal depth d we have V(a, f , E) = V(a, f , F). We get immediately

Remark: The two problems from above are closely related. Namely, if one represents sums in δ-optimal Σ∗ -extensions as suggested above, we actually try to solve B with f = (f ); see Exp. 6. Only if this fails, we construct a δ-optimal extension (E, σ) s.t. g ∈ E with σ(g) − g = f and δ(g) = δ(f ) + 1.

Theorem 4. Suppose (F(s), σ) is a Σ∗ -extension of (F, σ) with σ(s) = s + f and δ(s) = δ(F) + 1. Then the extension s is δ-complete iff ((1, −1), (f )) is F-complete.

4.

EXTENSION-STABLE REDUCTIONS

We sketch a reduction strategy presented in [7]. • With this reduction one can solve problem A if one can solve problem A in the ground field (G, σ) (see Base case I), and one can compute certain bounds (see Boundings). • Afterwards we show some properties of this reduction in Lemma 5, which is the starting point for further refinements. Namely, we modify the reduction strategy in Section 5 (see Remark 1) which finally enables us to solve C. Let (E, σ) with E = G(t1 ) . . . (te ) be a ΠΣ∗ -extension of (G, σ), K := constσ G, 0 6= a = (a1 , a2 ) ∈ E2 and f ∈ En . The reduction strategy for (a, f , E): If a1 a2 = 0, a basis is immediate. Hence suppose a ∈ (E∗ )2 . Base case I: If e = 0, take a basis of V(a, f , G). Denote H := G(t1 ) . . . (te−1 ), t := te ; suppose σ(t) = α t + β. Boundings: First a denominator bound is needed, i.e., a

Now the crucial observation is that problem B and the problem to represent sums can be reduced to problem C: Given a δ-optimal ordered ΠΣ∗ -ext. (F, σ) of (G, σ), a homogenous a ∈ F2 and f ∈ Fn . Find a Σ∗ -ext. (S, σ) of (F, σ) where (S, σ) is a δ-optimal ordered ΠΣ∗ -extension of (G, σ) by reordering and where (a, f ) is S-complete. • Representing sums in δ-optimal extensions: Suppose we have given a δ-optimal ordered ΠΣ∗ -extension (F, σ) of (G, σ) and given f ∈ F. Then by solving C we obtain a ΠΣ∗ -extension (S, σ) of (F, σ) which can be reordered to a δ-optimal ordered ΠΣ∗ -extension of (G, σ) and where ((1, −1), (f )) is S-complete. If there is a g ∈ S with σ(g) − g = f , we can represent the sum by g ∈ S; see Exp. 6.2. By

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d ∈ H[t]∗ such that for all c ∈ Kn and g ∈ H(t) with σa g = cf we have d g ∈ H[t]. Given such a d, define a0 = (a01 , a02 ) := (a1 /σ(d), a2 /d) q ∈ H[t]2 and f 0 := f q ∈ H[t]n for some q ∈ H(t)∗ ; more precisely, take a q such that the denominators are cleared and common factors are cancelled in a0 and f 0 . Since {(κi1 , . . . , κin , pi )}1≤i≤µ is a basis of V(a0 , f 0 , H[t]) iff {(κi1 , . . . , κin , pdi )}1≤i≤µ is a basis of V(a, f , H(t)), it suffices to find a basis of V(a0 , f 0 , H[t]). Next, need a degree bound b ∈ N0 ∪ {−1} for the polynomial solutions, i.e., a b s.t. V(a0 , f 0 , H[t]) = V(a0 , f 0 , H[t]b ) and f 0 ∈ H[t]l+b where l := max(deg(a01 ), deg(a02 )) ≥ 0. Set δ := b and fδ := f 0 . Then we go on with the Incremental reduction for (a0 , fδ ): Suppose that a0 = (a01 , a02 ) ∈ (H[t]∗ )2 with l := max(deg(a01 ), deg(a02 )) ≥ 0 and fδ = (f1 , . . . , fn ) ∈ H[t]n δ+l for some δ ∈ N0 ∪ {−1}; as given P from above. Then we look for all solutions g = δi=0 gi ti ∈ H[t]δ and c ∈ Kn with σaδ g = cfδ as follows. First derive the possible leading coefficients gδ in (H, σ), then plug in the resulting solutions into σa0 g = cfδ and look for the reP i maining g = b−1 i=0 gi t by recursion. More precisely, define

can be reordered to the ΠΣ∗ -extension (F, σ) of (G, σ) with F := G(x1 ) . . . (xr )(t1 ) . . . (te ) the polynomial d is a denominator bound of V(a, f , F). Similarly, b must also be a degree bound of V(a, f , G(x1 ) . . . (xr )(t1 ) . . . (te−1 )[te ]). We call a reduction of V(a, f , H(te )) to G extension-stable if all denominator and degree bounds within the reduction to G are extension-stable over G. Finally, we say that (G, σ) is depth-computable, in short δ-computable, if one can handle base case I for any 0 6= a ∈ G2 , f ∈ Gn , and for any ΠΣ∗ -extension (H(t), σ) of (G, σ) the following holds: one can hanlde base case II for f−1 ∈ H[t]n , and one can compute extension stable denominator and degree bounds. Given these properties a basis of V(a, f , E)) can be computed by our reduction. By the results in [3, Thm. 8.2], [2, Thm. 7.3] it follows that such extension-stable degree bounds exist. Moreover, they can be computed if (G, σ) is a ΠΣ∗ -field over a σ-computable constant field K. In this case Base case II can be solved by linear algebra methods and Base case I can be handled by applying our reduction again. Summarizing, we obtain

a ˜ δ := ([a01 ]l αδ , [a02 ]l ) and f˜δ := ([f1 ]δ+l , . . . , [fn ]δ+l ) (5) where 0 6= a ˜ δ ∈ H2 and f˜δ ∈ Hn ; [p]l gives the l-th coefficient of p ∈ H[t]. Afterwards the task is to find a basis B1 = {(ci1 , . . . , cin , wi )}1≤i≤λ of V(˜ aδ , f˜δ , H). We follow Reduction I: Apply Reduction strategy for (˜ aδ , f˜δ , H).

Theorem 7. Let (E, σ) with E := G(t1 ) . . . (te ) be a ΠΣ∗ extension of (G, σ). (1) There is an extension-stable reduction of (a, f , H(te )) to G for 0 6= a ∈ E2 and f ∈ En . (2) A ΠΣ∗ -field (G, σ) over a σ-computable K is δ-computable. Example 3. In the ΠΣ∗ -field from Exp. 1.1 there is the following extension-stable reduction (a, f , Q(t1 )(t2 )) to Q(t1 ) for a = (1, −1), f = σ(t2 /t21 ). Take for V(a, f , Q(t1 )(t2 )) the extension-stable denominator 1, for V(a, f , Q(t1 )[t2 ]) the extension-stable degree bound 2. We get the coefficient problems (((t1 + 1)2 , −(t1 + 1)2 ), f˜i ) with f˜2 = (0), f˜1 = (−2(t1 + 1), 1) and f˜0 = (t1 + 1); these are the Q(t1 )critical problems in our reduction.

If B1 = {} then c = 0 and g ∈ H[t]δ−1 are the only choices for σa0 g = cfδ . Hence, try to find a basis B2 of V(a, fδ−1 , H[t]δ−1 ) with fδ−1 := (0). Then the basis B1 can be reconstructed. Otherwise, if B1 6= {}, define C := (cij ) ∈ Kλ×n , g := (w1 tδ , . . . , wλ tδ ) ∈ tδ Hλ and fδ−1 := C fδ − σa0 g. (6) By construction, fδ−1 ∈ H[t]λδ+l−1 . Now we proceed as follows. Find all h ∈ H[t]δ−1 and d ∈ Kλ with σa0 (h + d g) = d C fδ which is equivalent to σa0 h = d fδ−1 , i.e., find a basis B2 of V(a, fδ−1 , H[t]δ−1 ). Then given B1 and B2 , a basis for V(a0 , f 0 , H[t]δ ) can be computed; for more details see [7]. To get B2 , we follow

Finally, we show the following generalization of [5, Prop. 1]. Lemma 5. Let (E(x), σ) be a ΠΣ∗ -extension of (F, σ) with E := F(t1 ) . . . (te ) and σ(x) = α x + β where α, β ∈ F; consider the reordered ΠΣ∗ -extension (F(x)(t1 ) . . . (te ), σ) of (F, σ). Let a ∈ E2 be homogeneous over E, f ∈ En and take an extension-stable reduction of (a, f , E) to F where S contains all F-critical problems. If for all (a0 , f 0 ) ∈ S we have V(a0 , f 0 , F) = V(a0 , f 0 , F(x)) then V(a, f , E) = V(a, f , E(x)). Moreover, there is an extension-stable reduction of (a, f , F(x)(t1 ) . . . (te )) to F(x) where all the F(x)critical problems are given by S.

Reduction II: Apply Incr. Reduction for (a0 , fδ−1 ). If δ = −1, we have reduced the problem to linear algebra. Base case II: Take a basis of V(a0 , f−1 , {0}) which equals to {k ∈ Kn | f−1 k = 0} × {0}. We call (a, f , H(t)) the reduction problem of V(a, f , H(t)). • Following Reduction II and Base case II one gets an incremental reduction of (a, f , H(t)). The incremental problems are ((a0 , fδ ), . . . , (a0 , f−1 )) and the coefficient problems are ((˜ aδ , f˜δ ), . . . , (˜ a0 , f˜0 )). (a, f , H(t)) is called the ˜ father-problem of (˜ ai , fi ) for 0 ≤ i ≤ δ. • Following Reduction I and Base case I one gets a tree of recursive reductions called a reduction of (a, f , H(t)) to G. We call a coefficient problem in the ground field (G, σ) within this reduction a G-problem. A G-problem (a0 , f 0 ) is critical if a0 is homogeneous over G and if for all its fatherproblems (ai , fi , G(t1 ) . . . (ti )) with 1 ≤ i ≤ e within the reduction the ai are homogeneous over G(t1 ) . . . (ti ). Next, we introduce reductions to F that are extension-stable. A denominator bound d ∈ H[te ]∗ of V(a, f , H(te )) or a degree bound b of V(a, f , H[te ]) is extension-stable over G if a is inhomogeneous over H(te ) or the following holds: For any ΠΣ∗ -extension (H(te )(x1 ) . . . (xr ), σ) of (H(te ), σ) that

Proof. The proof will be done by induction on the number of extensions. If e = 0, nothing has to be shown. Otherwise suppose that the lemma holds for the first e − 1 extensions with e ≥ 1. Let (F(t1 ) . . . (te )(x), σ) be a ΠΣ∗ extension of (F, σ) with σ(x) = α x + β, α, β ∈ F, and consider the reordered ΠΣ∗ -extension (F(x)(t1 ) . . . (te ), σ) of (F, σ). Denote E := F(t1 ) . . . (te−1 ), t := te and H := F(x)(t1 ) . . . (te−1 ) as shortcut. Let a ∈ E(t)2 be homogeneous over E(t), f ∈ E(t)n , take an extension-stable reduction of (a, f , E(t)) to F where S contains all F-critical problems, and suppose that V(a0 , f 0 , F) = V(a0 , f 0 , F(x)) for all (a0 , f 0 ) ∈ S. Then we show that V(a, f , E(t)) = V(a, f , H(t)). Moreover, as a by-product, we show that there is an extension-stable reduction of (a, f , H(t)) to F(x) with the F(x)-critical problems given by S. In the extension-stable reduction let d ∈ E[t]∗ be the denominator bound of the solution space V(a, f , E(t)). Since

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Condition A: Let (E, σ) with E := F(t1 ) . . . (te ) be a ΠΣ∗ extension of (F, σ) where δ(F) = d − 1 and δ(ti ) ≥ d. Let a ∈ E2 be homogeneous over E and f ∈ En , and suppose that all F-critical problems, say S = {(ai , fi )}1≤i≤k with ai = (ai1 , ai2 ), fi = (fi1 , . . . , firi ) ∈ Fri , of an extensionstable reduction of V((1, −1), f , E) to F are F-complete.

a is homogeneous over E(t), d ∈ H[t] is also a denominator bound of V(a, f , H(t)); by definition it is extensionstable. After clearing denominators and cancelling common factors, we get a0 = (a01 , a02 ) := (a1 /σ(d), a2 /d) q ∈ E[t]2 and f 0 := f q ∈ E[t]n for some q ∈ E(t)∗ in our reduction. Note that a0 is still homogeneous over E(t): we have σa0 h0 = 0 with h0 := h d ∈ H[t]∗ for some h ∈ E(t)∗ with σa h = 0. Now it suffices to show that V(a0 , f 0 , H[t]) = V(a0 , f 0 , E[t]). In the given reduction let b be the extensionstable degree bound of V(a0 , f 0 , E[t]). Therefore b is a degree bound of V(a0 , f 0 , H[t]); it is also extension-stable. Hence, we have to show V(a0 , f 0 , E[t]b ) = V(a0 , f 0 , H[t]b ). Let ((a0 , fb ), . . . , (a0 , f−1 )) be the incremental problems and ((˜ ab , f˜b ), . . . , (˜ a0 , f˜0 )) be the coefficient-problems in the incremental reduction. We show V(˜ ai , f˜i , E) = V(˜ ai , f˜i , H) for all 0 ≤ i ≤ b. First suppose that a ˜ i is inhomogeneous over E. Note that a ˜ i = ([a01 ]l αi , [a02 ]l ) by (5). Since σa0 h0 = 0, we get by coefficient comparison αk [a01 ]l σ(h00 ) + [a02 ]l h00 = 0 where k := deg(h0 ) and h00 ∈ E∗ is the leading coefficient of h0 . Hence (αk [a01 ]l , [a02 ]l ) ∈ (E∗ )2 is homogeneous over E. Since a ˜ i is inhomogeneous, i 6= k and α 6= 1, i.e., t is a Π-extension. Therefore, by Lemma 1.2 a ˜ i is inhomogeneous over E(x) and V(˜ ai , f˜i , E) = V(˜ ai , f˜i , E(x)). Thus a ˜ i is inhomogeneous over H and V(˜ ai , f˜i , E) = V(˜ ai , f˜i , H). In particular, there are no F-critical problems in (˜ ai , f˜i , E) to F and no F(x)-critical problems in (˜ ai , f˜i , H) to F(x). Otherwise, assume that a ˜ i is homogeneous over E. Then the extension-stable reduction of (a, f , E(t)) to F contains an extension-stable reduction of (˜ ai , f˜i , E) to F and all the Fcritical problems of the reduction of (˜ ai , f˜i , E) are given by a subset Si of S. Hence with the induction assumption it follows that V(˜ ai , f˜i , E) = V(˜ ai , f˜i , H) and the F(x)-critical ˜ problem in (˜ ai , fi , H) to F(x) are also Si . Since E[t]−1 = H[t]−1 = {0}, V(a, f−1 , E[t]−1 ) = V(a, f−1 , H[t]−1 ). Thus, we get an extension-stable reduction of (a, f , H(t)) to F(x) where the F(x)-critical problems are given by S. By construction, V(a0 , fi , E[t]i ) = V(a0 , fi , H[t]i ) for all i. Hence V(a, f , H(t)) = V(a, f , E(t)) = V(a, f , E(t)(x)).

5.

Proposition 2. Suppose that Condition A holds, and let (S, σ) with S = E(x1 ) . . . (xr ) be a ΠΣ∗ -extension of (E, σ) with maximal depth d − 1. Then V(a, f , E) = V(a, f , S). Moreover, for the reordered difference field (H(t1 ) . . . (te ), σ) with H = F(x1 ) . . . (xr ) there exists an extension-stable reduction of (a, f , H(t1 ) . . . (te )) to H with the H-critical problems S which are all H-complete. Proof. Since all F-critical problems are F-complete, we have V(ai , fi , F) = V(ai , fi , F(x1 )) = · · · = V(ai , fi , H). By applying Lemma 5 r times, it follows that there is an extension-stable reduction of (a, f , H(t1 ) . . . (te )) to H with the H-critical problems given by S; clearly they are H-complete. Moreover, V(a, f , E) = V(a, f , S). Theorem 9. Suppose that Cond. A holds with δ(ti ) = d. If (S, σ) is a Σ∗ -extension of (E, σ) with maximal depth d where for any 1 ≤ i ≤ k and 1 ≤ j ≤ ri there is a g ∈ D∗ with ai1 σ(g) − ai2 g = fij then (a, f ) is S-complete. Proof. Suppose that (a, f ) is not S-complete, i.e., there is a ΠΣ∗ -ext. (H, σ) of (S, σ) with maximal depth m ≤ d, a g ∈ H \ S and c ∈ Kn with σa g = cf . Let m be minimal. ˜ By [5, Lemma 1] we may refine this assumption to H = H(s) ∗ ˜ ˜ with δ(s) = m and σ(s) − s ∈ H where (H, σ) is a ΠΣ -ext. ˜ ˜ Subseof (S, σ) with maximal depth m−1 and g ∈ H(s)\ H. ˜ = E(x1 ) . . . (xr ) with δ(xi ) < d. Now conquently, write H sider the extension-stable reduction as claimed above and take the reordered ΠΣ∗ -ext. (F(x1 ) . . . (xr )(t1 ) . . . (tr ), σ) of ˜ := F(x1 ) . . . (xr ). Applying Prop. 2 we (F, σ); denote F ˜ 1 ) . . . (te )) to get an extension-stable reduction of (a, f , F(t ˜ with the F-critical ˜ ˜ F problems S which are all F-complete. By Lemma 5 together with V(a, f , E) ( V(a, f , H) it follows that there is an (a0 , f 0 ) ∈ S with f 0 ∈ Fν and V(a0 , f 0 , F) = ˜ ( V(a0 , f 0 , F(s)). ˜ V(a0 , f 0 , F) Therefore, there is a g 0 ∈ ˜ ˜ and c0 ∈ Kν with σ 0 g 0 = c0 f 0 . In particular, F(s) \F a ˜ g 0 ∈ H(s) \ S. By assumption on the Σ∗ -extension (S, σ) of (E, σ) there are gi ∈ S with σa0 gi = fi0 . Hence for h0 := c h ∈ S with h = (g1 , . . . , gν ) we have σa0 h0 = c0 f 0 , a contradiction to Prop. 1.1. Hence (a, f ) is S-complete.

SOLVING PROBLEM C

We will solve C (Theorem 8) by refining the reduction from above. Some special cases (Lemma 6) are immediate. Lemma 6. [8] Let (F, σ) be a ΠΣ∗ -ext. of (G, σ) with d := δ(F), a = (a1 , a2 ) ∈ F2 be homogeneous, f ∈ Fn and V := V(a, f , F). (1) If d = 0 or dim V = n + 1, then (a, f ) is Fcomplete. (2) If d = 1, constσ G = G and σ(g) − g ∈ G∗ for some g ∈ F, then (a, f ) is F-complete. (3) If δ(f ), δ(a) < δ(F) and (a, f ) is F-complete, then dim V = n + 1.

Example 4. With this result and Theorems 4 and 5 we can test if the extension t3 in Exp. 1.1 is δ-optimal: Take the reduction to Q(t1 ) from Exp. 3. By Lemma 6.2 the Q(t1 )critical problems are Q(t1 )-complete, i.e., Condition A holds. Take (Q(t1 )(t2 )(x01 ), σ) with σ(x01 ) = x01 + 1/(t1 + 1)2 . Since there are g ∈ Q(t1 )(t2 )(x01 ) with (t1 +1)2 σ(g)−(t1 +1)2 g = f for f ∈ {0, −2(t1 +1), 1, 1/(t1 +1)}, P := ((1, −1), (σ(t2 /t21 ))) is Q(t1 )(t2 )(x01 )-complete by Thm. 9. Since there is no g ∈ Q(t1 )(t2 )(x01 ) with σ(g) − g = σ(t2 /t21 ), P is Q(t1 )(t2 )complete by Thm. 5. Thus t3 is δ-complete by Thm. 4.

Theorem 8. Let (F, σ) be a δ-optimal ordered ΠΣ∗ -ext. of (G, σ), a ∈ F2 be homogeneous and f ∈ Fn . Then there is a Σ∗ -extension (S, σ) of (F, σ) where (S, σ) is a δ-optimal ordered ΠΣ∗ -ext. of (G, σ) by reordering and where (a, f ) is S-complete. It can be computed if (G, σ) is δ-computable. We proceed as follows. Using Lemma 5 from Section 4 we provide a sufficient condition (Condition A) in Proposition 2 that guarantees that the solution space cannot be increased by extensions with maximal depth d − 1. Given this result we can derive a criterion wether (a, f ) is S-complete for a given Σ∗ -extension (S, σ) of (E, σ); see Thm. 9.

Finally, we prove Thm. 8 by showing that such an extension (S, σ) supposed in Thm. 9 exists. More precisely, in Lemma 7 we show how we can construct an extension s.t. Condition A holds (see Alg. 1), and in Lemma 8 we show how we can construct an extension (S, σ) with the criterion

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in Thm. 9 (see Alg. 2). The resulting algorithms are applicable if (G, σ) is δ-computable. The corresponding proofs are done inductively/recursively: under the assumption that Theorem 8 holds for the depth level d − 1 we show the desired results for the depth level d.

there is an extension in which all coefficient problems have extension-stable reductions and where all D0 -critical problems are D0 -complete. Since b and d are extension-stable, we obtain an extension-stable reduction of (a, f , B0 (t)) to D0 where all D0 -problems are D0 -complete. If (G, σ) is δcomputable, Thm. 8 can be applied constructively. Hence such an extension can be computed; see Alg. 1.

Lemma 7. Suppose that Thm. 8 holds with the restriction that δ(F) = d − 1. Let (E, σ) be a δ-optimal ordered ΠΣ∗ extension of (G, σ) where E := F(t1 ) . . . (te ) with δ(F) = d−1 and δ(ti ) = d; let a ∈ E2 be homogeneous over E and f ∈ En . Then there is a Σ∗ -extension (S, σ) of (E, σ) with maximal depth d − 1 that can be reordered to a δ-optimal ordered ΠΣ∗ -extension (D(t1 ) . . . (te ), σ) of (G, σ) with δ(D) = d − 1 such that the following holds: there is an extension-stable reduction of (a, f , D(t1 ) . . . (te )) to D where all D-critical problems are D-complete. If (G, σ) is δ-computable, such an extension can be computed.

Algorithm 1. CompleteSubProblems(a, f , E, d) In: A δ-optimal ordered ΠΣ∗ -extension (E, σ) of a δ-computable (G, σ) where 0 ≤ d < δ(E) and E = F(t1 ) . . . (te ) with δ(F) = d, δ(ti ) > d; a homogeneous a = (a1 , a2 ) ∈ E2 , f ∈ En . An algorithm with the specification as Alg. 2. Out: (E0 , B, S). A δ-optimal ordered ΠΣ∗ -extension (E0 , σ) of (G, σ) with E0 := D(t1 ) . . . (te ) s.t. reordering of (E0 , σ) gives a Σ∗ -extension of (E, σ) with maximal depth d. A basis B of V. The D-critical problems S, all D-complete, of an extension-stable reduction from (a, f , D(t1 ) . . . (te )) to D. (1)IF e = 0 RETURN (E0 , B, {(a, f )}) after computing (E0 , B):=CompleteSolutionSpace(a, f , E). FI (2)Write H := F(t1 ) . . . (te−1 ). Compute an extension-stable denominator bound d ∈ H[te ]∗ of V(a, f , H(te )). Set a0 := (a1 /σ(d), a2 /d) ∈ H(te )2 , f 0 := f and clear the denominators and common factors. Compute an extensionstable degree bound b of V(a0 , f 0 , H[te ]). (3)FOR δ := b to 0 DO (4) Define 0 6= a ˜ δ ∈ H2 and f˜δ ∈ Hn as in (5). (5) IF a ˜ δ is inhomogeneous over H THEN compute a basis Bδ of V(˜ aδ , f˜δ , H) and set Sδ = {}, ELSE (H, Bδ , Sδ ) := CompleteSubProblems(˜ aδ , f˜δ , H, d). FI (6) Take the δ-optimal ΠΣ∗ -extension (H(te ), σ) of (H, σ) and define fδ−1 by (6) or fδ−1 := (0). (7)OD (8)Compute a basis B−1 of V(a0 , f−1 , {0}) (base case II). Given the bases Bi , compute for V(a, f , H[te ]b ) a basis S B = {(κi1 , . . . , κin , pi )}1≤i≤µ ; set S := bi=0 Si . pi (9)RETURN (H(te ), S, {(κi1 , . . . , κin , d )}1≤i≤µ ). FI

Proof. If e = 0, the lemma follows by using the depthrestricted version of Thm. 8. Otherwise suppose that the lemma holds for a ΠΣ∗ -extension (H, σ) of (F, σ) with H := F(t1 ) . . . (te−1 ), e ≥ 1. Now take a δ-optimal ΠΣ∗ -ext. (H(t), σ) of (H, σ) with δ(t) = d; let f ∈ H(t)n and a ∈ H(t)2 be homogeneous. Then we show that the lemma holds for (H(t), σ). Take an extension-stable denominator bound d ∈ H[t]∗ of V(a, f , H(t)). Set a0 := (a1 /σ(d), a2 /d) ∈ H(t)2 , f 0 := f and clear denominators and common factors s.t. a0 ∈ (H[t]∗ )2 and f 0 ∈ H[t]n . Take an extensionstable degree bound b of V(a0 , f 0 , H[t]). Now we show that there is a Σ∗ -ext. (S, σ) of (H(t), σ) with maximal depth d − 1 that can be reordered to a δ-optimal ordered ΠΣ∗ extension (D(t1 ) . . . (te−1 )(t), σ) of (G, σ) with δ(D) = d − 1 such that for all coefficient problems there is an extensionstable reduction of (a, f , D(t1 ) . . . (te−1 )) to D in which all D-critical problems are D-complete. If b = −1, nothing has to be shown. Otherwise, suppose that we have obtained such an extension that gives extension-stable reductions for the first u ≥ 0 coefficient problems in which all D-problems are D-complete. Denote B := D(t1 ) . . . (te−1 ) and let (˜ a, f˜) be the u + 1-th coefficient problem. If a ˜ is inhomogeneous, no additional D-critical problems appear. Hence our extension gives extension-stable reductions for the first u + 1 coefficient problems. Otherwise, if a ˜ is homogeneous, we can apply our induction assumption and get a Σ∗ -extension (S0 , σ) of (B, σ) with maximal depth d − 1, i.e., S0 = B(s1 ) . . . (sr ) with δ(si ) < d and with the following properties. We can reorder the extension to a δoptimal ordered ΠΣ∗ -extension (B0 , σ) of (G, σ) with B0 := D0 (t1 ) . . . (te−1 ) and δ(D0 ) < d such that all D0 -critical problems in (˜ a, f˜, B0 ) to D0 are D0 -complete. Then we apply Prop. 2 and it follows that also all the D0 -critical problems in the extension-stable reductions of the first u coefficientproblems are D0 -complete. In particular, the corresponding solution spaces are the same. Hence, we obtain an extension where the first u + 1 coefficient problems have extension-stable reductions where all D0 -critical problems are D0 -complete. Since (D(s1 ) . . . (sr ), σ) is a Σ∗ -extension of (D, σ), (B(t)(s1 ) . . . (sr ), σ) is a Σ∗ -ext. of (B(t), σ) by Lemma 3.2. Moreover, (S0 (t), σ) is a δ-optimal ΠΣ∗ -ext. of (S0 , σ): if t is a Π-ext., this follows by definition; otherwise, since (B(t), σ) is a δ-optimal ext. of (B, σ), this follows by Lemma 4. Since reordering below of t does not change this property, (B0 (t), σ) is a δ-optimal ordered ΠΣ∗ -ext. of (G, σ). Applying these arguments b + 1-times shows that

Example 5. We apply our algorithm for a = (1, −1), f = (σ(t3 /t31 )) with the δ-optimal Σ∗ -extension (Q(t1 )(t2 )(t3 ), σ) of (Q, σ) given in Exp. 1.1. Denote D = Q(t1 )(t2 ). We compute the denominator bound 1, the degree bound 2, and the first D-critical problem P2 := ((1, −1), (0)); it is D-complete. Hence, D is not extended. Next, we get the D-critical problem P1 := (a0 , (−2(1 + (t1 + 1)t2 ), 1)) with a0 = ((t1 + 1)3 , −(t1 + 1)3 ). We compute the δ-optimal Σ∗ -extension (D0 , σ) of (Q, σ) with D0 := Q(t1 )(t2 )(x1 ) and σ(x1 ) = x1 + 1 s.t. P1 is D0 -complete; see Exp 6.1. By Lemma 4 we (t1 +1)3 can take the δ-optimal extension (D0 (t3 ), σ) of (D0 , σ) and get P0 := (a0 , (1 + (t1 + 1)t2 , −x1 (1 + (t1 + 1)t2 ), 1)) as the last D0 -critical problem; like in Exp. 4 one can test that P0 is D-complete. Hence, we get an extension-stable reduction of (a, f , D0 (t3 )) to D0 with the D0 -complete problems Pi . A basis of V(a, f , D0 (t3 )) is {(0, 1)}. Lemma 8. Suppose that Thm. 8 holds with the restriction that δ(F) = d − 1. Let (E, σ) be a δ-optimal ordered ΠΣ∗ extension of (G, σ) where E := F(t1 ) . . . (te ) with δ(F) = d−1 and δ(ti ) = d; let a1 , . . . , an ∈ F2 be homogeneous and let f1 , . . . , fn ∈ F. Then there is a Σ∗ -extension (S, σ) of (F, σ) with maximal depth d which can be reordered to a δ-optimal ordered Σ∗ -extension of (G, σ) with the following property:

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there are gi ∈ S with σai gi = fi for all i. If (G, σ) is δcomputable, such an extension can be computed.

and f = (−2(1 + (t1 + 1)t2 ), 1) by applying Algorithm 2: (i) Alg. 1 computes for (a, f , Q(t1 )(t2 ), 1) the Q(t1 )-critical problems (a, f ) with f ∈ {(0, 0), (−2(t1 + 1)2 , 0, −2(t1 + 1)), ((t1 +1)2 , 1)}; they are all Q(t1 )-compete by Lemma 6.2. (ii) Next, the Σ∗ -extension (D, σ) of (Q(t1 )(t2 ), σ) with maximal depth 2 is computed s.t. for any g ∈ D we have (t1 + 1)3 σ(g)−(t1 +1)3 g = f for f ∈ {0, −2(t1 +1)2 , −2(t1 +1), 1}. 1 We get D = Q(t1 )(t2 )(x1 )(x01 ) with σ(x1 ) = x1 + (t1 +1) 3 and 0 0 1 σ(x1 ) = x1 + (t1 +1)2 . By Thm. 9, (a, f ) is D-complete. (iii) Finally, we compute for V(a, f , Q(t1 )(t2 )(x1 )(x01 )) the basis {(0, 0, 1), (0, 1, x1 )}. By Thm. 5 (a, f ) is Q(t1 )(t2 )(x1 )complete, i.e., we can remove the extension x01 . (2) We solve C for (Q(t1 )(t2 )(t3 ), σ), a = (1, −1) and f = (σ(t3 /t31 )) by applying Algorithm 2: (i) We run Alg. 1, see Exp. 5, and get the δ-optimal ordered Σ∗ -extension (D0 (t3 ), σ) of (Q, σ) with the D0 -critical problems P2 , P1 , P0 for the reduction (a, f , D0 (t3 )) to D0 . Note that for all f ∈ {0, 1, −2(1 + (t1 + 1)t2 ), x1 (1 + (t1 + 1)t2 )} there is a g ∈ D0 (t3 ) with (t1 + 1)3 σ(g) − (t1 + 1)3 g = f , except the last entry, say f 0 . (ii) We run Alg. 1 and obtain (D0 (t3 ), {(0, 1)}) for the input (((t1 + 1)3 , −(t1 + 1)3 ), (f 0 ), D0 (t3 ), 2). Next we construct 0 the Σ∗ -ext. (D0 (t3 )(x2 ), σ) with σ(x2 ) = x2 + (t1f+1)3 . (It is δoptimal by Prop 2; (a, f ) is D0 (t3 )(x2 )-complete by Thm. 9.) (iii) Finally, we get the solution in Exp. 1.1 by computing a basis for V(a, f , D0 (t3 )(x2 )). Summarizing, we have solved B for (σ(t3 /t31 )). In particular, we have represented the lhs of (2) in a δ-optimal Σ∗ -extension.

Proof. Suppose that the existence of such an extension (S, σ) of (F, σ) is proven for the first n ≥ 0 cases. Take an additional homogeneous a = (a1 , a2 ) ∈ F2 and f ∈ F. If there is a g ∈ S with σa g = f , we are done. Otherwise, by reordering of (S, σ) we get a δ-optimal ordered ΠΣ∗ -ext. (D(x1 ) . . . (xr ), σ) of (G, σ) with δ(D) = d − 1 and δ(xi ) = d. Take an h ∈ F∗ with σa h = 0. By Lemma 7 we can take a Σ∗ -ext. (S0 , σ) of (S, σ) with maximal depth d − 1 that can be reordered to a δ-optimal ordered ΠΣ∗ ext. (D0 (x1 ) . . . (xr ), σ) of (G, σ) with δ(D0 ) = d − 1 s.t. the following holds: there is an extension-stable reduction of ((1, −1), (−f /(ha2 )), D0 (x1 ) . . . (xr )) to D0 where all D0 critical problems are D0 -complete. If σa g = f for some g ∈ D0 (x1 ) . . . (xr ) = S0 , we are done. Otherwise, take the Σ∗ ext. (D0 (x1 ) . . . (xr )(x), σ) of (D0 (x1 ) . . . (xr ), σ) with σ(x) = x − f /(ha2 ) and δ(x) ≤ d. Then σa (hx) = f . By Prop. 2 the Σ∗ -ext. (D0 (x1 ) . . . (xr )(x), σ) of (D0 (x1 ) . . . (xr ), σ) is δoptimal; by reordering one gets a Σ∗ -ext. (S0 (x), σ) of (E, σ) with maximal depth d. Suppose that (G, σ) is δ-computable. Then such g, h can be computed and Lemma 7 becomes constructive. Hence also Lemma 8 gets constructive. Proof of Theorem 8. The proof will be done by induction on δ(F). If d = 0, (a, f ) is G-complete by Lemma 6.1. Now suppose that the theorem holds for (F, σ) with δ(F) = d−1, d > 0. Consider the δ-optimal ΠΣ∗ -ext. (E, σ) of (F, σ) with E := F(t1 ) . . . (te ) and δ(ti ) = d; let a ∈ E2 be homogeneous and f ∈ En . By Lemma 7 there is a Σ∗ -ext. (S, σ) of (E, σ) with maximal depth d − 1 which can be reordered to a δ-optimal ordered ΠΣ∗ -ext. (D(t1 ) . . . (te ), σ) of (G, σ) with the following property: there is an extension-stable reduction of (a, f , D(t1 ) . . . (te )) to D s.t. all D-critical problems, say S = {(ai , fi )}1≤i≤k with fi = (fi1 , . . . , firi ) ∈ Dri , are D-complete. Lemma 8 shows that there is a Σ∗ -ext. (S0 , σ) of (E, σ) with maximal depth d which can be reordered to a δ-optimal ordered ΠΣ∗ -ext. (D0 (t1 ) . . . (te ), σ) of (G, σ) s.t. σai = fij for all i, j. By Thm. 9 (a, f ) is D0 (t1 ) . . . (te )complete and hence S0 -complete. If (G, σ) is δ-computable, Lemmas 7 and 8 are constructive. This leads to Alg. 2. 

Improvements of Alg. 2: (1) Skip (3) if there are n+1 elements in B; see Lemma 6.1. (2) Modify (2) if δ(f ), δ(a) < d − 1: Write F = H(t1 ) . . . (te ) where δ(ti ) = d, δ(H) = d − 1, and set (H0 , B) := CompleteSolutionSpace(a, f , H). Afterwards, construct the ΠΣ∗ -extension (E, σ) of (H0 , σ) with E := H0 (t1 ) . . . (te ) by Lemma 3.2. By Lemma 6.3 (a, f ) is E-complete. (3) Similarly, speed up the computations in step (3) (Lemma 8): if δ(fi ), δ(a0 ) < d − 1, do all the computations in (H, σ). (4) Remove redundant extensions of (D, σ) in step (5) by applying Thms. 4 and 5; see Exp. 6.1.

Algorithm 2. CompleteSolutionSpace(a, f , F) In: A δ-optimal ordered ΠΣ∗ -ext. (F, σ) of a δ-computable (G, σ) with d := δ(F); 0 6= a = (a1 , a2 ) ∈ F2 and f ∈ Fn . An algorithm with the specification as Alg. 1. Out: (D, B). A δ-optimal ordered ΠΣ∗ -extension (D, σ) of (G, σ) s.t. reordering gives a Σ∗ -ext. of (F, σ) with maximal depth d and (a, f ) is D-complete; a basis B of V(a, f , D). (1)IF d = 0, compute a basis B of V(a, f , F); RETURN (F, B). (2)(E, B, S) := CompleteSubProblems(a, f , F, d − 1). (3)Following Lemma 8, construct a δ-optimal ordered ΠΣ∗ extension (D, σ) of (G, σ) s.t. reordering gives a Σ∗ -ext. of (E, σ) with maximal depth d and s.t. there are g ∈ D∗ with σa0 g = fi0 for all (a0 , f 0 ) ∈ S and all fi0 from f 0 . (4)IF E = D RETURN (E, B). (5)Compute a basis B 0 of V(a, f , D); RETURN (D, B 0 ). Remark 1. If one always skips line (3) (during the recursion), one obtains the reduction presented in Section 4. Example 6. Consider (Q(t1 )(t2 )(t3 ), σ) from Exp. 1.1. (1) We solve C for (Q(t1 )(t2 ), σ), a = ((t1 + 1)3 , −(t1 + 1)3 )

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Acknowledgments. I would like to thank the referees for their valuable comments and suggestions.

6.

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[1] M. Karr. Summation in finite terms. J. ACM, 28:305–350, 1981. [2] C. Schneider. Degree bounds to find polynomial solutions of parameterized linear difference equations in ΠΣ-fields. To appear in Appl. Algebra Engrg. Comm. Comput. [3] C. Schneider. A collection of denominator bounds to solve parameterized linear difference equations in ΠΣ-extensions. In Proc. SYNASC04, pages 269–282, 2004. Mirton Publishing. [4] C. Schneider. The summation package Sigma: Underlying principles and a rhombus tiling application. Discrete Math. Theor. Comput. Sci., 6:365–386, 2004. [5] C. Schneider. Symbolic summation with single-nested sum extensions. In J. Gutierrez, editor, Proc. ISSAC’04, pages 282–289. ACM Press, 2004. [6] C. Schneider. Product representations in ΠΣ-fields. Annals of Combinatorics, 9(1):75–99, 2005. [7] C. Schneider. Solving parameterized linear difference equations in terms of indefinite nested sums and products. To appear in J. Differ. Equations Appl. [8] C. Schneider. Finding telescopers with minimal depth for indefinite nested sum and product expressions (extended version). J. Kepler University, Linz. SFB-Report 2005-08, 2004. [9] D. Zeilberger. A fast algorithm for proving terminating hypergeometric identities. Discrete Math., 80(2):207–211, 1990.

Multivariate Power Series Multiplication ´ Eric Schost

´ LIX, Ecole polytechnique, 91128 Palaiseau, France

[email protected]

ABSTRACT

space, also called the degree degM of M , and possibly the regularity regM of M , which is defined later on. If V is a set, and f and g are maps V → R>0 , we say that f ∈ O(g) if there exists C ∈ R such that f (v) ≤ Cg(v) holds for all v in V . To make expressions involving nested logarithms well-defined in all cases, we write lg (x) = max(1, log2 (max(x, 1))). The notation f ∈ Olg (g) indicates the omission of logarithmic factors, that is, that there exists a constant α such that f is in O(glg α (g)). Previous work. In all that follows, we will distinguish the bilinear complexity, which estimates the number of algebra multiplications, and is also called rank, and the total complexity, which counts linear operations as well (precise definitions are given in the next section). We start by reviewing the previous results in one variable, i.e. when n = 1.

We study the multiplication of multivariate power series. We show that over large enough fields, the bilinear complexity of the product modulo a monomial ideal M is bounded by the product of the regularity of M by the degree of M . In some special cases, such as partial degree truncation, this estimate carries over to total complexity. This leads to complexity improvements for some basic algorithms with algebraic numbers, and some polynomial system solving algorithms.

Categories and Subject Descriptors I.1.2 [Computing Methodologies]: Symbolic and Algebraic Manipulation—Algebraic Algorithms

General Terms

• The rank of k[X1 ]/(X1d ) is at least 2d − 1 [34, 16], and if k has cardinality at least 2d − 1, this is also an upper bound. Hence, in this case, the rank of power series multiplication equals that of polynomial multiplication.

Algorithms, Experimentation, Theory

Keywords Power series, multiplication

1.

• For some multiplication algorithms with complexity more than linear (such as Karatsuba or Toom-Cook algorithms), truncated power series multiplication is faster than polynomial multiplication by a constant factor, regarding total and bilinear complexity [27, 18].

INTRODUCTION

We investigate the complexity of the multiplication of multivariate power series. We work modulo a 0-dimensional monomial ideal M in k[X1 , . . . , Xn ], where k is a field. This is no loss of generality, since the k-algebras k[X1 , . . . , Xn ]/M and k[[X1 , . . . , Xn ]]/M are isomorphic. Complexity conventions. Once k is fixed, M is specified by the number n of variables and the degrees of some generators. Thus, our complexity estimates are functions defined on some subsets of NN , which may vary, depending on M being arbitrary, or restricted to some special patterns, such as total or partial degree truncation (unless specified otherwise, the number of variables is not fixed). These estimates will be expressed directly in terms of the degrees of the generators of M , or in terms of some quantities attached to the algebra k[X1 , . . . , Xn ]/M : its dimension as a k-vector

When n ≥ 2, the situation is more complex, and results might depend on the shape of the quotient algebra. We will in the first place focus on the “general” case, for which no assumption is made on the ideal M . We will also mention two important special cases, partial and total degree truncation. • Truncating in partial degrees (d1 , . . . , dn ) amounts to compute modulo the ideal (X1d1 , . . . , Xndn ) of degree d1 · · · dn ; the support of such power series is a rectangular parallelotope. Section 4 gives applications of this truncation pattern. • Truncating in total degree d amounts to compute modulo the ideal generated ` by all ´ monomials of total degree d, which has degree d+n−1 ; the support of such power n series is a simplex. This truncation pattern is used in various versions of Newton-Hensel lifting algorithms, see [24] for references.

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In the case n = 2, the following results are known for these two special truncation patterns. For truncation in partial degrees (d, d), Bl¨ aser [5] gave a lower bound for the rank of

293

7 2 d −O(d), 3

and an upper bound of 3d2 +O(d) (for base fields of characteristic 0, containing all roots of unity), improving results of Sch¨ onhage [33]. For truncation in total degree d, Bl¨ aser [5] gave a lower bound for the rank of 54 d2 −O(d), and an upper bound of 32 d2 + O(d) (with the same restrictions on k), again improving results of [33]. Let now the number of variables be arbitrary. From now on, we denote by T the monomial basis of k[X1 , . . . , Xn ]/M . The direct approach to power series multiplication consists in expanding products. Then, obtaining the coefficient of a single monomial X1t1 · · · Xntn ∈ T in the output requires to perform 2(t1 + 1) · · · (tn + 1) operations. For partial degree truncation, the total cost is then approximately 21n (degM )2 ; for total degree truncation, the total cost is approximately 1 (degM )2 . In any case, the cost is at most quadratic in n! degM . A second approach is to perform the product as that of polynomials, and discard unwanted coefficients: the penalty induced by computing more terms than necessary can be compensated by the use of fast multiplication algorithms. To give complexity estimates for this approach, let VM denote the cardinality of the set of terms T × T: this is the number of monomials that appear when power series are multiplied as polynomials. Then over fields of characteristic 0, using Emiris and Pan’s sparse multiplication algorithm [15] yields a total complexity in Olg (nVM ). At worst, VM grows quadratically with degM . However, better estimates hold in many cases. For instance, for partial degree truncation, VM ≤ 2n degM ; thus in this case, the product modulo M can be done with a total complexity in Olg (2n degM ). Note however that this particular result on partial degree truncation actually holds more generally (it suffices that k is a ring), using Kronecker’s substitution [14]. Finally, fast algorithms are known for total degree truncation. Let M ⊂ k[X1 , . . . , Xn ] be the ideal generated by all monomials of total degree d. In [24], with Lecerf, we gave sharp estimates for the product modulo M in characteristic 0: the total complexity is in O(degM lg 3 (degM )lg lg (degM )), and the bilinear complexity is at most degM lg (degM ). An algorithm of Griewank’s using similar ideas was previously given in [4]; it has the same bilinear complexity, but the total complexity relies on results for evaluation / interpolation that are not detailed there. In [19], van der Hoeven extended the result of [24] to weighted total degree; he gave better estimates in [20], but the algorithm of [20, Section 5] does not seem to work as claimed (see [21], and in particular Remark 8, as to corrected estimates). Finally, Bl¨ aser gave in [5] the lower bound degM (3 − ε(degM )) for the rank, for fixed d and n → ∞, with ε(degM ) → 0.

multiplying the values, (iii) interpolating the result at ±ε. Thus, this algorithm requires two bilinear multiplications of degree 1 polynomials in ε, plus some linear operations. Generalizing this to an arbitrary ideal M will show that the rank of multiplication modulo M is essentially bounded by the product of the regularity of M by the degree of M . One would like a similar result to hold for total complexity; to obtain that extension, the missing elements are fast (linear-time) algorithms for some multivariate evaluation / interpolation questions (which generalize evaluation / interpolation at ±ε used above). Even though no such result is known for general M , we will however obtain suitable algorithms in the specific case of partial degree truncation. Acknowledgments. I thank Alin Bostan, Marc Giusti, Andreas Griewank, Joris van der Hoeven, Gr´egoire Lecerf and Bruno Salvy for fruitful discussions.

2. MAIN RESULTS Basic definitions. Let k be a field. Given n ≥ 1, we write k[X1 , . . . , Xn ] for the ring of polynomials in n variables over k. A term is a product X1α1 · · · Xnαn , with all αi ≥ 0; if S is a finite set of terms,PSpan(S) ⊂ k[X1 , . . . , Xn ] is the k-vector space of all sums t∈S ct t, with all ct in k. Let M be a monomial ideal in k[X1 , . . . , Xn ] of dimension 0, so that Q = k[X1 , . . . , Xn ]/M is a finite-dimensional k-vector space. Let T be the set of terms not in M : T is the monomial basis of Q, so, as a k-vector space, Q is isomorphic to Span(T). The degree of M , written degM , is the dimension of k[X1 , . . . , Xn ]/M over k, that is, degM = |T|. The regularity of M , written regM , is the first degree d for which the degree d component of the graded algebra Q is zero. Any term in T has a total degree less than regM . For 1 ≤ i ≤ n, let di be the least integer such that Xidi belongs to M (di is well-defined, since M has dimension 0). We will work under the following assumption. For all i, di is less than or equal to the cardinality of k. For 1 ≤ i ≤ n and 0 ≤ j < di , let ai,j be in k, with the requirement that ai,j = ai,j
for all j = j  . We will write A1 = [a1,0 , . . . , a1,d1 −1 ], . . . , An = [an,0 , . . . , an,dn −1 ]. Then, to the set T we associate AT ⊂ A1 × · · · × An given by AT = [(a1,c1 , . . . , an,cn ) | X1c1 · · · Xncn ∈ T] ; for definiteness, we may suppose that T is ordered in some way, so that AT inherits its order. The set of points AT is a parameter for our algorithms: in our computational model, any value depending on these points only will be available for free (the cost of computing the relevant such values could be estimated as well, using a more involved model). To illustrate this last definition, take for instance ai,j = j, if the characteristic of k allows this. Then, the set AT equals

Our contribution. We obtain two kinds of results: on the rank of the multiplication modulo an arbitrary 0-dimensional monomial ideal, and on the total complexity in some specific cases. The core of these results is to prove that power series modulo M can be approximately multiplied with a number of multiplications that equals the degree of M , using the idea of approximate algorithm introduced in [2, 1, 3]. In one variable, this approach is well-known, and can be easily illustrated for multiplication modulo X12 [3, 31]. If A and B ∈ k[X1 ] have degree at most 1, then computing AB modulo X12 −ε2 (ε is a new variable) and letting ε = 0 in the result yields AB modulo X12 . Now, the product AB modulo X12 −ε2 can be obtained by (i) evaluating A and B at ±ε, (ii)

[(c1 , . . . , cn ) | X1c1 · · · Xncn ∈ T] , so it equals the set of exponents of the terms in T. In any case, we have the equality |AT | = |T| = degM , and the set AT has the same “shape” as the above set of exponents. Model of computation. We will estimate both the rank (which counts bilinear multiplications) and the total complexity of the multiplication in Q. The former quantity is defined in the bilinear algorithm computational model, and the latter using linear straight-line programs.

294

Let R be a ring, and S be an R-algebra, which is also a free R-module. The inputs and outputs of our algorithms are given by means of their coordinates on an R-basis of S (all cases considered below come with a natural choice of a basis, which we will not mention explicitly). A bilinear algorithm of length  for the multiplication in S is the data of 2 R-linear forms f1 , . . . , f and h1 , . . . , h over S, and  elements w1 , . . . , w in S, such that the equality AB =

 X

occurs for n = 1: even with fast polynomial arithmetic, the algorithm of Theorem 1 then requires quadratic time, versus an almost linear complexity for the naive algorithm. Our algorithm is efficient when the number of variables is large; we will see that in a sense, it complements the previous approaches. Partial degree truncation. More complete results are obtained for multiplication truncated in partial degree, since in this case, sharp estimates on Eval(AT ) and Interp(AT ) are known in terms of univariate evaluation / interpolation. We will prove the following corollaries in Section 4 (following the convention given before, the constants in the big-O notation are independent of the number of variables).

fi (A)hi (B)wi

i=1

holds for all A and B in S. The rank of the multiplication map in S is the smallest length of such a bilinear algorithm. We will further say that a bilinear algorithm has total complexity L if  ≤ L, and if the linear map

Corollary 1. Let M be the ideal (X1d1 , . . . , Xndn ). Then the total complexity of the multiplication modulo M is in ! X M(di )lg (di ) O regM degM + M(regM ) degM . di i P Q Using regM ≤ i di and degM = i di , and Equations (1), the total complexity is thus in the complexity class ““ X ”“ Y ”” Olg (regM degM ) ⊂ Olg di di .

→ [h1 (B), . . . , h (B)], A → [f1 (A), . . . , f (A)], B P [λ1 , . . . , λ ] → i λi wi can be computed by a R-linear straight-line program of size at most L (see [11, Chapter 13] for the definition). Multiplication, evaluation and interpolation. Our algorithms rely on polynomial multiplication, evaluation and interpolation. We now give the corresponding notation. We let MBil and M be two maps N → N such that for any ring R and any integer s, the product of two polynomials of degree at most s in R[T ] can be computed by a bilinear algorithm of length MBil (s) and total complexity M(s). We also impose some standard growth conditions on these maps, see for instance [17, Chapter 8]. It follows from [12] that there exists a constant K such that one can take MBil (s) ≤ Ks lg (s), M(s) ≤ Ks lg (s)lg lg (s).

i

As mentioned above, this algorithm is useful when the number of variables is large with respect to the degrees. Combining it with the algorithm based on Kronecker’s substitution, we will get the following corollary. It shows that multiplication truncated in partial degree has an almost linear complexity in degM . Corollary 2. For any ε > 0, the total complexity of the product modulo (X1d1 , . . . , Xndn ) is in O((degM )1+ε ).

(1)

Next, we consider some questions of multivariate evaluation and interpolation, for the family of points AT defined above. We write Eval(AT ) for the minimal size of a linear straightline program that computes the evaluation map Span(T) P

→ →

i

Of course, estimates in Olg (degM ) would be desirable, but our methods do not seem to give such results. Applications. We will conclude this paper by two applications. First, we show how to speed up algorithms for the addition of algebraic numbers: over fields of small characteristic, the algorithm of [7] uses multivariate power series multiplication truncated in partial degree. The second application is the resolution of polynomial systems: Lecerf’s deflation algorithm for multiple roots [22] requires multiplications modulo suitable “gradients of ideals”, which are here similar to multiplications truncated in partial degree. Our algorithms improve the known results for both questions.

kAT [P (a) | a ∈ AT ],

and Interp(AT ) for the minimal size of a linear straight-line program that computes the inverse map (Section 3 gives the proof of invertibility). In our computational model, naive estimates for Eval(AT ) and Interp(AT ) are quadratic in degM . Main results: the general case. We can now give our main result (proved in Section 3) on the rank and the total complexity of multiplication modulo an arbitrary monomial ideal. The algorithm is parametrized by the choice of points AT : while the rank estimate does not depend on it, the total complexity does, since evaluation and interpolation on AT appear as “linear subroutines” of the underlying algorithm.

3. PROOF OF THEOREM 1 We now prove Theorem 1, still using the notation of Section 2. We first recall a deformation result of Macaulay [25]. Let g1 , . . . , gR be monomial generators of M such that gi does not divide gj , for i = j. For 1 ≤ r ≤ R, we write

Theorem 1. The rank of the multiplication in Q admits the upper bound MBil (regM ) degM . The total complexity is in O ((Eval(AT ) + Interp(AT ))regM + M(regM ) degM ) .

δ

δ

gr = X1 1,r · · · Xnn,r . Let A1 , . . . , An and AT be the sets of points introduced in the previous section, and let ε be a new indeterminate over k. Then, for 1 ≤ r ≤ R, we define the polynomial

The best results one could obtain with these techniques would be total complexity estimates in Olg (regM degM ). We already know that M(regM ) is in Olg (regM ). Thus, the main missing elements are linear-type estimates for evaluation and interpolation, that would hold for any choice of M . For the moment, I am not aware of such results. It should also be stressed that Theorem 1 does not improve the previous results in all situations. Its worst case

Gr =

n δi,r Y Y−1 (Xi − εai,j ), i=1

j=0

so that Gr is in k[ε, X1 , . . . , Xn ] ⊂ k(ε)[X1 , . . . , Xn ], and gr = Gr (0, X1 , . . . , Xn ). Up to the use of the parameter ε,

295

Multiplication modulo g1 , . . . , gR .

this definition is due to [25]. Together with generalizations, it is studied in more details in [26], who gives references to other occurrences in the literature. Let us fix a monomial order < that refines the total degree, in both k[X1 , . . . , Xn ] and k(ε)[X1 , . . . , Xn ]: in particular, for 1 ≤ r ≤ R, the leading term of Gr is gr . Then the following results are proved in [26] (see also [30]).

Input: A and B in Span(T). Output: The product AB modulo g1 , . . . , gR . 1. Let vA = [A(εa) | a ∈ AT ] and vB = [B(εa) | a ∈ AT ]. 2. Compute the pairwise product vC of vA and vB .

Proposition 1. The polynomials Gr (1, X1 , . . . , Xn ), 1 ≤ r ≤ R, form the Gr¨ obner basis in k[X1 , . . . , Xn ] of the ideal of the set of points AT ⊂ An (k). The polynomials obner basis in k(ε)[X1 , . . . , Xn ] of G1 , . . . , GR form the Gr¨ the ideal of the set of points [εa | a ∈ AT ] ⊂ An (k(ε)).

3. For i < 2 regM , compute Γi by interpolating on AT the degree-i coefficients of the entries of vC . Let Ci,0 be the homogeneous component of degree i of Γi . Return P i<2 reg M Ci,0 . Analysis. Evaluating the polynomial A ∈ k[X1 , . . . , Xn ] at · · · + Aδ , with [εa | a ∈ AT ] is done as follows. Let A = A0 + P i Ai homogeneous of degree i; then A(εa) = i≤δ Ai (a)ε . Thus, vA is obtained by evaluating all polynomials Ai on AT . Since A is in Span(T), δ is less than regM , and all Ai are in Span(T) as well; thus, vA is obtained by less than regM evaluations on AT . The same holds for vB . The entries of vA and vB have degree in ε less than regM , so all entries of vC are obtained for degM further multiplications in k[ε] in degree less than regM . Finally, C is recovered by performing at most 2 regM interpolations on AT . Evaluations and interpolations do not contribute to the bilinear complexity, which is thus degM times the bilinear cost of univariate polynomial multiplication in degree less than regM . This gives the first part of Theorem 1. To get the total complexity, one adds the costs of evaluation and interpolation, to the total cost of the polynomial multiplications. This concludes the proof.

The first part of the proposition shows that the map AT

→ Span(T) k [P (a) | a ∈ AT ] → P is well-defined, as claimed previously. The next intermediate result is to relate reductions modulo g1 , . . . , gR and G1 , . . . , GR . For A in k(ε)[X1 , . . . , Xn ], we write Rg (A) and RG (A) for its normal forms modulo respectively g1 , . . . , gR and G1 , . . . , GR . For A in k[ε, X1 , . . . , Xn ], we write deg A for its total degree in ε, X1 , . . . , Xn . Lemma 1. Let A ∈ k[X1 , . . . , Xn ] ⊂ k(ε)[X1 , . . . , Xn ]. Then RG (A) is in k[ε, X1 , . . . , Xn ] ⊂ k(ε)[X1 , . . . , Xn ], its specialization at ε = 0 is Rg (A) and deg RG (A) ≤ deg A. Proof. It suffices to prove the result for terms; we do it by induction for the order <. The property holds for all terms in T; let thus t ∈ / T be a term, such that the property holds for all terms less than t for the order <. The remainder Rg (t) is 0; let δ be a term and 1 ≤ r ≤ R be such that t = δgr , and let P = t − δGr . Then, P is in k[ε, X1 , . . . , Xn ], and is less than t for the order <. Besides, RG (P ) = RG (t) and deg P = deg t (since Gr is homogeneous in ε, X1 , . . . , XP n ).  PWrite P =  t

4. SPECIAL CASES AND APPLICATIONS Giving precise total complexity estimates requires results on the functions Eval and Interp, that is, on multivariate evaluation and interpolation. I do not know sharp results for this in general; however, it is easy to give estimates in the particular case of partial degree truncation. This section is devoted to present a standard algorithm for this task, and some applications and extensions. We first review previous results on univariate evaluation and interpolation. Let B be d pairwise distinct points in k. We will write EvalUni(B) for the minimal size of a linear straight-line program that computes the linear map P → [P (b) | b ∈ B], where P ∈ k[T ] has degree less than d; similarly, we write InterpUni(B) for the minimal size of a linear straight-line program that computes the inverse univariate interpolation map. The coefficients of these linear straight-line programs are functions of B: in our computational model, we do not take the cost of computing these coefficients into account. For arbitrary points, EvalUni(B) and InterpUni(B) belong to O(M(d)lg (d)), see [17] and references therein, as well as [8] for more recent algorithms. Better estimates are known in special cases, if the points B form a geometric progression [29, 6, 10] or are suitable powers of a root of unity [20].

We can then describe our multiplication algorithm. Given A and B in Span(T) ⊂ k[X1 , . . . , Xn ], it outputs C = Rg (AB). Let Cε = RG (AB). By Lemma 1, Cε can be written X Ci,0 + Ci,1 ε + · · · + Ci,ei εei , Cε = i≤µ

i in for some µ and ei ≥ 0, with Ci,j homogeneous of degree P k[X1 , . . . , Xn ]. The same lemma shows that C = i≤µ Ci,0 and that i + ei ≤ deg AB < 2 regM for all i, so µ < 2 regM . Let us evaluate A and B on all points εa, for a ∈ AT , and multiply these values. The last part of Proposition 1 shows the equality Cε (εa) = A(εa)B(εa) P for all a ∈ AT . For i < 2 regM , let Γi = j+=i Cj, ∈ k[X1 , . . . , Xn ]; then Ci,0 is the homogeneous component of degree i of Γi . P i Since Cε (εa) = i Γi (a)ε is known for all a ∈ AT , Γi is obtained by interpolation on AT . This yields the following algorithm.

4.1 Truncation in partial degree We consider now truncation in partial degrees (d1 , . . . , dn ); we prove Corollaries 1 and 2 and present an application to computation with algebraic numbers. The notation of Section 2 is still in use. Evaluation and interpolation on a rectangular grid. When truncating in partial degree, the basis T is the set of

296

all monomials X1t1 · · · Xntn , with 0 ≤ ti < di for 1 ≤ i ≤ n. The multivariate evaluation and interpolation problems we have to solve then become the following: Evaluation requires to evaluate a polynomial in Span(T) on the rectangular grid AT = A1 × · · · × An , with

discard unwanted coefficients. Using [14, Section 3.5], we obtain that for any ring R, any  and n, and any d+1 , . . . , dn , the total complexity of the product in the R-algebra

A1 = [a1,0 , . . . , a1,d1 −1 ], . . . , An = [an,0 , . . . , an,dn −1 ],

is at most D(d+1 , . . . , dn ) := KP  lg (P  )lg lg (P  ), where K is the constant introduced in Equation (1) of Section 2, and P  equals 2n− d+1 · · · dn . We can then prove the corollary itself. Let ε > 0, let n and d1 , . . . , dn be integers; let M = (X1d1 , . . . , Xndn ) and

d

+1 , . . . , Xndn ) R[X+1 , . . . , Xn ]/(X+1

and interpolation is the inverse operation. Fast solutions to these problems are well known. Let P ∈ Span(T) ⊂ k[X1 , . . . , Xn ]. To perform evaluation, we proceed with one variable at a time. We first compute the d1 polynomials

Q = k[X1 , . . . , Xn ]/(X1d1 , . . . , Xndn ).

P (a1,0 , X2 , . . . , Xn ), . . . , P (a1,d1 −1 , X2 , . . . , Xn )

Due to the k-algebra isomorphisms k[X]/(X)  k and k[X, Y ]/(X d , Y e )  k[X, Y ]/(X e , Y d ), we can without loss of generality suppose that 2 ≤ d1 ≤ · · · ≤ dn ; note then that degM = d1 · · · dn ≥ 2n . We define ω = 2ε , and  by the condition that d ≤ 2 ω < d+1 (we let d0 = 0 and dn+1 = ∞ to make  well-defined).

in k[X2 , . . . , Xn ]. Proceeding separately with each monomial in X2 , . . . , Xn , we see that the cost of this operation is EvalUni(A1 ) d2 · · · dn operations in k. Then, we evaluate these d1 polynomials on A2 , etc . . . Summing up, the whole cost of these operations is X EvalUni(Ai ) X EvalUni(Ai ) = degM . d1 · · · dn d di i i i

Lemma 2. The total complexity of the product in Q is at most 6CD, with C = C(d1 , . . . , d ) and D = D(d+1 , . . . , dn ).

Interpolation is performed in the same manner, interpolating one variable after the other; similarly, the total cost is X InterpUni(Ai ) X InterpUni(Ai ) = degM . d1 · · · dn di di i i

Proof. Let R = k[X1 , . . . , X ]/(X1d1 , . . . , Xd ), so that d+1 , . . . , Xndn ). We apply our alQ = R[X+1 , . . . , Xn ]/(X+1 gorithm for the product in R, and Kronecker’s substitution for the product in Q seen as an R-algebra. Going back to the definition of our computational model, we deduce that there exist c ≤ C, d ≤ D, R-linear forms λi , µi : Q → R and ρi ∈ Q, with 1 ≤ i ≤ d, k-linear forms fj , gj : R → k and wj ∈ R, with 1 ≤ j ≤ c, such that for all A, B in Q one has X ` ´` ´ AB = (fj ◦ λi )(A) (gj ◦ µi )(B) wj ρi ;

See for instance [35, Chapter 14.1] for the details of this algorithm. Using the most general estimates on EvalUni and InterpUni, we deduce the following estimates: ! X M(di )lg (di ) Eval(AT ), Interp(AT ) ∈ O degM . di i (Of course, these estimates can be improved for the special choices of A1 , . . . , An mentioned above.) This suffices to prove Corollary 1. Indeed, plugging these estimates into those of Theorem 1, we deduce that the total complexity of multiplication truncated in partial degrees (d1 , . . . , dn ) is in the requested class ! X M(di )lg (di ) + M(regM ) degM . O regM degM di i

i≤d,j≤c

furthermore, the R-linear map A → [λ1 (A), . . . , λd (A)], B P → [µ1 (B), . . . , µd (B)], (2) [η1 , . . . , ηd ] → i ηi ρi can be evaluated by a R-linear straight-line program Γ of size D, and the k-linear map → [g1 (b), . . . , gc (b)], a → [f1 (a), . . . , fc (a)], b P [η1 , . . . , ηc ] → j ηj wj

Combining algorithms. To conclude our discussion, we prove Corollary 2: For any ε > 0, the total complexity of the product modulo M = (X1d1 , . . . , Xndn ) is in O((degM )1+ε ). The idea behind the proof is the following. The algorithm of Corollary 1 is efficient when the number of variables is large compared to the truncation degrees, whereas the naive algorithm using Kronecker’s substitution is efficient when the degree is large compared to the number of variables. Combining these two strategies leads to the proof of Corollary 2. First we give some preliminary formulas, which explicit the above statements. Corollary 1 shows that there exist L and α in R such that for all  and all d1 , . . . , d , the total complexity of the product in the k-algebra

(3)

can be evaluated by a k-linear straight-line program of size C. Since cd ≤ 6CD, it remains to estimate the cost of evaluating the map P A → (fj ◦ λi )(A), B → (gj ◦ µi )(B), [η1,1 , . . . , ηc,d ] → ηi,j wj ρi . In view of the multiplication algorithm of [12], Γ can be taken with constants in the image of Z in k; then, the above linear map can be read off the tensor product of the maps (2) and (3). In view of the sizes of the matrices involved, Lemma 13.7.5 in [11] shows that this tensor product can be computed by a linear straight-line program of size at most 3dC + 3cD ≤ 6CD. Using the inequalities 2n ≤ degM and d1 +· · ·+d ≤ degM , this bound becomes 6KL(d1 + · · · + d )∆2n− degM , with ∆ = lg α+1 ((degM )2 )lg lg ((degM )2 ). Next, we write the inequalities d1 + · · · + d ≤ nd ≤ n2 ω ≤ 2 ω lg (degM ) and (2 ω )n− ≤ d+1 · · · dn ≤ degM , ε so that 2n− ≤ (degM ) 2 . In particular, (d1 + · · · + d )∆ is polylogarithmic in degM , so it admits the upper bound

k[X1 , . . . , X ]/(X1d1 , . . . , Xd ) is at most C(d1 , . . . , d ) := LSP lg α (SP ), with S = d1 +· · ·+ d and P = d1 · · · d . On the other hand, we can use Kronecker’s substitution to multiply series as polynomials, and

297

ε

Corollary 3. Let f, g be as above, and suppose that all roots of f ⊕ g have multiplicity less than p. Then f ⊕ g can be computed in O(Dlg 2 (D)lg 2 lg (D)) operations in k.

Nε (degM ) 2 , for some constant Nε . Putting all these estimates together finishes the proof, since it shows that the total complexity of the product in Q is upper bounded by 6KLNε (degM )1+ε .

This algorithm was implemented using Shoup’s C++ NTL library, available at www.shoup.net; timings were obtained on an AMD 64 3500+, running in 64 bit mode. Figure 1 illustrates this algorithm taking F3 for base field, with deg f = deg g in abscissae and time (in seconds) in ordinates; we distinguish the time for power series multiplication from the total running time. Since k = F3 , the multivariate power

Application: computing with algebraic numbers. Let f and g be monic polynomials in k[T ], of degrees m and n. We are interested in computing their composed sum f ⊕ g, which is the polynomial of degree D = mn given by Y` ´ f ⊕g = T − (α + β) , α,β

160

the product running over all the roots α of f and β of g, counted with multiplicities. See [7] for a review of some applications of this basic operation on algebraic numbers. In characteristic 0, the fastest known approach for computing f ⊕ g is due to Dvornicich and Traverso [13]. The key idea is that the exponential generating series of the power sums of f ⊕ g is the product of the exponential generating series of the power sums of f and g. Using fast conversions between coefficients and power sums, computing f ⊕ g then has complexity O(M(D)), which is almost linear in D. Suppose now that k has characteristic p < D. Then the previous approach does not apply anymore. The first difficulty is that a polynomial is not determined by its power sums. The second difficulty is that the relation between the power sums of f , g and f ⊕g is no more valid: the above generating series cannot be defined, since they require division by p. We first discuss how to handle the first problem: if the multiplicities of all roots of f ⊕ g are less than p, given its first 2D power sums, one can recover f ⊕ g in time „ „ « „ «« D D O M(D) + p M lg , p p

120 100 80 60 40 20 0 0

series are truncated in partial degree 3 in all variables, and the number of variables grows with deg f . We compared our multiplication algorithm to the naive one for this specific task: ours becomes more efficient for 7 variables or more.

4.2 Gradient of ideals Let again M = (X1d1 , . . . , Xndn ), and T the terms not in M . Following [22], the gradient ∇M of M is defined as the 0-dimensional monomial ideal generated by the monomials not in T+ , where T+ is the set of terms T ∪ ∪i≤n Xi T. Lecerf’s deflation algorithm [22], which yields a quadratic Newton iteration for systems with multiple roots, relies on computations modulo an ideal of the form ∇M . We obtain the following estimate for this task. Proposition 2. For any ε > 0, the total complexity of the product modulo the gradient of M = (X1d1 , . . . , Xndn ) is in O(n (degM )1+ε ). Proof. The set T+ is the disjoint union T ∪ ∪i≤n Ti , where Ti is the set of terms X1t1 · · · Xntn , with ti = di and 0 ≤ tj < dj for j = i. Let then A and B in Span(T+ ), with X X A = AT + Ai , B = BT + Bi ,

X nλ(i) (f ) i T, i! i∈Λ

then the following identity holds: Np (f ⊕ g) = Np (f )Np (g)

50 100 150 200 250 300 350 400 450 500 550

Figure 1: Composed sum over F3 .

in the computation tree model, using an elaboration by Pan [28] of an idea of Sch¨ onhage’s [32]. On the other hand, the first 2D power sums of f and g can still be computed in O(M(D)) operations in k. The article [7] gives a solution for the second problem. Let T = (Ti )i≥0 be an infinite set of indeterminates; let Λ be the set of finite sequences (of arbitrary length) i = (i0 , . . . , is ), with all i in {0, . . . , p−1}. For i ∈ Λ, we write i! = i0 ! · · · is !, Ti = T0i0 · · · Tsis and λ(i) = i0 + i1 p + · · · + is ps . For a univariate polynomial f ∈ k[T ], let ni (f ) be its ith power sum. Writing Np (f ) =

Multiplication Total

140

i≤n

i≤n

such that AT and BT are in Span(T), and Ai and Bi are in Span(Ti ) for all i. Computing AB mod ∇M is done by “expanding” the product. We firstP compute C = AT BT mod ∇M , which we write C = CT + i≤n Ci , with CT in Span(T) and Ci in Span(Ti ). The component CT is obtained by multiplication modulo M . To obtain Ci , we write BT = bi + Xi bi , with bi in k[X1 , . . . , Xi−1 , Xi+1 , . . . , Xn ]. Let ci = AT bi mod M ; then the coefficient of a monomial t in Ci is obtained as the coefficient of t/Xi in ci , so Ci is known as soon as ci is. Next, we compute AT Bi and Ai BT mod ∇M , for all i. This is done by writing Ai = X di αi and Bi = X di βi , so

mod (Tp ),

where (Tp ) denotes the ideal generated by all monomials of the form Tip in k[[T]]. Since 2D power sums of f ⊕ g are requested, we only need the coefficients of the monomials in T0 , . . . , Ts , where s = lg (2D)/lg (p). Given the first 2D power sums of f and g, this can be done by multiplying two multivariate power series involving at most lg (2D)/lg (p) variables and of degree less than p in each variable. Suppose for simplicity that p is constant; based on this discussion, the algorithm of [7] has complexity O(D1+1/p ) in this case. Corollary 1 yields the following improved bound.

298

along the X3 -axis, into the two parts T1 and T2 displayed on the rightmost picture. The analogue of the classical fast evaluation algorithm is to reduce P modulo defining ideals of the two corresponding sets of points, and go recursively. As to interpolation, an interesting question would be whether evaluation and interpolation are equivalent, as is essentially the case when n = 1 [9]. Even for such special cases, precise answers to these questions still have to be written.

all coefficients of the products above are known as soon as AT βi mod M and αi BT mod M are known. Finally, notice that for all i, j, Ai Bj is 0 mod ∇M . Summarizing, AB mod ∇M can be computed for 3n + 1 products modulo M and less than 3n degM additions. Corollary 2 ends the proof. Application: polynomial system solving. As mentioned above, computing modulo gradients of ideals can speed up the resolution of polynomial systems. Giving the main result of [22] would require to introduce extensive notation; we will simply repeat the main example given there. Let F = F1 , F2 , F3 in Q[X1 , X2 , X3 ], with F1 F2 F3

= = =

6. REFERENCES [1] D. Bini. Relations between exact and approximate bilinear algorithms. Applications. Calcolo, 17(1):87–97, 1980. [2] D. Bini, M. Capovani, F. Romani, and G. Lotti. O(n2.7799 ) complexity for n × n approximate matrix multiplication. Inf. Proc. Lett., 8(5):234–235, 1979. [3] D. Bini, G. Lotti, and F. Romani. Approximate solutions for the bilinear form computational problem. SIAM J. Comput., 9(4):692–697, 1980. [4] C. Bischof, G. Corliss, and A. Griewank. Structured second- and higher-order derivatives through univariate Taylor series. Opt. Meth. and Soft., 2:211–232, 1993. [5] M. Bl¨ aser. The complexity of bivariate power series arithmetic. TCS, 295(1-3):65–83, 2003. [6] L. I. Bluestein. A linear filtering approach to the computation of the discrete Fourier transform. IEEE Trans. Electroacoustics, AU-18:451–455, 1970. ´ Schost. Fast [7] A. Bostan, P. Flajolet, B. Salvy, and E. computation with two algebraic numbers, 2003. ´ Schost. Tellegen’s [8] A. Bostan, G. Lecerf, and E. principle into practice. In ISSAC’03, pages 37–44. ACM, 2003. ´ Schost. On the complexities of [9] A. Bostan and E. multipoint evaluation and interpolation. TCS, 329(2):223–235, 2004. ´ Schost. Polynomial evaluation and [10] A. Bostan and E. interpolation on special sets of points, J. Complexity, to appear. [11] P. B¨ urgisser, M. Clausen, and M. A. Shokrollahi. Algebraic complexity theory, volume 315 of Grund. der Math. Wiss. Springer, 1997. [12] D. G. Cantor and E. Kaltofen. On fast multiplication of polynomials over arbitrary algebras. Acta Inform., 28(7):693–701, 1991. [13] R. Dvornicich and C. Traverso. Newton symmetric functions and the arithmetic of algebraically closed fields. In AAECC-5, volume 356 of LNCS, pages 216–224. Springer, 1989. [14] I. Z. Emiris and V. Y. Pan. Applications of FFT. In Algorithms and theory of computation handbook, pages 17–1–17–30. CRC, Boca Raton, FL, 1999. [15] I. Z. Emiris and V. Y. Pan. Symbolic and numeric methods for exploiting structure in constructing resultant matrices. JSC, 33(4):393–413, 2002. [16] C. M. Fiduccia and Y. Zalcstein. Algebras having linear multiplicative complexities. J. Assoc. Comput. Mach., 24(2):311–331, 1977. [17] J. von zur Gathen and J. Gerhard. Modern computer algebra. Cambridge University Press, 1999.

2X1 + 2X12 + 2X2 + 2X22 + X32 − 1, (X1 + X2 − X3 − 1)3 − X13 (2X13 + 5X22 + 10X3 + 5X32 + 5)3 − 1000X15 .

Then x∗ = (0, 0, −1) ∈ Q3 is a root of this system. Given a prime p, and the point (0, 0, p − 1) ∈ F3p , we are interested in computing approximations of x∗ in the p-adic topology. However, x∗ is a root of multiplicity 18 of F , so the Newton iterator fails to converge with quadratic speed to this root. Lecerf’s deflation algorithm enables one to recover a quadratic convergence rate; it turns out that each step of the modified Newton iteration now requires to compute modulo the gradient of the ideal (X1 , X23 , X34 ). In general, one has to compute modulo the gradient of an ideal M = (X1d1 , . . . , Xndn ), where degM is upper bounded by the multiplicity of the root to approximate. Lecerf’s algorithm uses a quadratic power series multiplication algorithm, so each lifting step has a complexity quadratic in the multiplicity; our algorithm reduces this to an almost linear complexity (Lecerf’s result also counts the cost of handling exponents, which is not taken into account in our model). Note that this lifting algorithm is also a crucial part of a more general equidimensional decomposition algorithm [23].

5.

CONCLUSION, OPEN PROBLEMS

Our approach requires to solve specific but well-known [30] evaluation / interpolation problems. Obtaining sharp complexity estimates for these tasks is of independent interest. Before considering the general case, one may want to look at special cases first. We saw in Section 4 how to solve these problems in quasi-linear time for partial degree truncation, which corresponds to evaluation / interpolation on a rectangular grid. Another important special case is that of truncation in total degree, which corresponds to evaluation / interpolation on a simplex. For the moment, a fast algorithm (of linear complexity) for this purpose is missing. To conclude, let us illustrate possible approaches on another particular case, illustrated on Figure 2. The monomial support T is on the leftmost picture; it consists in the settheoretic difference of two cubes in 3-space. The goal is to evaluate or interpolate a polynomial P with this support on a set of points which form the same pattern. A natural idea T1 T

Split

T2

Figure 2: Splitting a monomial support. is to apply a divide-and-conquer algorithm, by splitting T

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[28] V. Y. Pan. New techniques for the computation of linear recurrence coefficients. Finite Fields and their Applications, 6(1):93–118, 2000. [29] L. R. Rabiner, R. W. Schafer, and C. M. Rader. The chirp z-transform algorithm and its application. Bell System Tech. J., 48:1249–1292, 1969. [30] T. Sauer. Lagrange interpolation on subgrids of tensor product grids. Math. Comp., 73:181–190, 2004. [31] A. Sch¨ onhage. Partial and total matrix multiplication. SIAM J. Comput., 10(3):434–455, 1981. [32] A. Sch¨ onhage. Fast parallel computation of characteristic polynomials by Leverrier’s power sum method adapted to fields of finite characteristic. In Automata, languages and programming, volume 700 of LNCS, pages 410–417. Springer, 1993. [33] A. Sch¨ onhage. Bivariate polynomial multiplication patterns. In AAECC-11, volume 948 of LNCS, pages 70–81. Springer, 1995. [34] S. Winograd. Some bilinear forms whose multiplicative complexity depends on the field of constants. Math. Systems Theory, 10(2):169–180, 1976/77. [35] R. Zippel. Effective Polynomial Computation. Kluwer Academic Publishers, 1993.

[18] G. Hanrot and P. Zimmermann. A long note on Mulders’ short product. JSC, 37(3):391–401, 2004. [19] J. van der Hoeven. Relax, but don’t be too lazy. JSC, 34(6):479–542, 2002. [20] J. van der Hoeven. The Truncated Fourier Transform and applications. In ISSAC’04, pages 290–296. ACM, 2004. [21] J. van der Hoeven. Notes on the Truncated Fourier Transform, preprint, 2005. [22] G. Lecerf. Quadratic Newton iteration for systems with multiplicity. Found. of Comput. Math., 2(3):247–293, 2002. [23] G. Lecerf. Computing the equidimensional decomposition of an algebraic closed set by means of lifting fibers. J. Complexity, 19(4):564–596, 2003. ´ Schost. Fast multivariate power [24] G. Lecerf and E. series multiplication in characteristic zero. SADIO Electronic Journal, 5(1), 2003. [25] F. S. Macaulay. Some properties of enumeration in the theory of modular systems. Proc. London Math. Soc., 26:531–555, 1927. [26] F. Mora. De nugis Groebnerialium 2: Applying Macaulay’s trick in order to easily write a Gr¨ obner basis. AAECC, 13(6):437–446, 2003. [27] T. Mulders. On short multiplications and divisions. AAECC, 11(1):69–88, 2000.

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Partial Degree Formulae for Rational Algebraic Surfaces ∗ Sonia Perez-D´ ıaz ´

J. Rafael Sendra

Dpto de Matematicas. ´ Universidad de Alcala´ E-28871 Madrid. Spain

Dpto de Matematicas. ´ Universidad de Alcala´ E-28871 Madrid. Spain

[email protected]

[email protected]

ABSTRACT

(see Chapter IV in [18]), or analyzing the Hilbert polynomial (see Chapter 9 in [4]). This question, i.e. the determination of the degree, is specially interesting when working with unirational hypersurfaces given by means of a rational parametrization, since the degree of the hypersurface can be used to approach some other problems as for instance the construction of implicitization algorithms based on interpolation techniques; note that in this case the degree gives information on the interpolation space to be used (see [7]). Motivated by this fact, some authors have addressed this problem. For instance, in [16], [17] the authors analyze the question for the case of algebraic curves. For the surface case, interesting contributions have also been presented. In [13], one may find bounds for the degree, in [2] a method based on the degree of rational maps and on the base points of the parametrization is presented, in [1] and [3] one computes the degree by analyzing the base points and by means of syzygies, etc. In addition, the degree problem has also been studied for the case of special computer aided geometric constructions as offseting (see [6] and [11]). As we have already mentioned, the degree of the hypersurface gives information on the interpolation space for the implicitization problem. Nevertheless, the space of interpolation can be decreased if the partial degrees, w.r.t. each variable, of the implicit equation are known. For the case of plane curves, formulae for the partial degrees are given in [17], and a successful applications of them can be found in [8], where an interpolating implicitization algorithm is presented. However, at least to the authors’ knowledge, the existing contributions for the case of parametric surfaces only deal with the total degree of the implicit equation. In this paper, we present formulae for the computation of the partial degrees of the implicit equation of a rational surface given by means of a proper parametrization; note that properness can be check using the results in [9]. Moreover, when the parametrization is not proper we give upper bounds. The formulae in this paper generalize the results in [17] to the surface case, and they are based on the computation of the degree of the rational maps induced by the projections, onto the coordinate planes of the three dimensional space, of the input surface parametrization. In addition, using the results presented in [10], the formulae simply involve the computation of the degree of univariate polynomials directed determined from the parametrization by means of some univariate resultants and some polynomial gcds.

In this paper, we present formulae for the computation of the partial degrees w.r.t. each variable of the implicit equation of a rational surface given by means of a proper parametrization. Moreover, when the parametrization is not proper we give upper bounds. These formulae generalize the results in [17] to the surface case, and they are based on the computation of the degree of the rational maps induced by the projections, onto the coordinate planes of the three dimensional space, of the input surface parametrization. In addition, using the results presented in [9] and [10], the formulae simply involve the computation of the degree of univariate polynomials directed determined from the parametrization by means of some univariate resultants and some polynomial gcds.

Categories and Subject Descriptors I.1.2 [Symbolic and Algebraic Manipulation]: Algorithms

General Terms Algorithms

Keywords Rational Surface Parametrization, Algebraic Surface, Partial Degrees of an Algebraic Surface, Rational Map Degree

1.

INTRODUCTION

The computation of the degree of an algebraic variety is an important question by its own. To approach this problem one may for instance compute the maximum number of points of intersection of the variety with a projective linear subspace in general position of complementary dimension ∗This work is partially supported by BMF2002-04402-C0201 (Curvas y Superficies: Fundamentos, Algoritmos y Aplicaciones), and GAIA II (IST-2002-35512).

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. ISSAC’05, July 24–27, 2005, Beijing, China. Copyright 2005 ACM 1-59593-095-7/05/0007 ...$5.00.

301

2.

NOTATION AND DEGREE NOTIONS

the total degrees of its rational components. Similarly for degti (P( t )).

In this section we introduce the notation and terminology that will be used throughout this paper, and we discuss on the different notions of degree related to the problem. We consider a rational surface V over an algebraically closed field K of characteristic zero, as well as a rational parametrization   p1 ( t ) p2 ( t ) p3 ( t ) P( t ) = , ∈ K( t )3 , q1 ( t ) q2 ( t ) q3 ( t )

• We denote by degree(φP ) the degree of the rational map φP (for further details on this notion see e.g. [18] pp.143, or [5] pp.80); similarly, for degree(φπi,j (P) ). As an important result on degree(φP ), we recall that the birrationality of φP , i.e. the properness of P( t ), is characterized by deg(φP ) = 1 (see [5] and [18]). Also, we recall that the degree of a rational map can be seen as the cardinality of the fibre of a generic element (see Theorem 7, pp. 76 in [18]). We will use this characterization of the degree for our reasoning. For this purpose, we introduce the following notation. If φ : V1 −→ V2 is a rational dominant map of irreducible varieties of the same dimension, for every P ∈ V1 , we denote by Fφ (P ) the fibre φ−1 (φ(P )); i.e. Fφ (P ) = {Q ∈ V1 | φ(Q) = φ(P )}.

of V, where t = (t1 , t2 ) and gcd(pi , qi ) = 1 for i = 1, 2, 3. Moreover, we denote by F (x1 , x2 , x3 ) the implicit equation of V. We assume w.l.o.g. that none pi ( t )/qi ( t ) is constant. Note that, if pi ( t )/qi ( t ) = λ ∈ K, then F (x1 , x2 , x3 ) = xi − λ and the problem is trivial. In addition, for i, j ∈ {1, 2, 3}, with i < j, we denote by πi,j the (i, j)-projection mapping in the three dimensional space. That is, πi,j (x1 , x2 , x3 ) = (xi , xj ). Furthermore, associated to the parametrization P( t ) we consider the induced rational map:

We observe that there does not exist a direct relationship between the different degree notions introduced
above. For instance, let P(t1 , t2 ) = t1 t2 , t2 + t21 , t1 + t2 . Then, the implicit equation of V is

φP : K2 −→ V ⊂ K3 ; t 7−→ P( t ). Similarly, for i, j ∈ {1, 2, 3}, with i < j, we consider the pair of rational functions   pi ( t ) pj ( t ) πi,j (P( t )) = , ∈ K( t )2 , qi ( t ) qj ( t )

F (x1 , x2 , x3 ) = 2x1 x2 +x21 −3x1 x3 −x3 x2 +x33 +x1 +x22 −x23 x2 .

and we also denote by φπi,j (P) the rational map induced by πi,j (P( t )). Associated to the parametrization P( t ) we introduce the following polynomials

and

However, deg(V) = 3, degx1 (V) = 2, degx2 (V) = 2, degx3 (V) = 3 tdeg(P( t )) = 2, degt1 (P( t )) = 2, degt2 (P( t )) = 1,

Gi ( t , h ) = pi ( t )qi ( h )−pi ( h )qi ( t ) ∈ K[ h ][ t ], i ∈ {1, 2, 3}, and G4 ( t ) = lcm(q1 , q2 , q3 ) ∈ K[ t ], where h = (h1 , h2 ) is a generic element of K2 . We denote by F the algebraic closure of the field K( h ). We observe that since we have assumed that the components of P( t ) are not constant, then Gi is not identically zero. Indeed, for G4 is clear; now let us assume that Gi , with i < 4, is identically zero, then pi ( h )qi ( t ) = pi ( t )qi ( h ). Thus, qi ( t ) divides pi ( t ). But, since gcd(pi , qi ) = 1, one deduces that qi ( t ) and pi ( t ) are both constant, which is impossible. Related to the problem we are dealing with, several different concepts of degree appear. More precisely, we will use the following:

degree(φP ) = 1. We will see in Section 4 that the connection is given by the degree of the rational maps φπi,j (P) .

3.

• For a polynomial G ∈ K[x1 , x2 , x3 ] we denote by tdeg(G) the total degree of G and by degxi (G) the degree of G w.r.t. xi . • If R ∈ K(t1 , t2 ) is a rational function, we denote by degti (R) the maximum of the degrees w.r.t. ti of the denominator and the denominator of R(t1 , t2 ), expressed in reduced form. Similarly for tdeg(R).

COMPUTATION OF THE DEGREE OF A RATIONAL MAP

In the next section we will see that the degree of the rational maps φπi,j (P) play an important role in the partial degree formulae for V. For this reason, in this section we show how to compute the degree of the rational map induced by a pair of bivariate rational functions. All results in this section are either included in [9] or [10], or are easily deduced from there. Therefore, we omit proofs in this section, and for further details we refer to [9] and [10]. For this purpose, in the sequel we consider
R( t ) = r1 ( t ), r2 ( t ) ∈ K( t )2 , where ri ( t ) = ri1 ( t )/ri2 ( t ), and gcd(ri1 , ri2 ) = 1 for i ∈ {1, 2}. Associated with R( t ), we have the rational map φR , and the polynomials (compare to Gi in the previous section) Hi ( t , h ) = ri1 ( t )ri2 ( h )−ri1 ( h )ri2 ( t ) ∈ K[ h ][ t ], i ∈ {1, 2}, and H3 ( t ) = lcm(r12 , r22 ). We denote Wih the algebraic set defined over F by the polynomials Hi ( t , h ) ∈ F[ t ], i ∈ {1, 2}, and by W3 the algebraic set defined over K by H3 ( t ). In addition, for every α ∈ K2 , and i ∈ {1, 2}, we denote by Hiα ( t ) the polynomial Hi ( t , α ), and by Wiα the algebraic set defined over K by Hiα ( t ). We assume that U = gcd(H1 ( t , h ), H2 ( t , h )) = 1. Note that this implies that {∇r1 , ∇r2 } are linearly independent

• We denote by deg(V) the degree of the surface V, that is deg(V) = tdeg(F ). In addition, we define the partial degree of V w.r.t. xi as degxi (V). Note that even though the degree of a surface is invariant under linear changes of coordinates, the partial degree does depend. • We define the total degree of the parametrization P( t ), and we denote it by tdeg(P( t )), as the maximum of

302

as vectors in K[t1 , t2 ]2 . Indeed; let us assume that {∇r1 , ∇r2 } are linearly dependent. Thus, the dimension of the tangent space of the rational function R at any point is 1; that is, R parametrizes a curve. Then, R( t ) = M (N ( t )), with M ∈ K(t)2 and N ∈ K( t ). Therefore, the factor (N ( t ) − N ( h )) divides U ( t , h ) which contradicts the hypothesis. We observe that in particular, this implies that r1 , r2 are both not constant, and ensures that W1α and W2α are plane curves. We also assume that none of the projective curves defined by each of the numerators and denominators of components of R passes through the point at infinity (0 : 1 : 0). Note that under these conditions, if HiH (t1 , t2 , w, h1 , h2 ) denotes the homogenization of Hi (t1 , t2 , h1 , h2 ) as a polynomial in K[ h ][ t ], it holds that HiH (0, 1, 0, h ) 6= 0 for i ∈ {1, 2}. This in particular implies that, for i ∈ {1, 2}, the leading coefficients of Hi w.r.t. t2 , only depends on h , and then the polynomials Hi do not factor in K[t1 ]. In these conditions, the following results hold.

Remark 1. Observe that if U 6= 1, then TR = SR = 0. Corollary 1. There exists a non–empty open subset ΩR ⊂ K2 , such that for every α ∈ ΩR deg(φR ) = Card(FR ( α )) = degt1 (SRα (t1 )) = degt2 (TRα (t2 )). Moreover, it holds that deg(φR ) = degt1 (SR (t1 , h )) = degt2 (TR (t2 , h )).

4.

Theorem 1. There exists a non–empty open subset ΩR ⊂ K2 , such that for every α ∈ ΩR it holds that,

 1. FR ( α ) = t ∈ K2 Hi α ( t ) = 0, i ∈ {1, 2}, H3 ( t ) 6= 0 .

That is, FR ( α ) = (W1α ∩ W2α ) \ (W1α ∩ W2α ∩ W3 ).

Theorem 3. (Plane Curve Case) Let C be a rational affine plane curve, let f (x1 , x2 ) ∈ K[x1 , x2 ] be its defining polynomial, and let P(t) be a rational parametrization in reduced form of C. Then it holds that

2. FR ( α ) = {(a, b) ∈ K2 | SRα (a) = 0, Ma α (b) = 0}, where SRα (t1 ) = Maα (t2 ) =

Rest2 (H1α , H2α ) , ContentZ (Rest2 (H1α , H2α + ZH3 )) gcdK[t2 ] (H1α (a, t2 ), H2α (a, t2 ))

gcdK[t2 ] (H1α (a, t2 ), H2α (a, t2 ), H3 (a, t2))

(i) P(t) is proper iff degt (P(t)) = max{degx1 (f ), degx2 (f )}.

.

(ii) If P(t) is proper, then degx1 (C) = degx1 (f ) = degree(φπ2 (P) ) = degt (π2 (P(t)),

3. FR ( α ) ⊆ {(a, b) ∈ K2 | SRα (a) = 0, TRα (b) = 0}, where SRα (t1 ) =

Rest2 (H1α , H2α ) , ContentZ (Rest2 (H1α , H2α + ZH3 ))

TRα (t2 ) =

Rest1 (H1α , H2α ) . ContentZ (Rest1 (H1α , H2α + ZH3 ))

and degx2 (C) = degx2 (f ) = degree(φπ1 (P) ) = degt (π1 (P(t)). Proof. See Theorem 5 in [17]. Theorem 3 seems to indicate that, under the assumption of properness, degx1 (V) (resp. for x2 , and for x3 ) is given by the degree of the rational map induced by π2,3 (P( t )) (resp. π1,3 (P( t )), and π1,2 (P( t )). In this section, we prove that this is the case. We start with the following two technical lemmas.

Theorem 2. The following statements hold: 1. FR ( h ) =

n

t ∈ F2 Hi h ( t ) = 0, i ∈ {1, 2},

PARTIAL DEGREE FORMULAE

In this section we state the formulae for computing the partial degrees of V from the parametrization P( t ). These formulae generalize the corresponding results for plane algebraic curve stated in [17]. We start recalling the result for curves given in [17]. For this purpose, for i ∈ {1, 2} we denote by for πi the iprojection mapping of the plane; that is πi (x1 , x2 ) = xi . In addition, we observe that if R(t) ∈ K(t) is a univariate rational function then tdeg(R) = degt (R) = degree(φR ), where φR : K → K; t 7→ R(t) is the rational map induced by R(t). Furthermore, if P(t) is a plane curve parametrization, then tdeg(P(t)) = degt (P(t)). In this situation, the result is stated as follows

o H3 ( t ) 6= 0 .

That is, FR ( h ) = (W1h ∩ W2h ) \ (W1h ∩ W2h ∩ W3 ).

2. FR ( h ) = {(A, B) ∈ K2 | SR (A, h ) = 0, MA (B, h ) = 0}, where

Lemma 1. Let H( t , h ) ∈ K[ h ][ t ]\K[ h ] be a polynomial without factors in K[ t ]. Then, there exists A = (A1 , A2 ) either in (F\K)×K or in K×(F\K) such that H(A, h ) = 0.

SR (t1 , h ) = pp h (Rest2 (H1 ( t , h ), H2 ( t , h ))),

Proof. First, we write H as H( t , h ) = C( h )H ∗ ( t , h ), where C( h ) is the content of H w.r.t. t . Note that, since H ∈ K[ h ][ t ] \ K[ h ] and it does not have factors in K[ t ] then H ∗ depends on both t and h . Let us first assume that, concerning to t , H ∗ depends only on one of the variable ti , say on t2 ; similarly if it depends only on t1 . Then, we write the polynomial H ∗ as a univariate polynomial in t2 as:

and MA (t2 , h ) = pp h (gcd (H1 (A, t2 , h ), H2 (A, t2 , h ))), F[t2 ]

and pp h denotes the primitive part w.r.t. h . 3. FR ( h ) ⊆ {(A, B) ∈ K2 | SR (A, h ) = 0, TR (B, h ) = 0}, where

H ∗ ( t , h ) = an ( h )tn 2 + · · · + a0 ( h ).

SR (t1 , h ) = pp h (Rest2 (H1 ( t , h ), H2 ( t , h ))),

Observe that H ∗ , as univariate polynomial in F[t2 ], can not have all its roots in K, since it would imply that H ∗ has a factor in K[ t ], and therefore H also, which is a contradiction. Then, if A2 is a root of H ∗ in F \ K, the lemma holds taking (A1 , A2 ) where A1 is any element of K.

and TR (t2 , h ) = pp h (Rest1 (H1 ( t , h ), H2 ( t , h ))), and pp h denotes the primitive part w.r.t. h .

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Now, we assume that H ∗ does depend on both t1 and t2 . In this situation, we write the polynomial H ∗ as a univariate polynomial in t2 as:

where Gi ( t , h ) = pi ( t )qi ( h ) − pi ( h )qi ( t ). Then, it holds that H( t , h ) 6= 1.

H ∗ ( t , h ) = an (t1 , h )tn 2 + · · · + a0 (t1 , h ),

Proof. First, we observe that, since pi /qi are not constant, then N ( t ) is not constant. Moreover, reasoning as in Section 2, one has that G1 , G2 are non-zero polynomials, and hence H is well-defined. Now, we consider the polynomial

and each coefficient aj (t1 , h ) as a univariate polynomial in t1 : mj

aj (t1 , h ) = bmj ,j ( h )t1

+ · · · + b0,j ( h ).

F ( t , h ) = n1 ( t )n2 ( h ) − n1 ( h )n2 ( t ) ∈ K[ h ][ t ].

Moreover, for every A1 ∈ K, we consider the polynomial PA1 (t2 , h ) := H ∗ (A1 , t2 , h ) ∈ K[ h ][t2 ]. Then, we prove that for almost all A1 ∈ K the polynomial PA1 (t2 , h ) has degree n in t2 . Indeed, let us assume that there exist infinitely many values of A1 in K such that an (A1 , h ) is zero. This implies that an (t1 , h ), as univariate polynomial in F[t1 ], has infinity many roots, and therefore an (t1 , h ) must be zero, which is impossible since it is a leading coefficient. Let Ω1 be the non-empty open subset of K satisfying the above property. Now, we see that we may assume w.l.o.g. that H ∗ is primitive w.r.t. t2 . Indeed, let M (t1 , h ) be the content of H ∗ w.r.t. t2 . Since H ∗ is primitive w.r.t. t and it cannot have factors in K[ t ], it follows that M is either constant or it depends on t1 and h . However, if it is not constant, reasoning on M as in the first part of the proof, one sees that the lemma holds. Thus, we may assume that H ∗ is primitive w.r.t. t2 . In this situation, by well-known results in computer algebra on the specialization on gcds (see e.g. Lemma 3 in [17]), one deduces that there exists a non-empty open subset Ω2 of K such that for A1 ∈ Ω2 , the polynomial PA1 (t2 , h ) is primitive w.r.t. t2 . Now, we see that for almost all values of A1 in Ω1 ∩ Ω2 , the univariate polynomial PA1 (t2 , h ) ∈ F[t2 ] has at least one root in F \ K. Indeed, let us assume that there exists infinitely many values of A1 in Ω1 ∩ Ω2 such that PA1 (t2 , h ) ∈ F[t2 ] has all its roots in K. For these values, since PA1 (t2 , h ) is primitive w.r.t. t2 , it follows that PA1 (t2 , h ) ∈ K[t2 ]. Then, for every coefficient ak (t1 , h ) of H ∗ (note that degt1 (ak ) = mk ) we take A1,0 , . . . , A1,mk ∈ K, all different, such that all roots of PA1,j (t2 , h ) are in K. Then, since PA1,j (t2 , h ) ∈ K[t2 ], for every k it holds that ak (A1,j , h ) ∈ K. Then, expressing these equalities in matrix form, one gets that ¯b · V ∈ Kmk +1 , where V is the Vandermonde matrix associated to A1,0 , . . . , A1,mk , and ¯b is the vector (b0,k ( h ), . . . , bmk ,k ( h )). Thus, since V is regular with entries in K, it follows that ¯b ∈ Kmk +1 . Therefore, ak (t1 , h ) ∈ K[t1 ]. But this implies that H ∗ does depend only on t and it is not constant, which is a contradiction because H does not have factors in K[ t ]. Let Ω3 be the non-empty open subset of K satisfying the above property. Then, for A1 ∈ Ω1 ∩Ω2 ∩Ω3 (note that since K is irreducible, and Ωi are non-empty open subsets, then Ω1 ∩ Ω2 ∩ Ω3 6= ∅) and A2 ∈ F \ K being one root of PA1 (t2 , h ), it holds that H(A1 , A2 , h ) = 0. Lemma 2. Let R( t ) =



p1 ( t ) p2 ( t ) , q1 ( t ) q2 ( t )

Observe that F ( t , h ) 6∈ K. Indeed, if F ( t , h ) = a ∈ K, then since F ( h , h ) = 0, we get that a = 0. Thus, we deduce that n1 ( t )n2 ( h ) = n1 ( h )n2 ( t ), which is impossible since gcd(n1 , n2 ) = 1 and at least one of the ni is not constant. Now, let us assume that H( t , h ) = 1. Taking into account Lemma 7 in [10], we deduce that there exists a non-empty open subset Ω1 of K2 , such that for every α ∈ Ω1 , it holds that gcd(G1 ( t , α ), G2 ( t , α )) = 1. Also, let Ω2 = { α ∈ K2 | p1 (α)p2 (α)q1 ( α )q2 ( α ) 6= 0}. Ω2 is open and non-empty since p1 p2 q1 q2 is a non-zero polynomial; note that pi /qi is not constant. Now, we observe that for α ∈ Ω2 the polynomials Gi ( t , α ) are non-zero. Indeed, let α ∈ Ω2 be such that Gi ( t , α ) = 0, then pi ( t )qi ( α ) = pi ( α )qi ( t ), and since pi ( α ) 6= 0 and qi ( α ) 6= 0 and one of the polynomials pi ( t ), qi ( t ) is not constant, it follows that gcd(pi ( t ), qi ( t )) 6= 1, which is a contradiction. In this situation, let α ∈ Ω1 ∩ Ω2 , then, we prove that W α = { β ∈ K2 | G1 ( β , α ) = G2 ( β , α ) = 0} is either empty or it contains finitely many elements. Indeed, if any of the polynomials Gi ( t , α ) is constant, since the constant cannot be zero, one has that W α = ∅; otherwise the statement follows from B´ezout Theorem. Now, we consider the non-empty open subset of K2 , Ω3 = { α ∈ K2 | n2 ( α )q1 ( α )q2 ( α ) 6= 0} and for every α ∈ Ω1 ∩ Ω2 ∩ Ω3 we introduce the subset of ? = U α \ V, where K2 defined as W α U α = { β ∈ K2 | F ( β , α ) = 0}, and V = { β ∈ K2 | n1 ( β ) = n2 ( β ) = 0}. We note that F ( t , α ) cannot be constant, because in this case substituting t = α in F ( t , α ) one gets that this constant is zero, and reasoning as above one reaches a contradiction. Therefore, U α is the curve defined over K by F ( t , α ). On the other hand, using that n1 ( t )/n2 ( t ) is not constant one deduces that either V = ∅ (if n1 ( t ) or n2 ( t ) is constant) or by B´ezout’s theorem that Card(V ) < ∞. More? over, since V ⊂ U α , one deduces that Card(W α ) = ∞. Fur? ? ⊂ W α . Indeed, let β 0 ∈ W α . thermore, let us see that W α Since nj ( β 0 ) 6= 0, and nj ( α ) 6= 0, for j = 1, 2, we have that N ( β 0 ) = N ( α ). Taking into account that qj ( α ) 6= 0, for j = 1, 2, we deduce that M (N ( α )) is defined, and since N ( β 0 ) = N ( α ), we also have that M (N ( β 0 )) is defined, and R( β 0 ) = M (N ( β 0 )) = M (N ( α )) = R( α ). Thus, G1 ( β 0 , α ) = G2 ( β 0 , α ) = 0, which implies that ? ⊂ W α . But, this is impossiβ 0 ∈ W ? , and then W α ble because Card(W ?α ) = ∞. Therefore, we conclude that H( t , h ) 6= 1.



∈ K( t )2 ,

where p1 /q1 , p2 /q2 6∈ K, let N ( t ) = n1 ( t )/n2 ( t ) ∈ K( t ), and let M (t) ∈ K(t)2 , all rational functions in reduced form, such that R( t ) = M (N ( t )). Let H( t , h ) = gcd(G1 ( t , h ), G2 ( t , h )),

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In the following theorem we see how to characterize, by means of gcds, when the surface V is a cylinder over one of the coordinate planes of K3 . Note that when V is such a cylinder, Theorem 3 provides the partial degrees of V.

surface. Therefore W2 is a rational plane curve, and hence irreducible . Let h(x1 , x2 ) ∈ K[x1 , x2 ] be the implicit equation of W2 . Then, h(P( t )) = h(π1,2 (P( t ))) = 0, which implies that F = h because the polynomials F and h are irreducible. Therefore, degx3 (F ) = 0.

Theorem 4. (Cylinder Characterization) It holds that

Remark 2. In order to apply Theorem 4, one needs to compute gcds in 4 variables. Nevertheless, taking into account Lemma 7 in [10] one may deduce that there exists a non-open Zariski subset Ω of K2 such that for α ∈ Ω it holds that

1. V is a cylinder over the x1 x2 -plane (i.e. degx3 (F ) = 0) if and only if gcd(G1 , G2 ) 6= 1. 2. V is a cylinder over the x1 x3 -plane (i.e. degx2 (F ) = 0) if and only if gcd(G1 , G3 ) 6= 1. 3. V is a cylinder over the x2 x3 -plane (i.e. degx1 (F ) = 0) if and only if gcd(G2 , G3 ) 6= 1.

tdeg(G1 ( t , h ), G2 ( t , h )) = tdeg(G1 ( t , α ), G2 ( t , α )). Therefore, one may also consider an heuristic algorithm to apply Theorem 4 that uses gcds of bivariate polynomials.

Proof. We prove Statement 1. Reasoning similarly, one proves Statements 2 and 3. Let H = gcd(G1 , G2 ). First, we assume that degx3 (F ) = 0. Then F defines a plane curve C in K2 , and V is the cylinder over C. Moreover, since F (P( t )) = F (π1,2 (P( t )) = 0, one deduces that C is rational; note that there exists a ∈ K such that π1,2 (P(a, t)) or π1,2 (P(t, a)) is not a point in K2 . In these conditions, let M (t) ∈ K(t)2 be a proper parametrization of C (see [14]), and let I(x2 , x2 ) denote the inverse of M (t). Then



The computations show that gcd(G1 , G2 ) = 1, gcd(G1 , G3 ) = 1, gcd(G2 , G3 ) = t1 + 2t2 − h1 − 2h2 Therefore, by Theorem 4, we conclude that V is a cylinder over the x2 x3 -plane. Furthermore, since P( t ) is proper, by Theorem 3, one has that deg(Fx2 ) = 3, and deg(Fx3 ) = 2. In fact, F = x2 2 + 5 x2 x3 + 3 x2 + 5 x3 2 + 7 x3 + 3 + x3 3 .

π1,2 (P( t )) = M (N ( t )), where N ( t ) :=

Example 1. Let V be the surface parametrized by P(t1 , t2 ) =  t1 2 1 + t1 + 2 t2 t1 + 2 t2 − 1 − t1 2 − 4 t1 t2 − 4 t2 2 , , . t1 + t2 (t1 + 2 t2 )3 (t1 + 2 t2 )2

n1 ( t ) = I(π1,2 (P( t ))). n2 ( t )

Once we have analyzed the special case of cylinders, we deal with the general situation. Therefore, we assume that V is not a cylinder over any of the coordinate planes. Thus gcd(Gi , Gj ) = 1. In addition, we observe that the polynomials Tπi,j (P( t )) and Sπi,j (P( t )) , introduced in Theorem 2, are not identically zero (see Corollary 1). We start with the following lemma. For this purpose, we observe that P(K2 ) is a constructible subset of V (see [5]), and therefore there exist non-empty open subsets included in it.

Furthermore, we observe that no component of π1,2 (P( t )) can be constant, since otherwise the corresponding polynomial Gi is zero and we have excluded this case. In this situation, the hypotheses of Lemma 2 hold and therefore H( t , h ) 6= 1. Conversely, let H( t , h ) 6= 1. First, we prove that H( t , h ) does not have factors depending only on the variable t . Indeed, if this is the case then there exists a non-constant polynomial L( t ) dividing G1 ( t , h ). Now, observe that gcd(L( t ), q1 ( t )) = 1 since otherwise it would imply that gcd(p1 ( t ), q1 ( t )) 6= 1, which is impossible. Then, we take α ∈ K2 such that L( α ) = 0 and q1 ( α ) 6= 0; note that this is always possible by B´ezout’s theorem because gcd(L, q1 ) = 1. In this situation, one gets that p1 ( h )/q1 ( h ) = p1 ( α )/q1 ( α ); that is p1 ( h )/q1 ( h ) is constant, case that has been excluded. Similarly, one may prove that H( t , h ) ∈ K[ h ][ t ] \ K[ h ]. Therefore, the hypotheses of Lemma 1 hold for H. Thus, we take A = (A1 , A2 ) ∈ (F \ K) × K (similarly if A ∈ K × (F \ K)) such that H(A, h ) = 0. Moreover, let us see that qi (A) 6= 0, for i = 1, 2. Indeed, let us assume that q1 (A) = 0; similarly if q2 (A) = 0. Then, since G1 (A, h ) = 0, one deduces that p1 (A) = 0. Thus, since the tj –coordinates of the intersection points of the polynomials p1 , q1 are given by Resti (p1 , q1 ) ∈ K[tj ], j, i ∈ {1, 2}, j 6= i, we deduce that A ∈ K2 which is a contradiction. Now, from the equalities Gi (A, h ) = 0 for i = 1, 2, and taking into account that qi (A) 6= 0, we deduce that

Lemma 3. Let V not be a cylinder over the x1 x2 plane, and let V0 ⊂ P(K2 ) be a non-empty open subset of V. Then, there exists a non-empty open subset Ω of K2 such that for all α ∈ Ω it holds that 1. P( α ) is defined. 2. If P( α ) = (β1 , β2 , β3 ), then the intersection of the line, of equation {x1 = β1 , x2 = β2 }, with V is included in V0 . Proof. For P = (a, b, c) ∈ V, we denote by LP the line of equation {x1 = a, x2 = b}. Let C1 = V \ V0 . If C1 = ∅ then Ω = K2 . Thus, let C1 6= ∅. We consider the set −1 C2 = V ∩ π1,2 (π1,2 (C1 )), and let C = {P ∈ V | LP 6⊂ V0 }. Observe that C2 consists in the points (a, b, c) ∈ V such that there exists γ ∈ K and (a, b, γ) ∈ V0 ; i.e. (a, b, γ) ∈ V but (a, b, γ) 6∈ C1 . Let us see that C ⊂ C2 . Indeed, let P = (a, b, c) ∈ C, then P ∈ V and there exists γ ∈ K such that (a, b, γ) ∈ LP but (a, b, γ) 6∈ V0 . Therefore, (a, b) ∈ −1 π1,2 (π1,2 (C1 )), and hence P ∈ C2 . Thus

π1,2 (P(A1 ( h ), A2 )) = π1,2 (P(h1 , A2 ))◦A1 ( h ) = π1,2 (P( h )).

C ⊂ C2 ⊂ C2 .

In this situation, let W1 and W2 be the Zariski closures of {π1,2 (P( h )) | h ∈ K2 } and {π1,2 (P(h1 , A2 )) | h1 ∈ K}, respectively. Then, because of the above equality one has that W1 ⊂ W2 . Moreover, observe that π1,2 (P(h1 , A2 )) 6∈ K2 , because it would implies that π1,2 (P( h )) = π1,2 (P(h1 , A2 )) ◦ A1 ( h ) ∈ K2 which is impossible since P( h ) parametrizes a

Let V1 = V \ C2 , and let Ω = P −1 (V0 ∩ V1 ). Note that Ω is an open subset of K2 since V0 ∩ V1 is open. In this situation, if α ∈ Ω then P( α ) ∈ V0 ⊂ P(K2 ), and therefore it is defined, and P( α ) ∈ V1 , which implies that P( α ) 6∈ C2 , and hence P( α ) 6∈ C. Thus LP( α ) ⊂ V0 .

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In order to finish the prove, we have to show that Ω is not empty, which is equivalent to show that V1 6= ∅ because V0 6= ∅ and V is irreducible. Let us assume that V1 = ∅. −1 Then C2 = V. This implies that π1,2 (π1,2 (C1 )) = V. Thus, π1,2 (V) ⊂ π1,2 (C1 ). Since C1 6= ∅ then π1,2 (C1 ) 6= ∅. Moreover, since dim(C1 ) ≤ 1 then dim(π1,2 (V)) ≤ 1. If this dimension is 0, then π1,2 (P( t )) is a point, which is impossible because V is a surface. Let us assume that dim(π1,2 (V)) = 1. Then, π1,2 is not finite (see Theorem 7, pp. 76 in [18]), and therefore for almost all (a, b) ∈ π1,2 (V) the univariate polynomial F (a, b, x3 ) has infinitely many solutions. We write F (x1 , x2 , x3 ) as

that by definition degx3 (V) = degx3 (F ) and that by Corollary 1, it holds that degree(φR ) = degt1 (SR ) = degt2 (TR ). Then, we prove that degx3 (V) = degree(φR ). For this purpose, we will construct a non-empty open subset of K2 as the intersection of several open subsets. Let   Card(FR ( α )) = deg (SRα (t1 )) = t1 Ω1 = α ∈ K2 . = degt1 (SR (t1 )) = degree(φR ). By Corollary 1, it holds that Ω1 is open and non-empty. We also consider the non-empty open subset Ω2 = { α ∈ K2 | G4 ( α ) 6= 0}. Also, let   2 R( α ) is defined and Ω3 = α ∈ K , ∂F Resx3 (F, ∂x )(R( α )) · `(R( α )) 6= 0 3

n−1 F = fn (x1 , x2 )xn + · · · + f0 (x1 , x2 ). 3 + fn−1 (x1 , x2 )x3

Then, we deduce that for every (a, b) ∈ π1,2 (V) it holds that fi (a, b) = 0, for i = 1, . . . , n. Since π1,2 (V) contains infinitely many points, by B´ezout Theorem, we get that there exists h(x1 , x2 ) ∈ K[x1 , x2 ] such that h divides fi , for i = 1, . . . , n. Therefore, h divides F , and since F is irreducible, we obtain that F = h. But this is impossible, because V is not a cylinder over the x1 x2 plane.

where `(x1 , x2 ) denotes the leading coefficient of F w.r.t. x3 . Let us see that Ω3 is non-empty. Indeed, it is clear that the first condition is open and non-empty. Furthermore ` is not identically zero, since V is not a cylinder over the x1 x2 plane and therefore degx3 (F ) ≥ 1. Moreover, `(R( t )) is not identically zero, since otherwise P( t ) would parametrize a cylinder over the x1 x2 . In addition, ∂F Resx3 (F, ∂x ) ∈ K[x1 , x2 ] is not identically zero because F 3 is irreducible and degx3 (F ) ≥ 1, and reasoning as above one ∂F has that Resx3 (F, ∂x )(R(t)) is not identically zero. Con3 sequently Ω3 is non-empty. Also, let Ω4 be the non-empty open subset of Lemma 3, taking as V0 as a non-empty open subset of V which points are reachable via P( t ) with injectivity. Finally, let Ω5 be the non-empty open subset introduced in Lemma 4, statement 2 in [10], when applied to R( t ) and q3 ( t ). In this situation, we introduce the nonempty open subset of K2 :

Remark 3. Note that Lemma 3 can be similarly stated if V is not a cylinder over the other coordinate planes. Remark 4. We observe in Lemma 3 it is necessary to exclude cylinders. For instance, take V as the cylinder of equation x21 + x22 = 1, and   2t1 t21 − 1 1 P( t ) = . , , t21 + 1 t21 + 1 t2 In the following theorem, we present the formulae for the partial degrees of V. This theorem generalizes Theorem 3, and the partial degrees are given in terms of the polynomials Sπi,j (P( t )) (resp. Tπi,j (P( t )) ), introduced in Theorem 2. We remark that these polynomials can be computed by using Theorems 1 and 2, and Corollary 1.

Ω = Ω 1 ∩ · · · ∩ Ω5 . Now we take an element α ∈ Ω, and let (β1 , β2 , β3 ) = P( α ); note that P( α ) is defined because α ∈ Ω2 . We consider the intersection of V and the line L(β1 ,β2 ) of equation {x1 = β1 , x2 = β2 }. Since α ∈ Ω3 , one has that `(β1 , β2 ) 6= 0 and hence degx3 (F (β1 , β2 , x3 )) = degx3 (F ). Let us say that m := degx3 (F ) and s := degree(φR ). Moreover, since

Theorem 5. (General Case) Let V not be a cylinder over any of the coordinate planes (i.e. gcd(Gi , Gj ) = 1), and let Sπi,j (P( t )) , Tπi,j (P( t ))

Resx3 (F,

be the polynomials introduced in Theorem 2. Then, it holds that

∂F )(β1 , β2 ) 6= 0, ∂x3

and `(β1 , β2 ) 6= 0, one has that all roots of F (β1 , β2 , x3 ) are different; let us call them {r1 , . . . , rm }. Since α ∈ Ω4 the m points {(β1 , β2 , ri )}i=1,...,m are reachable by P( t ). Let t i ∈ K2 such that P( t i ) = (β1 , β2 , ri ). Then { t 1 , . . . , t m } ⊂ FR ( α ). Moreover, since the points {(β1 , β2 , ri )}i=1,...,m are all different then Card({ t 1 , . . . , t m }) = m. Thus, since α ∈ Ω1 then m ≤ s. Let us assume that m < s, then there exists γ ∈ FR ( α ) \ { t 1 , . . . , t m }. Since α ∈ Ω5 , and γ ∈ FR ( α ), one has that q3 (γ) 6= 0. That is P(γ) is defined. Moreover, since the points (β1 , β2 , ri ) are reachable with injectivity and γ 6= t i , one has that L(β1 ,β2 ) ∩ V contains more than m points. Therefore, the polynomial F (β1 , β2 , x3 ) has more than m roots, which is impossible because degx3 (F (β1 , β2 , x3 )) = m because α ∈ Ω3 . Now, let us prove statement (2). Let us assume that the hypothesis holds for degx3 (V). The reasoning is similar for the two other cases. Also, let  ?  p1 ( t ) p?2 ( t ) p?3 ( t ) , , P ?( t ) = q1? ( t ) q2? ( t ) q3? ( t )

(1) If P(t1 , t2 ) is proper then degxk (V) = degxk (F ) = degree(φπi,j (P) ) = degt1 (Sπi,j (P( t )) ) = degt2 (Tπi,j (P( t )) ) where i, j, k ∈ {1, 2, 3}, i < j and i 6= k 6= j. (2) If there exist i, j, k ∈ {1, 2, 3}, with i < j and i 6= k 6= j, and such degxk (V) = degxk (F ) = degree(φπi,j (P) ) = degt1 (Sπi,j (P( t )) ) = degt2 (Tπi,j (P( t )) ) then, P( t ) is proper. Proof. Let us prove statement (1). We prove the result for degx3 (V); for x1 and x2 the reasoning is similar. We denote π1,2 (P( t )) by R( t ) ∈ K( t )2 . First of all, we remark

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be a proper parametrization of V (see [12]), and let us denote π1,2 (P ∗ ( t )) by R∗ ( t ) and π1,2 (P( t )) by R( t ). Then, there exists M ( t ) ∈ K( t )2 such that P ? (M ( t )) = P( t ); in fact M ( t ) = I(P( t )) where I(x1 , x2 , x3 ) denotes the inverse of P ∗ ( t ). We observe that it holds that R? (M ( t )) = R( t ). Now, since P ? ( t ) is a proper parametrization, by Statement (1), one has that degx3 (V) = degree(φR? ). Therefore, since by hypotheses we have that degx3 (V) = degree(φR ), one gets that

over the coordinate planes. Now, we consider the rational functions R1,2 ( t ) = 



,

R1,3 ( t ) = 

degree(φR ) = degree(φR? ).

24t2 + (t1 + t2 )3 + 10(t1 + t2 )2 + 25t1 (t1 + t2 )2 + 5t1 + 10t2 , (−t1 − t2 )(t1 + 5 + t2 ) t2



R2,3 ( t ) =   (t1 + t2 )2 + 5t1 + 6t2 (t1 + t2 )2 + 5t1 + 10t2 , . t1 + t2 t2

Since the degree is multiplicative with respect to composition, and R? (M ( t )) = R( t ), we deduce that degree(φM ) = 1. Hence M ( t ) is invertible, and since P ? ( t ) is proper and P ? (M ( t )) = P( t ), one concludes that P( t ) is proper.

Applying Corollary 1, we get that d1,2 = 2, d1,3 = 1, d2,3 = 1. Thus, by Theorem 5, Statement 1, we deduce that degx3 (F ) = 2, degx1 (F ) = 1 and degx2 (F ) = 1. In fact,

Applying Theorem 5, one deduces the following corollaries.

F = x1 x23 − 9x3 x1 + 20x1 + 4 − x3 + 25x2 − 10x2 x3 + x2 x23 .

Corollary 2. (Upper Bounds) Let V not be a cylinder over any of the coordinate planes (i.e. gcd(Gi , Gj ) = 1), and let

Example 3. Let V be the offset to the elliptic paraboloid −x1 + x22 + x23 = 0, parametrized by (see Theorem 2 in [15]): P(t1 , t2 ) = (p1 /q1 , p2 /q2 , p3 /q3 ), where

Sπi,j (P( t )) , Tπi,j (P( t ))

p1 = 1−2t22 −6t21 −4t42 t41 +8t42 t21 +4t22 t21 +2t22 t41 +4t42 +15t41 +4t42 t12 1 − 4 t6 −8t2 t6 +2t2 t8 −20t6 +15t8 +t12 −6t10 −t6 − 4t42 t81 +8t42 t10 −16t 1 2 1 2 1 2 1 1 1 1 1 2 1 2 10 2 6 12 6t62 t21 − 15t62 t41 − 20t62 t61 − 15t62 t81 − 6t62 t10 1 − t2 t1 + 4t1 t2 − 2t1 2t2 , q1 = (t22 +2t22 t21 +2t21 −t41 +t22 t41 −1)2 (1−2t21 +t41 +t22 +2t22 t21 +t22 t41 ), p2 = −(−1 + t21 )2 t2 (−1 + 2t21 − t41 + 3t22 + 6t22 t21 + 3t22 t41 ), q2 = q3 = (t22 + 2t22 t21 + 2t21 − t41 + t22 t41 − 1)(1 − 2t21 + t41 + t22 + 2t22 t21 + t22 t41 ), p3 = −2(−1 + t21 )t1 t2 (−1 + 2t21 − t41 + 3t22 + 6t22 t21 + 3t22 t41 ).

be the polynomials introduced in Theorem 2. Then, it holds that degxk (V) = degxk (F ) ≤ degree(φπi,j (P) ) = degt1 (Sπi,j (P( t )) ) = degt2 (Tπi,j (P( t )) ) where i, j, k ∈ {1, 2, 3}, i < j and i 6= k 6= j.

Applying Corollary 1, we get that d1,2 = 12, d1,3 = 12, d2,3 = 8. Thus, by Theorem 5, Statement 2, we deduce that P( t ) is not proper. Observe that by Corollary 2, we get the upper bounds degx3 (F ) ≤ 12, degx1 (F ) ≤ 8, and degx2 (F ) ≤ 12. In fact, F (x1 , x2 , x3 ) = −25 + 40x1 + 9x21 − 40x31 + 16x41 + 28x22 + 6x1 x22 − 32x22 x31 − 47x42 − 40x1 x42 + 16x21 x42 + 16x62 + 28x23 + 6x1 x23 − 32x31 x23 − 94x22 x23 − 80x22 x1 x23 + 32x23 x21 x22 + 48x23 x42 − 47x43 − 40x1 x43 + 16x21 x43 + 48x43 x22 + 16x63 .

Proof. The proof follows as the proof of statement 1 in Theorem 5 but taking Ω4 as in Lemma 3 applied to any non-empty open subset V0 ⊂ P(K2 ). Note in the proof of statement 1 in Theorem 5, properness has only been used to prove that m cannot be strictly smaller that s. Corollary 3. Let V not be a cylinder over any of the coordinate planes (i.e. gcd(Gi , Gj ) = 1), and let Sπi,j (P( t )) , Tπi,j (P( t ))

5.

be the polynomials introduced in Theorem 2. If there exist i, j, k ∈ {1, 2, 3}, with i < j and i 6= k 6= j, and such degree(φπi,j (P) ) = degt1 (Sπi,j (P( t )) ) = degt2 (Tπi,j (P( t )) ) = 1 then, P( t ) is proper. Proof. It follows from Theorem 5 and Corollary 2 We illustrate the theorem by some examples. In these examples, for simplicity, we denote πi,j (P( t )) by Ri,j ( t ), and degree(φRi,j ) by di,j . Example 2. Let V be parametrized by P( t ) =  24t2 + (t1 + t2 )3 + 10(t1 + t2 )2 + 25t1 , (−t1 − t2 )(t1 + 5 + t2 ) (t1 + t2 )2 + 5t1 + 6t2 (t1 + t2 )2 + 5t1 + 10t2 , t1 + t2 t2

24t2 + (t1 + t2 )3 + 10(t1 + t2 )2 + 25t1 (t1 + t2 )2 + 5t1 + 6t2 , (−t1 − t2 )(t1 + 5 + t2 ) t1 + t2



.

Applying Theorems 5 and 6 in [10], we get that P( t ) is proper, and by Theorem 4, we check that V is not a cylinder

307

REFERENCES

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[7] Kotsireas I. S., (2004), Panorama of Methods for Exact Implicitization of Algebraic Curves and Surfaces. Geometric Computation. Falai Chen and Dongming Wang (eds.). Lecture Notes Series on Computing, Vol. 11, Chapter 4, World Scientific Publishing Co., Singapore. [8] Marco, A., Mart´ınez J.J., (2001) Using polynomial interpolation for implicitizing algebraic curves. Computer Aided Geometry Design, Vol. 18, pp. 309-319. [9] P´erez-D´ıaz S., Schicho J., Sendra J.R. (2002). Properness and Inversion of Rational Parametrizations of Surfaces. Applicable Algebra in Engineering, Communication and Computing, Vol. 13, pp.29-51. [10] P´erez-D´ıaz, S., Sendra, J.R., (2004) Computation of the Degree of Rational Surface Parametrizations. Journal of Pure and Applied Algebra. Vol. 193/1-3. pp. 99-121. [11] San Segundo F., Sendra J.R (2005). Degree Formulae of Offsets Curves. Journal of Pure and Applied Algebra. Vol. 195/3. pp. 301-335. [12] Schicho J. (1998), Rational Parametrization of Surfaces. J. of Symb. Comp. Vol. 26, pp. 1-9.

[13] Schicho J. (1999). A degree bound for the parametrization of a rational surface. Journal of Pure and Applied Algebra. Vol. 145, pp. 91-105. [14] Sederberg T.W. (1986), Improperly Parametrized Rational Curves. Computer Aided Geometric Design, Vol. 3, pp. 67-75. [15] Sendra J., Sendra J.R., (2000). Rationality Analysis and Direct Parametrization of Generalized Offsets to Quadrics. Applicable Algebra in Engineering, Communication and Computing. Vol. 11, pp. 111-139. [16] Sendra J.R., Winkler F. (2001) .Computation of the degree of a Rational Map. Proc. ISSAC–2001, B. Mourrain (ed.), pp. 317-322. [17] Sendra J.R., Winkler F. (2001). Tracing Index of Rational Curve Parametrizations. Computer Aided Geometric Design, Vol. 18, no. 8, pp. 771-795. [18] Shafarevich I.R. (1994). Basic algebraic geometry Schemes; 1 Varieties in projective space. vol. 1.Springer-Verlag, Berlin, New York.

308

Computing the Rank and a Small Nullspace Basis of a Polynomial Matrix∗ Arne Storjohann

Gilles Villard

School of Computer Science, U. Waterloo Waterloo Ontario N2L 3G1 Canada

´ CNRS, LIP, Ecole Normale Superieure de Lyon ´ 46, Allee ´ d’Italie, 69364 Lyon Cedex 07, France

http://www.scg.uwaterloo.ca/~astorjoh

http://perso.ens-lyon.fr/gilles.villard

ABSTRACT

The cost of the algorithm is O˜(MM(n, d)) = O˜(nω d) operations in K. If M is m × n of rank r, a more precise and rank-sensitive expression of the cost is O˜(nmrω−2 d) (see Theorem 7.3). The soft-O notation O˜ indicates missing logarithmic factors α(log n)β (log d)γ for three positive real constants α, β, γ. We mention previous works on the subject in Section 2. Our main idea is to combine matrix lifting techniques [29, 30], minimal bases computation and matrix fraction reconstruction [1, 15, 16], together with a degree/dimension compromise for keeping the cost of the computation as low as possible. Within the target complexity, lifting used alone only allows to obtain few vectors of large degrees, while minimal bases used alone only leadx to an incomplete set of vectors of small degrees. Our study extends the knowledge of the interaction between matrix multiplication and other basic linear algebra problems on matrices over K[x]. The interaction is quite well known for linear algebra over an abstract field. For instance we refer to the survey [8, Chapter 16] for a list of problems on matrices in Kn×n that can be solved in O(nω ) or O˜(nω ) operations in K. Only recent results give an analogous view (although incomplete) of the situation for polynomial matrices. It is known that the following problems can be solved with O˜(MM(n, d)) operations: linear system solution, determinant, order d approximants, Smith normal form, and, for a non-singular matrix, column reduction [15, 29, 30]. It is possible to compute the inverse of a generic matrix in essentially optimal time O˜(n3 d) [16]. We may also consider the problem of computing the Frobenius normal form, thus in particular the characteristic polynomial, of a square matrix. It does not seem to be known how to calculate the Frobenius form in time O˜(MM(n, d)). The best known estimate O˜(n2.7 d) is given in [21] (see also [18]) with ω = 2.376 [11]. Hence, we augment the above list of problems solved in O˜(MM(n, d)) with the certified computation of the rank and a nullspace basis. This improvement is made possible by combining in a new way the key ideas of [15, 16, 29]. For the rank, the target complexity O˜(MM(n, d)) was only attainable by a Monte Carlo (non-certified) approach consisting in computing the rank of M (x0 ) for x0 a random value in K. In obtaining a certified value of the rank and a nullspace basis within the target complexity, a difficulty is related to the output size. For M ∈ K[x]2n×n of degree d and rank n, Gaussian elimination (fraction free or using evaluation/interpolation) leads to a basis of n vectors of degrees nd in K[x]2n in the worst-case, hence to an output size in Θ(n3 d). A complexity in O˜(nω d) must therefore rely on a different strategy.

We reduce the problem of computing the rank and a nullspace basis of a univariate polynomial matrix to polynomial matrix multiplication. For an input n × n matrix of degree d over a field K we give a rank and nullspace algorithm using about the same number of operations as for multiplying two matrices of dimension n and degree d. If the latter multiplication is done in MM(n, d) = O˜(nω d) operations, with ω the exponent of matrix multiplication over K, then the algorithm uses O˜(MM(n, d)) operations in K. For m×n matrices of rank r and degree d, the cost expression is O˜(nmrω−2 d). The soft-O notation O˜ indicates some missing logarithmic factors. The method is randomized with Las Vegas certification. We achieve our results in part through a combination of matrix Hensel high-order lifting and matrix minimal fraction reconstruction, and through the computation of minimal or small degree vectors in the nullspace seen as a K[x]-module. Categories and Subject Descriptors: I.1[Symbolic and Algebraic Manipulation]: Algorithms; F.2.1[Analysis of Algorithms and Problem Complexity]: Numerical Algorithms and Problems—Computations on matrices General Terms: Algorithms Keywords: linear algebra, polynomial matrix, matrix rank, nullspace basis, minimal polynomial basis

1.

INTRODUCTION

Two n × n univariate polynomial matrices over a field K, whose entries have degree d at most, can be multiplied in MM(n, d) = O˜(nω d) operations in K [7, 9] where ω is the exponent of matrix multiplication over K [8, Chapter 15]. For M ∈ K[x]n×n of degree d we propose an algorithm that uses about the same number of operations for computing the rank r of M over K(x), and n − r linearly independent vectors Ni in K[x]n such that Ni M = 0, 1 ≤ i ≤ n − r. ∗An extended version including proofs is available as the research report [31] (http://www.ens-lyon.fr/LIP/Pub).

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. ISSAC’05, July 24–27, 2005, Beijing, China. Copyright 2005 ACM 1-59593-095-7/05/0007 ...$5.00.

309

We propose a sort of elimination scheme based on minimal polynomial bases. A minimal basis of the nullspace as K[x]-module is a basis with lowest possible degrees (all necessary definitions are given in Section 3). For M ∈ K[x]2n×n as above, the total size of a minimal basis of the nullspace is in O(n2 d) (see Theorem 3.3). However, it is not known how to reduce the problem of computing such a basis to that of polynomial matrix multiplication. In the same context, minimal bases have been already used for computing the inverse of a polynomial matrix in [16], but only the generic case has been solved. Indeed, for a generic M ∈ K[x]2n×n , the degrees in a minimal basis of the nullspace are all equal to the input degree d, and somehow, a basis is easy to compute in O˜(MM(n, d)) operations [16, Section 4]. In the general case, the vector degrees in a minimal basis may be unbalanced, they range between 0 to nd. Known methods whose cost is essentially driven by the highest degree do not seem to allow our objective (see Section 2). Our solution presented in Section 7 is to slightly relax the problem, and to compute a small degree—rather than minimal—nullspace basis in a logarithmic number of steps. We rely on the fact that even in the unbalanced degree case, the sum of the degrees remains bounded by nd (Theorem 3.3). Intuitively, at step k for 1 ≤ k ≤ log2 n, we compute about n/2k nullspace vectors of degrees less than 2k d. Algorithm Nullspace2n (Section 7), for the whole nullspace, calls at most log2 n times Algorithm Nullspace minimal vectors (Section 6), for nullspace vectors of bounded degree δ, with increasing degree thresholds δ. To keep the cost as low as possible, the degree increase requires to reduce the dimensions of involved matrices in the same proportion (see (14)). We refer to an analogous degree/dimension compromise in [30, Section 17] for computing the Smith normal form, and in [16, Section 2] for matrix inversion.

factorization in Section 2) then gives Hp = S −1 Np = BA−1 P.

(3)

We prove that “good” choices of P imply that S, denominator matrix for Hp , is also a denominator matrix for H (Proposition 4.1) and that vectors in the nullspace of M can be recovered (Proposition 5.4). Indeed, the computation of S[H − Im−n ] mod xδ+1 gives row vectors in the nullspace of degrees bounded by δ (Proposition 6.4). From a candidate Monte Carlo value r0 for the rank, in log2 n steps of compression/uncompression (and choices of δ and p) combined with matrix lifting/matrix fraction reconstruction, we are able to compute candidate vectors for a nullspace basis. A final multiplication certifies that the rank is correct (i.e., r0 = r) and that a nullspace has actually been computed (Section 7.2). Although for each degree threshold δ we compute a minimal basis for the nullspace of (2), the compression strategy unfortunately does not lead to a minimal basis for the whole nullspace of (1). However, we prove that small degree vectors are obtained (Proposition 7.1). Our algorithms are randomized of Las Vegas kind—always correct, probably fast. Randomization is linked to the compression stages where the matrices P are chosen at random. We also use random matrices Q over K for linear independence preconditioning [10], or random evaluation points x0 in K. Our results are proven for symbolic points x0 and matrices P and Q. By evaluation [12, 34, 27], the same results hold with high probability for random x0 , P and Q if K has enough elements (Remark 7.4). The cost estimates might increase by poly-logarithmic factors in the case of small fields (with the introduction of an algebraic extension). We skip the details here, and refer for instance to the techniques used in [10, 19, 20] and to the references therein. Complexity estimates. We study the cost of the algorithms by bounding the number of field operations in K on an algebraic random access machine. In [15] and [30], ad hoc cost functions have been defined for matrix polynomial problems that can be reduced recursively to matrix polynomial multiplication:

Algorithm overview. For a general view of the process, including successive compressions of the problem into smaller problems for reducing dimensions, consider   A M= ∈ K[x]m×n (1) B

log2 d

with A square and non-singular. The rows of the matrix [BA−1 −Im−n ] give a basis of the nullspace of M . However, as noticed previously a direct calculation of BA−1 would be too expensive. Now, note that if [BA−1 − Im−n ] = S −1 N , for S and N two appropriate polynomial matrices, then the rows of S[BA−1 − Im−n ] = N are also in the nullspace. Considering polynomial matrices N and S instead of [BA−1 −Im−n ] will take advantage of minimal bases properties, and allow us to manipulate smaller degrees. For computing a nullspace basis we proceed the following way. We deal with a small number of submatrices of the initial input for reducing the problem to   A M= ∈ K[x](n+p)×n , 1 ≤ p ≤ n, (2) B

MM0 (n, d) =

X

2i MM(n, 2−i d)

(4)

i=0

and log2 n

MM(n, d) =

X

4i MM(2−i n, d) + n2 (log n)B(d)

(5)

i=0

where B(d) is the cost for solving the extended gcd problem for two polynomial in K[x] of degree bounded by d. The two techniques we combine are lifting and matrix fraction reconstruction. Lifting for computing the expansion of BA−1 to the order η (see Step (c) in Section 6) has cost O(MM(n, d) + log(η/d)dpη/ndeMM(n, d))

(6)

operations in K [30, Proposition 15]. From the latter expansion of BA−1 we will solve (3) (see Step (e) in Section 6) in

n×p

and introduce compressing matrices P ∈ K[x] . The successive choices of p are guided by the compromise with the degree. For a given p, we start with a matrix lifting/fraction reconstruction phase. We compute an expansion of H = BA−1 in K[[x]]p×n using [29, 30] to sufficiently high order η, and “compress” it to Hp = BA−1 P ∈ K[[x]]p×p . A reconstruction phase [1, 15] (see also the comments about coprime

O(MM0 (p, η) + ηMM(p))

(7)

operations in K [15, §2]. These rather technical complexity notations has been proposed for capturing the reduction to polynomial matrix multiplication. Both (6) and (7) concern algorithms that work in a logarithmic number of stages, with matrices whose dimensions and degrees are changing [15, 30].

310

We will keep both (6) and (7) in O˜(MM(n, d)) by retricting pη to O(nd) (degree/dimension compromise). For simplifying the cost results in this paper we consider either that MM(n, d) = O(nω M(d)) [9], or, when the field K has at least 2d + 1 elements [6, 7], MM(n, d) = O(nω d + n2 M(d)). Here M(d) is the number of operations in K required for multiplying two polynomials in K[x] of degree d. Hence, taking M(d) = O(d log d log log d) [9], and B(d) = O(M(d) log d) [22, 26], we assume that MM0 (n, d) = O(MM(n, d) log d), MM(n, d) = O((MM(n, d) + n2 B(d)) log n).

approaches—via state-space realizations or resultants— for the problem of coprime matrix fraction description or coprime factorization [17, Chapter 6]. We see from (3) that we will use a solution to the latter problem a logarithmic number of times. If all matrices involved are of degree d, then we use the σ-basis algorithm of [1], and the corresponding reduction to polynomial matrix multiplication of [15]. A solution of the coprime factorization in case of unbalanced degree, in a way similar to the block-Toeplitz approach, is faced with the question of saving a factor n in the cost. Known algorithms seem to have a cost driven only by the highest degree in the factorization, rather than by the sum of the involved degrees as we propose. Our work is a derivation of an elimination scheme using minimal bases directly on polynomial matrices. Our compression/uncompression strategy can be compared to the techniques used for the staircase algorithm of [3, 25] for preserving a special structure. We somehow generalize the latter to the case of polynomial matrices for reducing the factorization problem with input BA−1 to the polynomial matrix multiplication.

(8)

If the assumption (8) is not made then some of our cost results that use MM(n, d) are not valid. However, we state our algorithms in terms of polynomial matrix multiplication; precise complexity estimates in terms of the cost functions (4) and (5) could be derived with some extra care.

2.

PREVIOUS WORKS

The rank and a basis for the nullspace of a matrix M ∈ K[x]m×n of degree d and rank r may be computed by fraction free Gaussian elimination in O˜(nmrω−1 d) operations in K [28, Chapter 2]. The same asymptotic estimate may also be obtained using evaluation/interpolation techniques such as Chinese remaindering [14, Section 5.5]. Therefore, compared to these classical approaches, we improve the cost by a factor n in the worst-case (2n × n full column-rank matrix). An elimination strategy specific to polynomial matrices is given in [24] that improves, asymptotically in the dimensions, on O˜(nmrω−1 d) , and computes the rank by a deterministic algorithm in O(nmrd2 ) operations in K. But how to incorporate matrix multiplication, and generalize the approach to computing the nullspace, is not known. An alternative to the matrix over the polynomials approach above is to linearize the problem. A first type of linearization is to consider a degree one matrix of larger dimension with the same structural invariants (see the definition of the Kronecker indices in Section 3) [4]. A degree one matrix is a matrix pencil and an important literature exists on the topic. A minimal nullspace basis of a pencil may be computed through the calculation of the Kronecker canonical form. To our knowledge, the best known complexity for computing the Kronecker form of an m × n pencil is O(m2 n) [3, 23, 25]. Taking into account the dimension increase due to the linearization we may evaluate that computing a minimal basis of M would cost O((md)2 (nd)) = O(m2 nd3 ). This approach is superior to ours concerning the quality of the output basis which is minimal. However, it is unclear how it can lead to the reduction to polynomial matrix multiplication that we establish. A second alternative and different linearization of the problem is to associate to M a generalized Sylvester matrix (i.e., a block-Toeplitz matrix [5]) or another type of resultant. This has been heavily used for control theory problems and in linear algebra. A polynomial vector of degree δ in the nullspace of M may be obtained from the nullspace of a block-Toeplitz of dimension about nδ. This leads to costs too high by a factor of n when the degrees in a minimal nullspace basis are unbalanced. We are not aware of an approach based on successive compression here that would allow to save a factor n and to introduce polynomial matrix multiplication. These two types of linearization correspond to two main

3.

PRELIMINARIES

We give here some definitions and results about minimal bases [13] and matrix fraction descriptions that will be used in the rest of the paper. For a comprehensive treatment we refer to [17, Chapter 6]. For a matrix M ∈ K[x]m×n of rank r and degree d, we call (left) nullspace the K(x)-vector space of vectors v ∈ K(x)m such that vM = 0. We will compute a basis of that space. The basis will be given by m − r linearly independent polynomial vectors, and is related to the notion of minimal basis of the nullspace seen as a K[x]-module. Definition 3.1. A basis N1 , . . . , Nm−r ∈ K[x]m with degrees δ1 ≤ . . . ≤ δm−r of the nullspace of M seen as a K[x]-module is called a minimal basis if any other nullspace 0 basis with degrees δ10 ≤ . . . ≤ δm−r satisfies δi0 ≥ δi for 1 ≤ i ≤ m − r. In the rest of the text, basis will usually refer to the vector space while minimal basis will refer to the module. The degrees δ1 , . . . , δm−r are structural invariants of the nullspace. They are called the minimal indices of the nullspace basis. The minimal indices of a nullspace basis of M are called the (left) Kronecker indices of M . A full row-rank polynomial matrix N in K[x]l×m is called row-reduced if its leading row coefficient matrix has full rank l. It is called irreducible if its rank is full for all (finite) values of x (i.e., Il is contained in the set of K[x]-linear combinations of columns of N ). These two definitions are used for characterizing minimal bases; we refer to [17, Theorem 6.5-10] for the proof of the following. Theorem 3.2. The rows of N ∈ K[x](m−r)×m , such that N M = 0, form a minimal basis of the nullspace of M if and only if N is row-reduced and irreducible. A key point for keeping the cost of the computation low is the degree transfer between M and a minimal nullspace basis N . The McMillan degree of M of rank r is the maximum of the degrees of the determinants of r×r submatrices of M [17, Exercise 6.5-9]. Theorem 3.3.PThe Kronecker indices of M ∈ K[x]m×n of rank r satisfy m−r i=1 δi ≤ McMillan-deg M.

311

exists a matrix P of degree less than d − 1 in K[x]n×p such that

The reader may refer to [2, Theorem 5.1] for the latter bound. As discussed in the introduction, Gaussian elimination is far too pessimistic when it results in a nullspace basis with size in Θ(n3 d). Theorem 3.3 shows that there exist minimal bases with size in O(n2 d) whose computation should be cheaper. We will use minimal bases in relation with left or right matrix fraction descriptions. For H ∈ K(x)l×m , a left fraction description, i.e. a pair (S, N ) such that H = S −1 N , with S ∈ K[x]l×l and N ∈ K[x]l×m , is irreducible (or coprime), if any non-singular polynomial matrix and left common divisor U of S and N (S = U S 0 and N = U N 0 for polynomial matrices S 0 and N 0 ) is unimodular. We call (left) denominator matrix of H any non-singular polynomial matrix S such that SH is a polynomial matrix N , i.e. such that (S, N ) is a (left) description of H. Analogous definitions hold on the right.

Hp = BA−1 P = CT −1 = S −1 Np ∈ K[x]p×p

where CT −1 is a right irreducible description with T ∈ K[x]p×p of degree less than dν/pe ≤ (n/p)d + 1, and where S −1 Np is a left irreducible description. The proof of Proposition 4.1 [31] relies on the correspondence between the formalism of minimum generating polynomials introduced in [32, 33] and [21, Section 2], and matrix fraction denominators. Remark 4.2. Proposition 4.1 establishes fraction properties for a symbolic P . As a consequence of [31, Lemma 4.1], and [32, Corollary 6.4] or [21, Section 2], the same properties hold for a random matrix P unless it forms a zero of a fixed polynomial (for given A and B) of degree O(nd) in O(nd) variables over K.

¯ − S], such that N ¯M = Lemma 3.4. The rows of N = [N 0, with S non-singular, form a basis for the nullspace of M ¯ is irreducible. as a K[x]-module if and only if S −1 N

5.

In Section 6 we will focus on computing only vectors of degrees bounded by a given δ in a nullspace minimal basis. We define their number κ(= κ(δ)) = max{1 ≤ i ≤ m − r s.t. δi ≤ δ} (the Kronecker indices are arranged in increasing order). Corresponding vectors are called κ first minimal vectors in the nullspace.

4.

COMPRESSED MINIMAL BASES

We will compute a small basis for the nullspace of the input matrix as a set of successive minimal bases of matrices like in (2). The latter minimal bases are computed in two main steps. We first compute an expansion of Hp = BA−1 P and reconstruct a corresponding fraction (3) with denominator S. Then, if P is such that Hp satisfies (11b), we know that S[BA−1 − Ip ] is a polynomial matrix N , which by construction satisfies N M = 0. In the spirit of the scalar polynomial case and of [1] for the matrix case, the reconstruction may be done via Pad´e approximation, and through the computation of particular bases of the nullspace of [−Ip HpT ]T . Indeed we have the equivalence between S −1 Np = Hp and [Np S] · [−Ip HpT ]T = 0. Hence the purpose of this section is to identify the bases of the nullspace of [HpT − Ip ]T that actually lead to minimal bases N for M . Through a conditioning of M let us first specify the location of the leading degree terms in the latter bases (see Theorem 3.3).

MATRIX FRACTIONS AND NULLSPACE

We consider a matrix M = [AT B T ]T ∈ K[x](n+p)×n of degree d as in (2) with A square n × n and invertible. Our study here and in next section focuses on the case p ≤ n which is the heart of the method, and where all difficulties arise for establishing a cost sensitive to p (see Remark 6.6). The rows of H = [H −Ip ] = [BA−1 −Ip ] form a nullspace basis of M . Hence, for N a minimal nullspace basis, there exists a transformation S in K(x)p×p such that SH = N . With the special shape of H we deduce that S is a polynomial matrix in K[x]p×p whose columns are given by the last p columns of N . This leads to the following left matrix fraction description of H: ¯ − S] = S −1 N. (9) H = [H − Ip ] = [BA−1 − Ip ] = S −1 [N

Lemma 5.1. For M as in (2) there exist a matrix Q ∈ K(n+p)×(n+p) such that the McMillan degree of the top n × n submatrix of QM is equal to the McMillan degree of QM (and of M ). This implies that if N is a minimal basis of the nullspace of QM , then S = N·,n+1..n+p is row-reduced with row degrees the Kronecker indices δ1 , . . . , δp .

¯ and S −1 N must be irreThe left fraction description S −1 N ducible by Lemma 3.4. For reducing the cost of our approach we will introduce a (random) column compression Hp of H given by Hp = HP = BA−1 P ∈ K(x)p×p

(11a) (11b)

(10)

Remark 5.2. The property given by the multiplication by Q in Lemma 5.1 will hold for a random Q over K (compare to Remark 4.2).

with P ∈ K[x]n×p . In order to be appropriate for computing the nullspace of M , Hp must keep certain invariants of BA−1 . Next proposition establishes that there exists a P such that—on the ¯ P ) = S −1 Np remains irleft—the description Hp = S −1 (N reducible. With the same P the proposition also shows the existence—on the right—of a description whose denominator matrix has relatively small degree.

In next sections, nullspace vectors v T for M are easily obtained from nullspace vectors wT for QM , indeed v T = wT Q satisfies v T M = wT QM = 0. This conditioning of M —and implicitly of N —will alllow us to compute S, and then deduce N , from a shifted minimal basis for the nullspace of [−Ip HpT ]T . Shifted bases are defined as usual minimal bases by changing the notion of degree. For t¯ a fixed multiindex in Zm , the t¯-degree of a vector v in K[x]m is t¯-deg v = max {deg vi − t¯i }. (12)

Proposition 4.1. Let A ∈ K[x]n×n be non-singular of degree less than d and determinantal degree ν, and let B ∈ K[x]p×n . Assume that S ∈ K[x]p×p is any denominator of a left irreducible fraction description of BA−1 . Then there

1≤i≤m

312

Definition 5.3. A basis of a K[x]-submodule of K[x]m , given by the rows of a matrix N , is called t¯-minimal if N is row-reduced with respect to the t¯-degree. Equivalently, N · ¯ x−t is row-reduced with respect to the usual degree (see [2, Definition 3.1]).

Step (g) applies Proposition 5.4 for partly reconstructing the nullspace of M , and Steps (h) and (i) certify the outputs. Algorithm Nullspace minimal vectors (M ,δ) Input: M ∈ K[x](n+p)×n of degree d, a degree threshold δ, M has full column-rank. Output: κ = max{1 ≤ i ≤ p s.t. δi ≤ δ} independent Ni ∈ K[x]n+p of degree δi in the nullspace of M . (a) M := QM for a random Q ∈ K(n+p)×(n+p) ; (b) M := M (x + x0 ) for x0 random in K; A := M1..n,1..n ; if det A(0) = 0 then fail; /* probably rank M < n */ B := Mn+1..n+p,1..n ; η := δ + d + dnd/pe; (c) H := expansion of BA−1 mod xη ; (d) Hp := HP for P random in K[x]n×p , deg P ≤ d − 1; t¯ = [(d − 1)p , 0p ] = [d − 1, . . . d − 1, 0, . . . , 0] ∈ N2p ; (e) L := [Np S] := a σ-basis with respect to t¯ for [−Ip HpT ]T of order η; (f) κ := nb of rows of [Np S] of t¯-degree at most δ; select κ rows Si of S by increasing degrees; (g) Ni := Si [H − Ip ] mod xδ+1 , 1 ≤ i ≤ κ; Ni (x) := Ni (x − x0 )Q, 1 ≤ i ≤ κ; λ := #{Ni s.t. Ni M = 0}; (h) if λ 6= κ then fail; /* certification of κ */ N (δ) := the κ × (n + p) matrix formed by the Ni ’s; (i) if N (δ) is not row-reduced then fail; /* minimality */ else return κ and Ni , 1 ≤ i ≤ κ. 

For t¯ = [0, . . . , 0] the definition corresponds to the usual definition of minimal bases. The value t¯ = [(d − 1)p , 0p ] below, where (d − 1)p and 0p respectively denote the values d − 1 and 0 repeated p times, is chosen from the degree d − 1 of the compression matrix P of Proposition 4.1. This value forces the row reduction in the last columns of the bases. Proposition 5.4. Let M ∈ K[x](n+p)×n be of full rank such that the matrix S, formed by the last p columns of a minimal basis N for its nullspace, is row-reduced with row degrees the Kronecker indices δ1 , . . . , δp . Assume that P ∈ K[x]n×p satisfies (11). Let t¯ = [(d − 1)p , 0p ] ∈ N2p . Then [Np S] is a t¯-minimal basis for the nullspace of [−Ip HpT ]T ¯ − S] is a minimal if and only if N = S[BA−1 − Ip ] = [N basis for the nullspace of M . Most of the proof of Proposition 5.4 is technical for proving the minimality [31], the main argument comes from (11b) which allows to go from a basis to another. For compressing matrices P which satisfy (11b), Proposition 5.4 establishes strong links between the nullspace of [−Ip HpT ]T and the one of M . In particular, [−Ip HpT ]T and M have the same Kronecker indices. For any given δ, there is a oneto-one correspondence between the vectors of t¯-degree δ in the nullspace of [−Ip HpT ]T , and those of degree δ in the nullspace of M . This is seen from the “S” common part of the bases.

6.

The partial reconstruction of Np and S (i.e., of a t¯-minimal basis at Step (e), and of the denominator matrix at Step (f)) is done using a minimal nullspace basis expansion—or σbasis [1]. We generalize [2, §4.2] and [15, §3] especially for the partial computation aspects.

NULLSPACE MINIMAL VECTORS

Definition 6.1. Let G be in K[[x]]q×p . Let t¯ be a fixed multi-index in Zq . A σ-basis of (matrix-)order d with respect to t¯ for G is a matrix polynomial L in K[x]q×q such that:

We still consider a full column-rank matrix M of degree d. Let δ be a fixed integer and κ(= κ(δ)) be the number of vectors of degree less than δ in a minimal basis N of the K[x]-nullspace of M . In this section we study the cost for computing κ such vectors. Algorithm Nullspace minimal vectors below starts with lifting on a compressed matrix (Proposition 4.1). Then it partially (subject to the degree threshold) computes a denominator matrix S through a partial t¯-minimal basis computation. Using Proposition 5.4 the target nullspace vectors are finally obtained. We prove the algorithm and its cost in the rest of the section. Step (a) is the conditioning seen in Section 5 to ensure the degree dominance of the last p columns of N . Together with the randomized compression of Step (d) studied in Proposition 4.1 this will allow the computation of S at Step (e). Step (b) is a randomized choice for working with a matrix A non-singular at x = 0. The latter condition is required for computing at Step (c) the expansion of BA−1 by lifting [29, 30]. Step (e) partly reconstructs a description S −1 Np from a truncated expansion of Hp . The computation is explained in Lemma 6.3 below, and the selection of small degree rows at Step (f) is justified. Our approach for the reconstruction is very close to the column reduction of [15, §3]. A degree less than δ in S corresponds to a t¯-degree (see (12)) less than δ in [Np S] (the compression using P increases the degree in Np by d − 1), and to a degree less than δ in N .

i) L(x)G(x) ≡ 0 mod xd ; d ii) every v ∈ K[x]q such that v(x)G(x) Pq = O(x ) admits a unique decomposition v T = α L where, for i=1 i i 1 ≤ i ≤ q, Li is the ith row of L, and αi ∈ K[x] is such that deg αi + t¯-deg Li ≤ t¯-deg v.

The reader may notice that we have slightly adapted the notion of order of the original Definition 3.2 of [1] for a fully matrix point of view. We also use the notion of shifted degree (see [2]) equivalently to the notion of defect used in [1, Definition 3.1]. The following shows that a σ-basis to sufficiently high order contains a minimal basis. Lemma 6.2. Let us assume that a minimal nullspace basis of G has κ vectors of t¯-degree at most δ, and consider a σbasis L with respect to t¯. For an approximation order greater than δ + 1, at least κ rows in L have t¯-degree at most δ. Next lemma identify the situation where a σ-basis will give the exact information we need. We assume that we are in the situation of Proposition 5.4, in particular S in the minimal bases has row degrees δ1 , . . . , δp , the Kronecker indices of M and of [−Ip HpT ]T . We fix a value δ and define κ = max{1 ≤ i ≤ p s.t. δi ≤ δ}, and t¯ = [(d − 1)p , 0p ] ∈ N2p .

313

Lemma 6.3. Let us assume we are in the situation of Proposition 5.4. Let L be a σ-basis for [−Ip HpT ]T , with respect to t¯, and of order of approximation at least η = δ + d + dnd/pe. Then exactly κ rows of L have t¯-degree at most δ, are in the nullspace of [−Ip HpT ]T , and have t¯degrees δ1 , . . . , δκ .

Conversely, few vectors of large degrees (small p and large δ) are computed from a high-order lifting and reconstruction with matrices of small dimensions. Remark 6.6. The random compression P is relevant for p < n. Everything we saw is valid for any p, however, when p ≥ n, one may work directly with Hp = H at Step (d).

Our proof of Lemma 6.3 is a modification of the one of [15, Lemma 3.7] for taking into account the degree threshold. We use (11a) for fixing the approximation order.

7.

Corollary 6.5 which uses for (13) a compromise between p and δ, does not directly allow a low-cost computation of large degree vectors in a nullspace of large dimension. For the latter situation, and for computing a whole set of linearly independent vectors in the nullspace of a matrix M in K[x](n+q)×n , we need to successively restrict ourselves to smaller nullspace dimensions (while increasing the degree). Here we take the notation m = n + q for M as in (1). We keep the notation p for submatrices (2), and successive compressions, as in Sections 4-6 .

Proposition 6.4. Let M ∈ K[x](n+p)×n be of full columnrank with Kronecker indices δ1 , . . . , δp . Algorithm Nullspace minimal vectors with inputs M and δ ∈ N returns the quantity κ = max{1 ≤ i ≤ p s.t. δi ≤ δ}, and κ first minimal vectors of the nullspace of M . The algorithm is randomized, it either fails or returns correct values (Las Vegas fashion). For establishing Proposition 6.4 [31] we first verify that if the random choices of x0 , Q and P work as expected then the result is correct. This is obtained from Lemma 6.3 for the nullspace of [−Ip HpT ]T , and using the nullspace correspondence of Proposition 5.4 for the nullspace of M . Then, the certification of the outputs uses Lemma 6.2 and the test (h) for checking the value of κ. Using σ-bases properties we also show that the reduceness at Step (i) is equivalent to the minimality. Note that in any case, the computation of λ ensures that the returned vectors are in the nullspace. The algorithm may fail because the computed value κ is too large. This will happen for bad choices of P , when the nullspace of the compressed matrix (see (11b)), and the approximating σ-basis (see (11a)), does not reflect the nullspace of M correctly. Additionally, even for correct values of κ, the minimality may not be ensured without the test at Step (i). Indeed, a bad choice of Q, depending on P , may lead to a row reduction in the non-dominant part of the basis (see Lemma 5.1), and to a loss of minimality (see Proposition 5.4). In the latter case, a correctly computed value of κ may lead to a smaller value λ after the truncation (g) of a non-minimal vector. The cost of Algorithm Nullspace minimal vectors is essentially given by (6) for the lifting and (7) for the σ-basis i.e the reconstruction phase. It may be stated for arbitrary values of p and δ [31]. We rather give below a complexity estimate in the special case pδ/nd = O(1). The latter corresponds to the degree/dimension compromise that we will realize for the whole nullspace computation in Section 7. We also use (8) for simplifying the reduction to polynomial matrix multiplication.

7.1 Full column-rank and n < m ≤ 2n case Let M ∈ K[x](n+q)×n with 1 ≤ q ≤ n be of degree d and rank n. The way we restrict ourselves to smaller nullspaces is derived from the following observation. Let C be in K(n+p)×(n+q) with 1 ≤ p ≤ q. If CM ∈ K[x](n+p)×n also has full column-rank, then let δ1 , . . . , δp be its Kronecker indices, and with the degree threshold P δ = 2nd/p take κ = max{1 ≤ i ≤ p s.t. δi ≤ δ}. Since p1 δi ≤ nd, at most nd/δ = p/2, hence bp/2c, vectors in a minimal basis of the nullspace of CM may have degrees more than δ, therefore κ ≥ dp/2e. From at least p/2 minimal vectors D1 , . . . , Dκ ∈ K[x]n+p of degrees at most 2nd/p in the nullspace of CM , we obtain κ corresponding vectors Ni = Di C ∈ K[x]n+q in the nullspace of M . Algorithm Nullspace2n , proven in Proposition 7.1 below, uses above observation a logarithmic number of times. For computing the whole nullspace, the algorithm generates a sequence of decreasing dimensions p at Step (h). Following the observation, each time the algorithm passes through the “while loop” the dimension is divided by at least two, hence at most O(log2 q) stages are necessary. This corresponds to O(log2 q) calls to Nullspace minimal vectors with input CM (for different matrices C). Each time the dimension is decreased, the degree threshold is increased in the same proportion at Step (b), we preserve the invariant pδ/(nd) = 2

(14)

that will be used for applying the cost estimate (13). The proof of Proposition 7.1 [31] checks that q vectors in the nullspace are actually computed. Their linear independency is ensured on the fly, and relies on the initial conditioning with Q for working with a top n × n non-singular submatrix. The vectors for updating the nullspace are computed at Step (e) and Step (f) in the nullspace of MI,1..n , ¯ with I¯ = {1, 2, . . . , n, i1 , i2 , . . . , ip }. This is done through the construction of the index selecting matrix C at Step (c) which selects the corresponding rows of M . The choice of the indices {i1 , i2 , . . . , ip } at Step (a), complements the index choices at Step (g) that are kept in I at Step (h) for previous stages, and will provide the linear independency by construction. Another perhaps simpler strategy for ensuring independency could be based on randomization.

Corollary 6.5. Let M ∈ K[x](n+p)×n be of full columnrank and degree d with 1 ≤ p ≤ 2n, and let d ≤ δ ≤ nd. Minimal independent vectors in the nullspace of M , of degrees the Kronecker indices less than δ, can be computed by a randomized Las Vegas (certified) algorithm in O(MM(n, d) log(nd) + n2 B(d) log n + nM(nd))

NULLSPACE BASIS COMPUTATION

(13)

operations in K when pδ/nd = O(1). We see that computing minimal vectors in the nullspace at essentially the cost of multiplying two polynomial matrices relies on the compromise between p and δ. Many vectors of small degrees (large p and small δ) are computed using lifting to a limited order and large matrix reconstruction.

314

Algorithm Nullspace2n (M ) Input: M ∈ K[x](n+q)×n of degree d, M has full column-rank and 1 ≤ q ≤ n. Output: q “small” linearly independent polynomial vectors in the nullspace of M . M := QM for a random Q ∈ K(n+q)×(n+q) ; if det M1..n,1..n (x0 ) = 0 for x0 random in K then fail; I = {}; p := q; while #I < q (a) {i1 , . . . , ip } := {n + 1, . . . , n + q} \ I; (b) δ := 2nd/p; (c) construct C ∈ K(n+p)×(n+q) with Ci,i := 1, 1 ≤ i ≤ n, Cn+j,ij := 1, 1 ≤ j ≤ p, and Ci,j := 0 otherwise; ¯ := CM ∈ K[x](n+p)×n ; (d) M (e) {κ, {Di , 1 ≤ i ≤ κ}} := ¯ , δ); Nullspace minimal vectors (M (δ) Ni = Di C, 1 ≤ i ≤ κ; (δ) (f) N (δ) := the matrix formed by the Ni ’s; (g) J := κ column indices greater than n + 1 such that N1..κ,J is non-singular; (h) I := I ∪ J, p := p − κ; (i) N := [N T (N (δ) )T ]T ; /* nullspace update */ N := N Q; return Ni , 1 ≤ i ≤ q.

Since Algorithm Nullspace minimal vectors is called O(log q) times, and since pδ/nd = 2, (15) is a consequence of (13).

7.2

General case

We now briefly give an idea of our method with a general matrix M ∈ K[x]m×n of degree d. The details will be found in [31, Section 7.2]. We compute the rank r of M and m − r linearly independent and “small” polynomial vectors in the nullspace. Our strategy first uses Monte Carlo (non-certified) randomized techniques for computing a value r0 ≤ r, equal to r with high probability (see Remark 7.4), and a full column˜ ∈ K[x]m×r0 of degree d whose nullspace conrank matrix M tains the nullspace of M . This requires, for Step (a) below, the computation of the rank of M (x0 ) for x0 a random field ˜ = M R for a random n × r0 value, and the multiplication M matrix R over K. We then apply the results of previous sections for computing m − r0 candidate independent vec˜ . We finally test by multiplication tors in the nullspace of M whether these m − r0 vectors are actually in the nullspace of M . A positive answer implies that r ≤ r0 , therefore certifies that r = r0 , and that a correct nullspace representation has been constructed. For computing the candidate nullspace vectors, the case m ≤ 2r0 has been treated in Section 7.1. When m  r0 , the sum of the Kronecker indices is at most r0 d, hence at most r0 vectors may have degrees greater than d. We apply the technique of successive row indices selection of Section 7.1 for computing m − 2r0 independent vectors of degrees less than d. We work successively with ˜ (k) of M ˜ of size 3r0 × s = d(m − 2r0 )/r0 e submatrices M r0 , hence having at least r0 nullspace vectors of degree less than d. This requires s calls to Algorithm Nullspace minimal vectors at Step (c). We terminate by computing r0 vectors of possibly higher degrees using the case m = 2r0 , in one call to Algorithm Nullspace2n with input a 2r0 × r0 submatrix ˜ (s+1) of M ˜ at Step (d). M



Each of the times the algorithm passes through the “while loop”, the sum of the degrees of the computed vectors is bounded by the sum nd of the Kronecker indices (these vec¯ ). tors are minimal for the nullspace of the submatrix M Hence the sum of the degrees in output is less than nddlog2 qe. Proposition 7.1. Let M ∈ K[x](n+q)×n with 1 ≤ q ≤ n be of full column-rank. Algorithm Nullspace2n computes q linearly independent polynomial vectors in the nullspace of M . If M has degree d then the sum of the degrees of the output vectors is less than nddlog2 qe. The algorithm is randomized, it either fails or returns correct values (Las Vegas fashion).

Algorithm Nullspace(M ) Input: M ∈ K[x]m×n of degree d. Output: r = rank M , m − r “small” linearly independent polynomial vectors in the nullspace of M . ˜ ∈ K[x]m×r0 ; (a) Compute r0 and M if m = r0 then return m and {}; (b) randomly ensure that the top r0 × r0 submatrix ˜ is non-singular or fail; of M s := d(m − 2r0 )/r0 e; (c) {Ni , 1 ≤ i ≤ m − 2r0 } := ˜ (k) , d) for 1 ≤ k ≤ s; Nullspace minimal vectors(M 0 (d) {Ni , 1 ≤ i ≤ min{m, 2r0 } − r0 } := ˜ (s+1) ); Nullspace2n (M N := the matrix formed by the Ni ’s and the Ni0 ’s; (e) if N M 6= 0 then fail; else return r0 and Ni , 1 ≤ i ≤ m − r0 . 

The computed vectors Di ’s are minimal in the nullspace of CM but the minimality is not preserved in general for the vectors Ni ’s in the nullspace of M . The output basis for the nullspace as K(x)-vector space may not be a basis for the K[x]-module. However, Proposition 7.1 shows that if the sum of the Kronecker indices is nd (the maximum possible), then the sum of the computed degrees is only within dlog2 qe times the optimum. We notice also that the vectors computed at the first stage are minimal vectors by Proposition 6.4 (C = In+q ), hence the algorithm reaches the optimum for a generic matrix M (the whole nullspace is computed with p = q). It would be interesting to study the loss of minimality compared to the Kronecker indices in the general case. Corollary 7.2. Let M ∈ K[x](n+q)×n be of full columnrank and degree d with 1 ≤ q ≤ n, q polynomial vectors whose degree sum is less than nddlog2 qe can be computed in

As in Section 7.1, the successive index selections for con˜ lead to linearly independent structing the submatrices of M nullspace vectors. The proof of Theorem 7.3 [31] below takes into account (13) for the s = O(m/r) calls to Nullspace minimal vectors, and (15) for the call to Nullspace2n .

O((MM(n, d) log(nd) + n2 B(d) log n + nM(nd)) log q) (15) operations in K by a randomized Las Vegas (certified) algorithm.

315

Theorem 7.3. Let M ∈ K[x]m×n be of degree d. The rank r of M and m − r linearly independent polynomial vectors in the nullspace of M can be computed in

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O(nmMM(r, d)/r2 + (m/r + log r)(MM(r, d) log(rd) + r2 B(d) log r + rM(rd))) hence O˜(nmrω−2 d) operations in K by a randomized Las Vegas (certified) algorithm. The degree sum of the computed nullspace vectors is less than rddlog2 re + (m − 2r)d. For m ≤ 2r we have already commented after Proposition 7.1 the quality of the degree sum bound rddlog2 re. For m  r, since the sum of the Kronecker indices is no more than rd, we see that the bound we propose in Theorem 7.3 is within a factor asymptotically m/r from the optimal. Remark 7.4. We did not detail the probability analysis. Random values in K occur for: the choice of the compressing matrix P for Proposition 4.1; the choice of Q in Lemma 5.1 and as linear independence conditioning; a random shifting or evaluation point x0 in all algorithms; the conditioning of ˜ in Section 7.2. Our algorithms are deterministic M into M if random values are replaced by symbolic variables. For a given input matrix M , the algorithm succeeds if the random values do not form a zero of a fixed polynomial over K of degree O(nd) in the latter variables. The probability of success is at least 1 − cnd/|R|, for a positive real constant c, if the random values are chosen from a subset R of cardinal |R| of K [12, 34, 27] (see also our comments in Introduction).

Concluding remarks We compute a K(x)-nullspace basis of an input matrix over K[x] as the union of few minimal K[x]-basis of submatrices of the input matrix. It remains to compute a minimal basis with an analogous complexity estimate. A possible direction of work here is to ensure the irreducibility of the output basis either on the fly or a posteriori. Subsequent work may also concern the applicability of our compression/uncompression scheme to other problems such as questions about matrix approximants or block structured matrices. Computing a nullspace basis is added to the recent list of problems that can be solved in about the same number of operations as for multiplying two matrix polynomials. We hope that this will help in making progress for the characteristic polynomial [18, 21], and for (non-generic) matrix inversion [16].

8.

REFERENCES

[1] B. Beckermann and G. Labahn. A uniform approach for the fast computation of matrix-type Pad´ e approximants. SIAM J. Matrix Anal. Appl., 15(3):804–823, July 1994. [2] B. Beckermann, G. Labahn, and G. Villard. Normal forms for general polynomial matrices. RR LIP 2002-1, Laboratoire LIP, ´ ENS Lyon, France, 2002. To appear in J. Symb. Comput. [3] T. Beelen and P.M. Van Dooren. An improved algorithm for the computation of Kronecker’s canonical form of a singular pencil. Lin. Alg. Appl., 105:9–65, 1988. [4] T. Beelen and P.M. Van Dooren. A pencil approach for embedding a polynomial matrix into a unimodular matrix. SIAM J. Matrix Anal. Appl., 9(1):77–89, Jan. 1988. [5] R.R. Bitmead, S.Y. Kung, B.D.O. Anderson, and T. Kailath. Greatest common divisors via generalized Sylvester and Bezout matrices. IEEE Trans. Aut. Control., 23(6):1043–1047, 1978. [6] A. Bostan. Algorithmique efficace pour des op´ erations de base ´ en calcul formel. PhD thesis, Ecole Polytechnique, Palaiseau, France, Dec. 2003.

316

Approximation of Dynamical Systems using S-Systems ∗ Theory: Application to Biological Systems Laurent Tournier Laboratoire de Modelisation ´ et Calcul, 51 rue des Mathematiques, ´ B.P. 53, 38041 Grenoble cedex 9, France.

[email protected]

ABSTRACT

tional tools. Therefore a new area emerged, called ”systems biology”. It involves different fields of applied mathematics, from computer algebra (see for instance [10]) to numerical computation ([7]). In the past decades, a lot of different frameworks have been developped to study behaviors of complex biochemical processes. Let us cite here three of them: discrete networks (see the work of R. Thomas [16]), piecewise linear systems (the so-called Glass networks [6], see also [5]) and sigmoidal switch systems ([13]). The main goal of all these approaches is to propose a (more or less) generic class of dynamical systems, either discrete or differential, that model some behaviors of complex interaction systems. Once this class is clearly defined, its mathematical relevance generally allows both theoretical and numerical analysis. We use in this paper the class of S-systems (see [1], [17], [18]). The basic idea of this model is to represent interactions between biochemical species with power-law dynamics. Their mathematical expression is quite general, but sufficiently simple to allow theoretical and practical investigations. We propose in this article a symbolic-numeric algorithm that is based upon S-systems theory. Its goal is to compute iteratively the positive equilibria of a n-dimensional system of ordinary differential equations (ODE). As it converges towards a steady state, it provides a S-system that approaches the original dynamics around this steady state. As we will see, the local approximation of some dynamics with power-laws can be made symbolically at any point of the phase space. It can also include treatment of symbolic parameters. However, iterating this process in order to converge towards equilibria needs a numerical computation, which prevents the use of pure symbolic tools to the end. In the following, we give a definition of the S-system class as it can be found in the litterature (see for instance [17]). We then propose a symbolic-numeric algorithm that computes an iteration leading to the positive equilibria of a dynamical system. We will see an application of this algorithm on a biological example in dimension 14. We finally conclude with some remarks on our algorithm and some future works.

In this article we propose a new symbolic-numeric algorithm to find positive equilibria of a n-dimensional dynamical system. This algorithm uses a symbolic manipulation of ODE in order to give a local approximation of differential equations with power-law dynamics (S-systems). A numerical calculus is then performed to converge towards an equilibrium, giving at the same time a S-system approximating the initial system around this equilibrium. This algorithm has been applied to a real biological example in 14 dimensions which is a subsystem of a metabolic pathway in Arabidopsis Thaliana. Categories and Subject Descriptors: I.1.2 [Computing Methodologies]: Symbolic and algebraic manipulation - Algorithms G.1.5 [Mathematics of Computing]: Numerical Analysis Roots of Nonlinear Equations - Iterative Methods General Terms: Algorithms. Keywords: Power-law, S-systems, Dynamical Systems, Positive Equilibria, Stability.

1.

INTRODUCTION

The modelling and study of biological or biochemical systems has become an exciting challenge in applied mathematics. The complexity of real biological dynamical systems lies essentially in the non-linearities of the dynamics as well as in the huge dimension of systems, often leading to a numerical approach. However, as the understanding of cellular mechanisms grows, it has become obvious that the modelling step strongly needs symbolic tools in order to manipulate more and more information and data, and to improve computa∗This work is part of the CALCEL project, funded by ”La R´egion Rhˆ one-Alpes”, France.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. ISSAC’05, July 24–27, 2005, Beijing, China. Copyright 2005 ACM 1-59593-095-7/05/0007 ...$5.00.

2. S-SYSTEMS 2.1 Definition In this paragraph, we give a definition of the class of Ssystems (see for instance [1]).

317

Definition 2.1. A n-dimensional S-system S(α, β, G, H) is a dynamical system defined by the n differential equations: x˙i = αi

n Y

g

xj ij − βi

n Y

where b is the vector (β1 /α1 , . . . , βn /αn ). Taking the neperian logarithm, this equation leads to: (G − H) ln x = ln b

h

xj ij , i = 1 . . . n

(the logarithm is applied to all components of vector, i.e. ln x is the n-dimensional vector (ln x1 , . . . , ln xn )). Posing y = ln x, we are brought back to the resolution of a ndimensional linear system in y. We have therefore the following proposition:

j=1

j=1

(R∗+ )n

with α = (α1 , . . . , αn ) ∈ , β = (β1 , . . . , βn ) ∈ (R∗+ )n and G = (gij )i,j=1...n , H = (hij )i,j=1...n ∈ Mn (R). R∗+ denotes the set of strictly positive real numbers, Mn (R) denotes the set of real square matrices of order n.

Proposition 2.1. A S-system S(α, β, G, H) has a unique equilibrium x e in Ω (i.e. a positive equilibrium) if and only if the matrix (G − H) is invertible. x e can be calculated by the formula:

Let us introduce the vector field F defined on Ω = (R∗+ )n : 0 1 f1 (x1 , . . . , xn ) B C .. F (x) = @ A . fn (x1 , . . . , xn )

x e = b(G−H)

with:

fi (x1 . . . , xn ) = αi

n Y

g xj ij

n Y

− βi

,

The stability analysis of the equilibrium x e uses the study of the spectrum of JF (e x) (the jacobian of F in x e). The question we tackle here is to find some relationship between the stability of x e and some properties of the matrix G − H. As a motivating example, let us consider the one-dimensional case. A one-dimensional S-system is expressed by a single differential equation:

i = 1...n

F is C 1 and therefore locally lipschitz on the open Ω. CauchyLipschitz theorem ensures the existence and unicity of a maximal solution of S(α, β, G, H) in Ω, given any initial condition x(0) = x0 ∈ Ω. This definition of S-systems with power-law differential equations is strongly linked with equations of chemical kinetics. As an example, if we consider the following chemical pathway: k

k

1 2 A + 2B → C→ 3D + E

S(α, β, g, h) :

(1)

∂f (e x) = αe xg−1 (g − h) ∂x so the stability of x e depends directly on the sign of g − h: it is asymptotically stable if g−h < 0 and unstable if g−h > 0, regardless of parameters α and β. In the n-dimensional case, the stability depends on the sign of the real parts of the jacobian’s eigenvalues. Differentiating the functions fi (x1 . . . , xn ), we obtain, for i, j = 1 . . . n: n αi Y gik ∂fi x ek . (gij − hij ) (4) (e x) = ∂xj x ej

(capital letters designate species and small letters designate concentrations). Therefore in definition 2.1, coefficients αi and βi are sometimes called kinetic rates while gij and hij are called kinetic orders. S-systems are part of a broader formalism known as quasimonomial (QM) systems (see [2]). An interesting result shows that QM systems can be expressed in the form of Lotka-Volterra quadratic systems (see [2] for details).

k=1

We thus obtain a formula that links the jacobian of F in x e with the matrix G − H. However, it is not trivial to link the spectrum of JF (e x) with the spectrum of G − H. Let us recall here the definition of stability of matrices:

2.2 Equilibrium points The study of the phase portrait of a S-system S(α, β, G, H) begins with the search for equilibrium points in Ω. To find them, we have to solve the system: n Y

g

xj ij = βi

j=1

n Y

h

xj ij ,

i = 1...n

Definition 2.2. A real square matrix A of order n is said to be stable (resp. semi-stable) if all its eigenvalues λi , i = 1 . . . n, have a negative (resp. non positive) real part.

(2)

j=1

We could hope that the stability of matrix G − H was sufficient to deduce the stability of x e. However this is not true, as we can see in the following example. For n = 2, consider the S-system:  x˙ = 3xy 2 − 2x4 (S1) : y˙ = 4x3 y 4 − x5 y 3

In this paper we will use the following notation: Given a vector x ∈ (R∗+ )n and a real square matrix A = (aij )i,j=1...n , we define the vector xA ∈ (R∗+ )n by: (xA )i =

n Y

a

xj ij ,

x˙ = f (x) = αxg − βxh

where α, β > 0 and g, h ∈ R. The positive equilibrium x e of (S) exists and is unique if and only if g − h 6= 0. In this case, an obvious calculation leads to:

then the mass-action law applied to species C gives the equation: dc = k1 ab2 − k2 d3 e dt

αi

(3)

2.3 Stability analysis of the equilibrium h xj ij

j=1

j=1

−1

i = 1...n

j=1

we have: „ « „ « „ « „ « 3 2 1 2 4 0 α= ,β= ,G= ,H= 4 1 3 4 5 3

With this notation, we can express equation (2) as follows: xG−H = b

318

The matrix G − H is equal to: „ « −3 2 G−H = −2 1

can thus deduce that J is semi-stable and as x e is supposed hyperbolic, it is asymptotically stable. We can easily verify in the previous example that G − H is stable but not sign semi-stable (g22 − h22 = 1 > 0). Let us remark that the latter equation gives, in matricial notation:

Its characteristic polynomial is χ(λ) = (λ+1)2 so the matrix is stable. Since` G − H´ is invertible, there is a unique equilibrium: x e = 32 .We can calculate the two eigenvalues λ1 and , 256 3 9 λ2 of JF (e x). We find that λ1 , λ2 > 0, implying that x e is an unstable node. As a result, in spite of the stability of matrix G − H, the equilibrium x e is unstable. The stability of G − H is therefore insufficient to deduce the stability of x e. We need a stronger property known as sign stability (see [9],[11]).

J = ΓP D−1 where Γ and D are diagonal matrices: 0 0 1 x e1 γ1 B B C . .. D=@ Γ=@ A, γn

∀ i, j = 1 . . . n , sgn(aij ) = sgn(bij ) signum function: +1 0 −1

. x en

C A

so sgn(det(J)) = sgn(det(P )). As we have supposed that P is invertible, we deduce that J is also invertible and does not have null eigenvalues. It could nevertheless have imaginary eigenvalues, which would make x e unstable. The hyperbolicity of x e is therefore an essential assumption in the latter proposition.

Definition 2.3. Two real square matrices of order n, A = (aij )i,j=1...n and B = (bij )i,j=1...n , have the same sign pattern if:

The function sgn is the classical 8 < ∀x ∈ R , sgn(x) = :

..

1

3. LOCAL APPROXIMATION OF DYNAMICAL SYSTEM USING S-SYSTEMS

if x > 0 if x = 0 if x < 0

In this part, we propose an algorithm for approximating the equilibria of a dynamical system using S-systems. Simultaneously, we obtain a S-system that approximates the initial system around the equilibrium.

Definition 2.4. A real square matrix A of order n is said to be sign stable (resp. sign semi-stable) if all the matrices that have the same sign pattern are stable (resp. semi-stable) in the sense of definition 2.2.

3.1 Monomial approximation of a positive vector field (see [18],[15],[17]). Let’s consider the positive vector field F : (R∗+ )n → (R∗+ )n . 1 0 f1 (x1 , . . . , xn ) C B .. F (x) = @ A . fn (x1 , . . . , xn )

In [9] we find a characterization of the sign semi-stability: Theorem 2.1 (Quirk-Ruppert-Maybee). A real square matrix A = (aij )i,j=1...n is sign semi-stable if and only if it satisfies the following three conditions: (i) ∀ i = 1 . . . n , aii ≤ 0

We will suppose F sufficiently smooth on (R∗+ )n . Let us define the following change of variables: y = ln x, and express the logarithm of F (x) as a function G of the new variable y:

(ii) ∀ i 6= j , aij aji ≤ 0 (iii) for each sequence of k ≥ 3 distinct indices i1 , . . . , ik , we have: ai(1)i(2) . . . ai(k−1)i(k) ai(k)i(1) = 0 (The third condition is equivalent to the fact that the directed graph associated to A admits no k-cycle for k ≥ 3)

ln F (x) = ln F (ey ) = G(y) The function G is sufficiently smooth on Rn . Given any arbitrary point y 0 ∈ Rn , let us write the Taylor expansion of gi (for i = 1 . . . n) in the neighborhood of y 0 at the first order:

With this notion, we can formulate the following proposition, which links the stability of the equilibrium x e of a Ssystem with the sign semi-stability of matrix G − H:

Proposition 2.2. Let consider a n-dimensional S-system S(α, β, G, H). We assume that G − H is invertible and we note x e the unique positive equilibrium of (S). We also assume that x e is hyperbolic (i.e. none of the eigenvalues of the jacobian of F in x e have zero real part). If the matrix G − H is sign semi-stable (i.e. if it verifies the three conditions of theorem 2.1) then, regardless of parameters α and β, the equilibrium x e is asymptotically stable.

∀i = 1 . . . n , gi (y) = gi (y 0 ) +

n X

(yj − yj0 )

j=1

∂gi 0 (y ) + o(k y − y 0 k) ∂yj

We introduce the functions g˜i (y) for i = 1 . . . n: ∀i = 1 . . . n , g˜i (y) = gi (y 0 ) +

n X

(yj − yj0 )

j=1

∂gi 0 (y ) ∂yj

and the functions f˜i = exp(˜ gi (y)):

Proof. Let us note J the Jacobian of F in x e and P the matrix G − H. The equation 4 yields:

f˜i (x)

∂fi γi (e x) = pij ∂xj x ej

def

=

=

eg˜i (y) e

gi (y 0 )

exp

n X j=1

Q with γi = αi n egkik . As γi > 0 and x ej > 0 for all i k=1 x and j, matrices J and P have the same sign pattern. We

=

319

egi (y

0

)

(yj −

∂gi yj0 ) ∂yj

0

(y )

!

« „ ∂gi 0 (y ) exp (yj − yj0 ) ∂yj j=1 n Y

As y = ln x and gi (y) = ln fi (x), we have:

Using (5) and (6), we obtain:

« i (y 0 ) n „ Y xj ∂yj ∂g

f˜i (x) = fi (x0 )

(Sx0 ) :

x0j

j=1

= = =

where:

∂ (ln(fi (ey ))) ∂yj 1 ∂ (fi (ey )) fi (ey ) ∂yj 1 yj ∂fi y e (e ) fi (x) ∂xj xj ∂fi (x) fi (x) ∂xj

with:

> > > : gij (x0 )

= fi (x0 ) =

∂fi 0 (x ) fi (x0 ) ∂xj

h

xj ij ,

i = 1...n

j=1

vi− (x0 )

n Y

(x0j )−gij

j=1 n Y

(7) (x0j )−hij

j=1

8 > > < gij

= =

x0j + vi (x0 ) x0j − vi (x0 )

∂vi+ 0 (x ) ∂xj − ∂vi (x0 ) ∂xj

xeq = b(G−H)

(x0j )−gij

j=1

x0j

=

n Y

(8)

If the matrix G − H is invertible, the system (Sx0 ) admits a unique equilibrium xeq ∈ (R∗+ )n :

i=1...n n Y

g

xj ij − βi

= vi+ (x0 )

> > : hij

j

8 > 0 > > < αi (x )

8 > > > > < αi > > > > : βi

and:

“ ” Therefore, we have defined a vector field Fe = f˜i i=1...n ! n Y g (5) Fe (x) = αi x ij j=1

n Y

j=1

and: ∂gi (y) = ∂yj

x˙ i = αi

−1

with b = (β1 /α1 , . . . , βn /αn ). This point xeq depends on the initial point x0 where we made our approximation. Let x1 = xeq be the new initial point where we make our next Sapproximation. Iterating this process produces a sequence of points x0 , x1 , . . . that converges towards a positive equilibrium of (S). A proof of the convergence is given in section 5. The main steps of this algoritm are summarized here:

(6)

The basic idea is to use the monomial vector field Fe as an approximation of F in a neighborhood of x0 .

Definition 3.1. Let F be a smooth n-dimensional vector field, F : (R∗+ )n → (R∗+ )n and x0 any vector of (R∗+ )n . We call S-approximation of F in x0 the vector field Fe defined by equations (5) and (6).

Algorithm 1 Search of an equilibrium point of system (S) Require: X = x0 V +, V − >0

The following proposition is basic for what follows:

Proposition 3.1. Let F be a positive vector field and Fe its S-approximation in x0 . The following equalities hold:

∈ (R∗+ )n (initial condition) : positive vector fields defined over (R∗+ )n : precision

Ensure: unless we fall in a degenerate case, we find a point y close to a positive equilibrium of (S) with the precision . Meanwhile, we obtain the S-system (Sy ) that approximate system (S) around this equilibrium.

• Fe (x0 ) = F (x0 )

∂ f˜i 0 ∂fi 0 (x ) = (x ) ∂xj ∂xj 0 (or, which is equivalent: JF (x ) = JFe (x0 ))

• ∀ i, j = 1 . . . n ,

repeat Y := X for i = 1 to n do for j = 1 to n do Xj ∂vi+ Xj ∂vi− gij := + ; hij := − vi (X) ∂Xj vi (X) ∂Xj end for n n Y Y αi := vi+ (X) (Xj )−gij ; βi := vi− (X) (Xj )−hij

The proof is easy and left to the reader.

3.2 Finding equilibria of a dynamical system We consider a n-dimensional dynamical system of the form: (S)

x˙ = V + (x) − V − (x)

j=1

where x lies in (R∗+ )n and V + , V − are positive vector fields. V + , V − : (R∗+ )n → (R∗+ )n . For i = 1 . . . n, the term vi+ (x) is the production term of the variable xi and vi− (x) the decay term of xi . We propose an algorithm for finding an equilibrium point of (S) that lies in (R∗+ )n . Meanwhile, we get a S-system that approximates the system (S) around this equilibrium. Given a point x0 in (R∗+ )n , we introduce the fields Ve + and − e V which are the S-approximations of the fields V + and V − in x0 . Let us consider the n-dimensional S-system: (Sx0 )

j=1

bi := βi /αi end for if det(G − H) 6= 0 then −1 X := b(G−H) else degenerate case: algorithm terminated → restart the algorithm with a new initial condition end if until k X − Y k<  Result := X

x˙ = Ve + (x) − Ve − (x)

320

4.

AN EXAMPLE WITH MULTIPLE POSITIVE EQUILIBRIA

The numerical S-system obtained is given by:  x˙ = 1.500 y −1 − x y˙ = 1.837 x−1.5 − y

We present here the application of our algorithm for a dynamical system having multiple positive equilibrium points. It is a system known as biological switch (see [3]). Let’s consider the two dimensional dynamical system: 8 3 > −x < x˙ = f1 (x, y) = 1 + y2 (9) > 6.75 : y˙ = f2 (x, y) = − y 3.375 + x3

• With initial condition x0 = (0.2, 1.5), algorithm finished in 4 iterations and found P 1 with a precision of 10−5 . The numerical S-system obtained is given by:  x˙ = 1.745 y −1.535 − x y˙ = 1.647 x−0.274 − y • With initial condition x0 = (2, 0.2), algorithm finished in 4 iterations and found P 3 with a precision of 10−5 . The numerical S-system obtained is given by:  x˙ = 2.352 y −0.132 − x y˙ = 3.879 x−2.6 − y

It represents the temporal evolution of two positive quantities x and y with linear decay and sigmoidal production (we Kn use here the Hill function H − (z) = n often used by K + zn biologists to model sigmoidal interactions). As we can see on figure 1, this system has three equilibrium points. The values of these points can be calculated: „ « „ « „ « 0.697 1.5 2.802 P1 ≈ P2 = P3 ≈ 1.818 1.0 0.266

5. ANALYSIS OF THE ALGORITHM 5.1 Correctness Let’s describe the first iteration of our algorithm. Let x0 ∈ (R∗+ )n . With formulae (7) and (8), we define the quantities αi (x0 ), βi (x0 ), gij (x0 ) and hij (x0 ). Let us assume that matrices G(x0 ) and H(x0 ) verify the condition: det(G − H) 6= 0. Thanks to this assumption, there exists a unique equilibrium point of the system (Sx0 ). We denote it x1 , and we define the function Ψ : (R∗+ )n → (R∗+ )n that, to each x0 ∈ (R∗+ )n associates the point x1 . Our algorithm computes the recurrent sequence:  0 x ∈ (R∗+ )n (I) xn+1 = Ψ(xn )

We can show that P 2 is unstable whereas P 1 and P 3 are stable (cf. [3]).

4

y3

This iterative process converges towards fixed points of Ψ. However we do not a priori know if all fixed points of Ψ are indeed limits of (I). In other words, we must find which fixed points are attracting. The correctness of the algorithm (1) is a consequence of the two following lemmas:

[2, 2]

2

P1 [.2, 1.5]

1

P2

Lemma 5.1. The equilibria of initial system (S) are the fixed points of the function Ψ

P3

[2, .2] 0 0

1

2

3

4

5

Lemma 5.2. Given a fixed point x ¯ of Ψ, there exists some initial points x0 that lead to x ¯ by the iteration (I). In other words, the positive equilibria of (S) are the attracting fixed points of Ψ.

x

Figure 1: The two curves represents the nullclines of system (9), i.e. the curves f1 (x, y) = 0 (dashed line) and f2 (x, y) = 0 (solid line). The central equilibrium P 2 can be shown to be unstable while the two others, P 1 and P 3 are stable. The arrows represent the three experimentations described in the text.

Proof. (First lemma) Let x ¯ ∈ (R∗+ )n such that the determinant of G(¯ x) − H(¯ x) is different from zero. (for convenience, we will omit the dependency in x ¯, and note G in place of G(¯ x)). Using equation (3), we have: Ψ(¯ x) = b(G−H)

Applying our program in Maple, we found three different initial conditions, each of which tending towards one of the three equilibrium points (see figure 1 and numerical results below). The convergence appears to be fast since we need only 4 iterations to approach the equilibria with a precision of 10−5 . We will discuss about the convergence speed in part 5.3.

−1

(10)

where b is the vector (βi /αi )i=1...n . Therefore: Ψ(¯ x) = x ¯ ⇐⇒ ⇐⇒ ⇐⇒

• With initial condition x0 = (2, 2), algorithm finished in 4 iterations and found P 2 with a precision of 10−5 .

⇐⇒

b(G−H) = x ¯ b=x ¯(G−H) n Y βi g −h ∀i = 1 . . . n, = x ¯j ij ij αi j=1 n n Y Y h g ∀i = 1 . . . n, βi x ¯j ij = αi x ¯j ij −1

j=1

321

j=1

Q Q g h By definition, αi n ¯j ij (resp. βi n ¯j ij ) is the Sj=1 x j=1 x approximation of V + (resp. V − ) in x ¯. Proposition 3.1 implies then: Ψ(¯ x) = x ¯

⇐⇒

With (10) we have, for i = 1 . . . n and x ∈ W : „ « ∂u−1 1 exp − uj (x) i (x) xi ∂xj j=1 ! n X uj (x) ∂u−1 i = xi exp − (x) xi ∂xj j=1

Ψi (x) = xi

V + (¯ x) = V − (¯ x)

Thus, the equilibria of (S) are the fixed points of the function Ψ.

Differentiating this (and omitting the dependency in x), we get, for k 6= i: ! " n # n X X ∂ 2 u−1 ∂Ψi uj (x) ∂u−1 i i uj = exp − ∂xk ∂xj ∂xk xi ∂xj j=1 j=1

In order to prove the second lemma, we will use the following fixed point criterion (known as Ostrowski’s theorem, see e.g. [12]): If the function Ψ is a contraction on the open set W and if x ¯ ∈ W is a fixed point of Ψ, then x ¯ is the unique fixed point of Ψ in W and it is attracting, that is to say, for all x0 ∈ W , the iteration (I) converges towards x ¯.

and # " n n X ∂ 2 u−1 ∂Ψi 1 X ∂u−1 i i × − = uj uj ∂xi ∂xj ∂xk xi j=1 ∂xj j=1

Proof. (Second lemma) Let x ¯ be a fixed point of Ψ. We assume that det(G(¯ x) − H(¯ x)) 6= 0. The continuity of the determinant implies that there exists a neighboorhood W of x ¯ in which det(G − H) 6= 0. To prove that x ¯ is attracting, we have to show that there exists an open neighbourhood of x ¯ in which Ψ is contracting, i.e. the spectral radius of the jacobian of Ψ is strictly less than one. Using (7) and (8) and posing: 8 + U = log(V + ) > > > − > = log(V − ) < U „ +« V + − > U = U − U = log > > V− > : P = G−H

exp −

Ψi (x) = xi

j=1

“ ” (−1) where pij

i,j=1...n

vj− (x) vj+ (x)

!p(−1) (x)

5.2 Stability analysis of approximate S-system

ij

We have proven that our algorithm is correct, i.e. that it converges towards an equilibrium point of our initial system. At the same time, it provides a sequence of S-systems that locally approximate the initial system. We show here that this sequence of systems is useful because it facilitates the study of the stability of the equilibrium. Consider the n-dimensional dynamical system: (

8 αq > > < βq Gq > > : Hq

Therefore P −1 = ∆−1 (Ju (x))−1 = ∆−1 (Ju−1 (x)) (u−1 is the reciprocal function of u) and so: =

dx = F (x) = V + (x) − V − (x) dt x ∈ (R∗+ )n

(11)

Algorithm 1 ensures that, given any initial condition x0 in (R∗+ )n , unless we fall in a degenerate case, we produce a sequence (xq )q∈N (with xq ∈ (R∗+ )n ) that tends towards a limit point x e ∈ (R∗+ )n which is an equilibrium of (11). More q precisely, x = Ψ(q) (x0 ). Meanwhile, at each step, it provides us with a S-system Sq (αq , βq , Gq , Hq ) which comes from the S-approximations of functions V + and V − in xq . Thus, we have:

in matricial notation: P = Ju (x)∆ where Ju (x) is the jacobian of the function U evaluated in x and ∆ is the diagonal matrix: 0 1 x1 B C .. ∆=@ A . xn

(−1)

∂xj

This implies that the spectral radius of JΨ (¯ x) is strictly less than one. Consequently, Ψ is a contraction on an open neighboorhood of x ¯. According to the aforementioned fixed point criterion, we can deduce that x ¯ is attracting. This concludes the proof of the second lemma and the correctness of the algorithm.

= gij „− hij « 1 ∂vi− 1 ∂vi+ − = xj v + ∂xj vi− ∂xj „ i+ −« ∂ui ∂ui = xj − ∂xj ∂xj ∂ui = xj ∂xj

pij

xi

!

JΨ (¯ x) = 0

is the inverse of the matrix P = G−H.

∀i, j = 1 . . . n ,

i

As we have shown that the fixed points of Ψ are the equilibria of (S), we deduce that ∀k = 1 . . . n, uk (¯ x) = 0, therefore:

Let’s calculate pij : pij

n X uj (x) ∂u−1 j=1

we obtain, for all x ∈ W :

n Y

n Y

1 ∂u−1 i xi ∂xj

= = = =

α(xq ) q β(x ` q ´) q = gij (xq ) gij i,j=1...n with: gij ` q´ q hij i,j=1...n with: hij = hij (xq )

where α, β, gij and hij are the functions defined in (6). If we assume that V + and V − are at least C 1 , we deduce that

322

these sequences converge, as q tends to 8 def > αq → α(e x) = > > > def < βq → β(e x) = def > G → G(e x ) = > q > > : def Hq → H(e x) =

e be the following S-system: Let (S) e : x˙i = α (S) ei

n Y

j=1

g ˜

xj ij − βei

n Y

∞, towards:

However, even if the speed of convergence is very fast, the algorithm’s behaviour is strongly dependent on the choice of initial point x0 . Indeed, if initial system has multiple positive equilibria, each of them has distinct basins of attraction. We cannot a priori know in which of these basins the point x0 is. We even cannot ensure that x0 actually lies in one of them. In fact, the study of basins of attractions of such iterations is a complex issue. The boundaries of such basins can be quite complicated, even fractals [8]. As an example, we launched our algorithm for the switch system (9) with initial conditions taken on a grid of ]0, 4]2 . To visualize the three basins, we associated a color to each equilibrium. Then we colored the points of our grid with respect to the equilibrium towards which they lead (see fig 2).

α e βe

e G e H

˜ h

xj ij , i = 1 . . . n

(12)

j=1

We want to know in which sense the system (12) approximates the system (11). An answer is given by the following proposition: Proposition 5.1. F is supposed C r (r ≥ 1). The equilibrium x e of (11) is an equilibrium of (12). Moreover, if x e is hyperbolic, then the flow generated by (12) is topologically conjugate to the flow generated by (11) in a neighborhood of x e.

Proof. The first assertion is obvious with proposition 3.1. Let us prove the second assertion: it is a direct consequence of the Hartman-Grobman theorem (see e.g. [19]). According to Hartman-Grobman theorem, a dynamical system is topologically conjugate to its linearized system. Proposition 3.1 shows that systems (11) and (12) have the same linearized dynamical systems in x e. By transitivity of the topological conjugation, (11) and (12) are so topologically conjugate around x e. This proposition implies that the stability of x e for system (12) is the same that the stability of x e for system (11). As an exemple, let us consider the following 2-dimensional dynamical system: 8 x x2 y 4 > > x˙ = − < 2+y (3 + x)(4 + y 3 ) (Ex) > 5x 2xy 3 > : y˙ = − 3+x (x + 1)(y + 2)

Figure 2: Basins of attraction of points P 1 (dark), P 2 (white) and P 3 (grey). We obtained these graphs by applying algorithm 1 for system (9) with initial conditions taken in a regular grid of ]0, 4].

We find the equilibrium point x e ≈ (1.2301, 1.6950) and the e − H: e matrix G „ « −0.709 −2.812 e−H e = G 0.261 −2.541

6. APPLICATION TO BIOLOGY As we said in section 2.1, S-systems are particularly appropriate to model biochemical pathways. Indeed, in a biochemical pathway, each ODE represents the evolution of the concentration of a chemical species. It is composed of the difference of two positive terms (production and decay of the species). In the example given above (equation (1)), we saw that the mass-action law gives, for a simple reaction, monomial terms. The structure of S-systems seems therefore particularly relevant in the modeling of biochemical systems. Neverthelesss, in most biochemical pathways, systems are obviously more complicated. Most of biochemical reactions depend on the concentration of appropriate enzymes that catalyse the reaction. The dynamics of simple E substrate-product enzymatic reaction: S → P is given by the Michaelis law: kES S˙ = − KM + S

e H e is sign semi-stable. Thanks to theorem 2.1, we see that G− According to proposition 5.1, the point x e, as equilibrium of (Ex) is hence stable.

5.3 Discussion about convergence

The algorithm described above computes the iterations of a vectorial function Ψ on an initial point x0 ∈ (R∗+ )n , in order to converge towards a fixed point of Ψ. As the jacobian of Ψ is the null matrix in those fixed points, we know that the convergence is very fast (up to four or five iterations in all the examples presented, for a precision of 10−4 or 10−5 ). As a matter of fact, we are in a case where the speed of convergence is the best possible. Indeed, if the function Ψ is K-contractant, one can easily verify that the convergence of the iteration is in K n (where n is the number of iterations). Since JΨ (e x) = 0, then we can find a neighborhood of x e wherein Ψ is K-contractant for any 0 < K < 1.

where k and KM are specific constants. The presence of such expressions implies rational functions in the equations.

323

[4] G. Curien, S. Ravanel, and R. Dumas. A kinetic model of the branch-point between the methionine and threonine biosynthesis pathways in arabidopsis thaliana. Eur. J. Biochem., 270(23):4615–4627, 2003. [5] H. de Jong, J.-L. Gouz´e, C. Hernandez, M. Page, S. Tewfik, and J. Geiselmann. Qualitative simulation of genetic regulatory networks using piecewise-linear model. Bull. Math. Biol., 66(2):301–340, 2004. [6] L. Glass. Combinatorial and topological methods is nonlinear chemical kinetics. J. Chem. Phys., 63, 1975. [7] A. Goldbeter. Biochemical oscillations and cellular rhythms. Cambridge University Press, 1996. [8] C. Grebogi and E. Ott. Fractal basin boundaries, long-lived chaotic transients, and unstable-unstable pair bifurcation. Phys. Rev. Lett., 50(13):935–938, 1983. [9] C. Jeffries, V. Klee, and P. Van Den Driessche. When is a matrix sign stable ? Can. J. Math., 29(2):315–326, 1976. [10] R. Laubenbacher. A computer algebra approach to biological systems. Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation (ISSAC), 2003. [11] J. Maybee and J. Quirk. Qualitative problems in matrix theory. SIAM Review, 11(1):30–51, 1969. [12] J.M. Ortega. Numerical Analysis: A second course. SIAM, 1990 (reprint of 1970 ed.) [13] T. Mestl, E. Plahte, and S.W. Omholt. A mathematical framework for describing and analyzing gene regulatory networks. J. Theor. Biol., 176:291–300, 1995. [14] M.A. Savageau. Alternative designs for a genetic switch: analysis of switching times using the piecewise power-law representation. Math. Biosci., 180:237–253, 2002. [15] M.A. Savageau and E.O. Voit. Recasting nonlinear differential equations as s-systems : a canonical nonlinear form. Math. Biosci., 87:83–115, 1987. [16] R. Thomas and M. Kaufman. Multistationarity, the basis of cell differentiation and memory. i. structural conditions of multistationarity and other non-trivial behaviour, and ii. logical analysis of regulatory networks in terms of feedback circuits. Chaos, 11:170–195, 2001. [17] E.O. Voit. Computational analysis of biochemical systems. Cambridge University Press, 2000. [18] E.O. Voit and M.A. Savageau. Accuracy of alternative representations for integrated biochemical systems. Biochemistry, 26:6869–6880, 1987. [19] S. Wiggins. Introduction to applied nonlinear dynamical systems and chaos. Springer Verlag, 1990.

Actually, biological enzymes may exhibit even more complex behaviours like cooperativity or allostery, that introduce terms such as Hill functions or composition of Hill functions. Because of these terms, it is not possible to give a global representation of such a system with a S-system. The idea of S-approximation is therefore to propose a local approximation of the dynamics of a complex biochemical pathway in the shape of a S-system. We are currently working in collaboration with G. Curien (see [4]) on a specific example. The goal of this work is to understand the metabolic system responsible for the synthesis of aminoacids in Arabidopsis Thaliana. So far, we have focused our study on a subsystem of 14 variables, with 9 symbolic parameters. In vivo, this system exhibits a stationnary behaviour. Giving realistic values of parameters and initial conditions, we managed, thanks to our algorithm, to find this positive equilibrium. We now have to study the Sapproximation of the system near this equilibrium, with different realistic sets of parameters. This work is in progress.

7.

CONCLUSION AND PERSPECTIVES

As we said in the introduction, a large part of research concerning the analysis of biological phenomena uses both symbolic and numerical techniques. The S-systems as we described represent a large class of systems, yet their simple mathematical expression allows symbolic manipulations, providing a practical framework of study. Algorithm 1, as presented here, needs numerical estimations of symbolic parameters. Nevertheless the technique of S-approximation (def 3.1) consists of symbolic manipulations (in particular, we use symbolic computation of partial derivatives). It can be calculated in any point of the phase space and can include symbolic parameters. S-approximation gives a computable and rather good approximation of ODE systems (see [18] for a comparison between power-law approximation and linearization). A very interesting idea is therefore to use the context information (given for instance by biologists) of a particular system in order to create a piecewise S-approximation of this system. This should provide a global approximation interpolating the system in some critical points in the phase space (see [14]).

8.

REFERENCES

[1] M. Antoniotti, A. Policriti, N. Ugel, and B. Mishra. Xs-systems : extended s-systems and algebraic differential automata for modeling cellular behavior. Proceedings of the International Conference on High Performance Computing, HiPC 2002, pages 431–442, 2002. [2] L. Brenig and A. Goriely. Universal canonical forms for time continuous dynamical systems. Phys. Rev. A, 40:4119–4121, 1989. [3] J.L. Cherry and F.R. Adler. How to make a biological switch. J. Theor. Biol., 203:117–133, 2000.

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Generalized Laplace Transformations and Integration of Hyperbolic Systems of Linear Partial ∗ Differential Equations Sergey P. Tsarev Department of Mathematics Krasnoyarsk State Pedagogical University Lebedevoi 89, 660049 Krasnoyarsk, Russia

[email protected] ABSTRACT

For linear partial differential operators (LPDOs) and the corresponding equations (LPDEs) the theory of factorization is much more difficult. To the best of our knowledge there are only a few theoretical results and only one algorithm for “naive” factorization of hyperbolic LPDO. In this introduction we will give a brief account of the previously obtained results and state our main result: existence of a recurrent procedure for non-trivial factorization and finding closed-form complete solutions of strictly hyperbolic systems of LPDEs in two independent variables with coefficients in an arbitrary differential field. Theoretically one may propose several very different definitions of factorization for LPDOs. The obvious “naive” ˆ= definition suggests to represent a given operator L P i1 ˆ i2 in ˆ ˆ x)Dx1 Dx2 · · · Dxn as a composition of lower|~i|≤m ai1 ···in (~ ˆ k with coefficients in some fixed ˆ=L ˆ1 . . . L order LODOs: L differential field. Unfortunately this definition does not enjoy good theoretical properties: a given LPDO may have several very different decompositions of this form, even the ˆ s may be different, as the number of irreducible factors L following example (attributed in [4] to E.Landau) shows: if

We give a new procedure for generalized factorization and construction of the complete solution of strictly hyperbolic linear partial differential equations or strictly hyperbolic systems of such equations in the plane. This procedure generalizes the classical theory of Laplace transformations of second-order equations in the plane.

Categories and Subject Descriptors I.1.2 [Computing Methodologies: Symbolic and Algebraic Manipulation]: Algorithms

General Terms Algorithms, Theory

Keywords Systems of linear PDEs, complete solutions, factorization

1.

INTRODUCTION

ˆ x + xD ˆy, Q ˆ=D ˆ x + 1, Pˆ = D 2 ˆ ˆ ˆ ˆ ˆ ˆy, R = Dx + xDx Dy + Dx + (2 + x)D

Factorization of linear ordinary differential operators (LODOs) is often used in modern algorithms for solution of the corresponding differential equations. In the last 20 years numerous modifications and generalizations of algorithms for factorization of LODOs with rational function coefficients were given (see e.g [5]). Such algorithms have close relations with algorithms for computation of differential Galois groups and closed-form (Liouvillian) solutions of linear ordinary differential equations and systems of such equations ([20]). We have a nice and relatively simple theory of factorization of LODOs.

(1)

ˆ=Q ˆQ ˆ Pˆ = R ˆ Q. ˆ On the other hand the second-order then L ˆ is absolutely irreducible, i.e. one can not factor it operator R into product of first-order operators with coefficients in any extension of Q(x, y). Still the “naive” definition of factorization may help to solve the corresponding LPDE in some cases; recently ([13]) an algorithm for such factorization for the case of hyperbolic LPDOs of arbitrary order was given. In [25, 26] the adequate theoretical definition of factorization and a factorization algorithm for the case of overdetermined systems with finite-dimensional solution space and rational function coefficients was given. For a single second-order LPDO in two independent variables

∗Partially supported by the RFBR grant 04-01-00130.

ˆ=D ˆ xD ˆ y − a(x, y)D ˆ x − b(x, y)D ˆ y − c(x, y) L

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. ISSAC’05, July 24–27, 2005, Beijing, China. Copyright 2005 ACM 1-59593-095-7/05/0007 ...$5.00.

(2)

we have a very old and powerful theory of Laplace transformations (not to be mixed with Laplace transforms!). We expose this nice theory in Section 2. Roughly speaking, an operator (2) is Laplace-factorizable if after several applications ˆ x - or D ˆ y -transformations), of differential substitutions (D

325

2.

which change the coefficients of (2) in a simple way, one obˆ (k) = (D ˆ y + b(k) )(D ˆx + tains a naively-factorable operator L ˆ (k) = (D ˆ x + a(k) )(D ˆ y + b(k) ). This phenomenon a(k) ) or L of non-trivial Laplace-factorization explains the existence of Landau example (1). The definition of Laplace-factorizable operators turns out to be very fruitful in applications, it was extensively used in classical differential geometry (see e.g. [8]) and actively studied in the last decade in the framework of the theory of integrable nonlinear partial differential equations [1, 21, 22]. This is one of the most powerful methods of integration (construction of the complete solution with the necessary number of functional parameters) of the corresponding second-order equations in the plane. To the best of our knowledge the only serious effort to generalize the classical theory of Laplace-factorization to operators of higher order (in two independent variables) was undertaken in [15] with rather obscure exposition but deep insight and a few enlightening remarks, and [19] where an interesting generalization of Laplace transformation to a particular case of higher-order equations with degenerate principal symbol was given. Our approach, exposed in Section 3, gives a new uniform and general treatment of this topic directly for n × n strictly hyperbolic systems in two independent variables. Several modern papers [9, 14] investigate the theory of multidimensional conjugate nets initiated in [8, t. 4]; this line of research is in fact still in the domain of second-order equations in two independent variables: the systems discussed in the cited references are overdetermined systems with operators (2) and solution spaces parameterized by functions of one variable. An interesting special case (operators (2) with matrix coefficients) was studied in [22, 23], unfortunately the results are limited to this particular case of higher-order systems. A proper theoretical treatment of the factorization problem might be expected in the framework of the D-module theory (see e.g [6] and a very good exposition of the appropriate basic results in [17]). Unfortunately even in this modern algebraic approach a “good” definition of factorization of LPDOs with properties similar to the properties of factorization of LODOs or commutative polynomials (decomposition of algebraic varieties into irreducible components or primary decompositions in Noetherian commutative rings) is not an easy task. Without going into fine theoretical details we refer to [24] where a variant of such “theoretically good” definition of generalized factorization of a single LPDO was given. As we have shown in [24], this definition generalizes the classical theory of Laplace-factorizable second-order operators. A drawback of this theoretical approach was lack of any factorization algorithm for a given LPDO. In the present paper we give a new procedure (generalized Laplace transformations) for generalized factorization and integration of strictly hyperbolic LPDOs of arbitrary order with two independent variables or systems of such LPDOs. Section 3 is devoted to the detailed exposition of this new procedure. In Section 4 we give an example of application of this procedure to a 3 × 3 system and construct its complete solution using the results of Section 3. After this a general scheme of generalized factorization and integration of a strictly hyperbolic system in the plane is given. We conjecture that this new procedure provides an algorithm for generalized factorization and closed-form complete solution precisely in the sense of [24] if we limit the complexity of the answer.

THE CLASSICAL HERITAGE: LAPLACE TRANSFORMATIONS

Here we briefly sketch this classical theory in a slightly different form suitable for our purpose. The exhaustive exposition may be found in [8, 10, 11]. An arbitrary strictly hyperbolic second-order equation with two independent variables ˆ = 0 and the operator Lu ˆ= L

2 X

ˆ xi D ˆ y2−i + a1 (x, y)D ˆ x + a2 (x, y)D ˆ y + c(x, y), (3) pi D

i=0

pi = pi (x, y), may be rewritten in characteristic form ˆ1 X ˆ 2 + α1 X ˆ 1 + α2 X ˆ 2 + α3 )u = (X ˆ 1 + α2 X ˆ 2 + α3 )u = 0, ˆ 1 + α1 X ˆ2 X (X

(4)

where αi = αi (x, y), the coefficients of the first-order charˆ i = mi (x, y)D ˆ x + ni (x, y)D ˆ y are found acteristic operators X ˆ ˆ (up to a rescaling Xi → γi (x, y)Xi ) from the characteristic equation m2i p0 − mi ni p1 + n2i p2 = 0 for the principal symˆ i do not commute we have bol of (3). Since the operators X to take into consideration in (4) and everywhere below the commutation law ˆ2 − X ˆ2 X ˆ 1 = P (x, y)X ˆ 1 + Q(x, y)X ˆ 2 . (5) ˆ1 , X ˆ2 ] = X ˆ1 X [X Using the Laplace invariants of the operator (4): ˆ 1 (α1 ) + α1 α2 − α3 , h=X

ˆ 2 (α2 ) + α1 α2 − α3 , k=X

ˆ in partially factorized we represent the original operator L form ˆ = (X ˆ 1 + α2 )(X ˆ 2 + α1 ) − h = (X ˆ 2 + α1 )(X ˆ 1 + α2 ) − k. (6) L ˆ = 0 is equivalent From this form we see that the equation Lu to any of the first-order systems   ˆ 2 u = −α1 u + v, ˆ 1 u = −α2 u + w, X X (S1 ) : ˆ ⇔ (S2 ) : ˆ 2 w = ku − α1 w. (7) X X1 v = hu − α2 v. Proposition 1. Any strictly hyperbolic LPDE is equivalent to a 2 × 2 first-order characteristic system  ˆ 1 u1 = α11 (x, y) u1 + α12 (x, y) u2 , X (8) ˆ X2 u2 = α21 (x, y) u1 + α22 (x, y) u2 , ˆ i = mi (x, y)D ˆ x + ni (x, y)D ˆy, X ˆ 1 6= γ(x, y)X ˆ 2 , and with X any such system with non-diagonal matrix (αij ) is equivalent to a second-order strictly hyperbolic LPDE. Proof. Transformation of a strictly hyperbolic LPDE into the form (8) is already given. The converse transformation is also simple: if for example α12 6= 0 then substitute ˆ 1 u1 − α11 u1 )/α12 into the second equation of the u2 = (X system (8). Proposition 2. If a 2 × 2 first-order system         v1 a11 a12 v1 b11 b12 v1 = + v2 x a21 a22 v2 y b21 b22 v2

(9)

with aij = aij (x, y), bij = bij (x, y) is strictly hyperbolic (i.e. the eigenvalues λk (x, y) of the matrix (aij ) are real and distinct), then it may be transformed into a system in characteristic form (8).

326

Proof. Let λ1 (x, y), λ2 (x, y) be the eigenvalues of (aij ) and p ~1 = (p11 (x, y), p12 (x, y)), p ~2 = (p P21 (x, y), p22 (x, y)) be the corresponding left eigenvectors: k pik akj = λi pij . ˆi = D ˆ x − λi D ˆ y and the new characterForm the operators X P P ˆ ˆ istic k . Then Xi ui = k (Xi pik )vk + P functions ui = k pik vP (aks − λi δks ) (vs )y + k pik ((vk )x − λi (vk )y ) = k,s pik   P P ˆ P P ˆ k,s pik bks vs+ k (Xi pik )vk= s vs k pik bks + (Xi pis ) = P k uk αik (x, y), so we obtain the characteristic system (8). The characteristic system (8), equivalent to (9), is deterˆ i → γi (x, y)X ˆi mined uniquely up to operator rescaling X and gauge transformations ui → gi (x, y)ui . It is easy to check that the gauge transformations to not change the ˆ 2 (α11 ) − X ˆ 1 (α22 ) − X ˆ1 X ˆ 2 ln(α12 ) − Laplace invariants h = X ˆ ˆ X1 (P ) + P α11 + α12 α21 + (α22 + X2 (ln α12 ) + P )Q (where P (x, y), Q(x, y) are the coefficients of the commutator (5)) and k = α12 α21 , they are just the Laplace invariants of the operator (4), obtained after elimination of u2 from (8). ˆ i change them multiplicaRescaling transformations of X tively h → γ1 γ2 h, k → γ1 γ2 k. From the proofs we see that ˆ = 0 with the operator (3) we obfor a fixed equation Lu tain two different (inequivalent w.r.t. the scaling and gauge transformations) characteristic systems (7) and from every fixed system (8) we obtain two different (inequivalent w.r.t. the gauge transformation u → g(x, y)u) hyperbolic LPDEs: one for the function u1 and the other for the function u2 . This observation gives rise to the Laplace cascade method ˆ = 0 with of integration of strictly hyperbolic LPDEs Lu operators (3): (L1 ) If at least one of the Laplace invariants h or k vanˆ factors (in the “naive” way) into ishes then the operator L composition of two first-order operators as we see from (6); if we perform an appropriate change of coordinates (x, y) → (x, y) (NOTE: for this we have to solve first-order nonlinear ODEs dy/dx = ni (x, y)/mi (x, y), cf. Appendix in [13]) one ˆ1 = D ˆ x, X ˆ2 = D ˆ y so we obtain the complete can suppose X solution of the original equation in quadratures: if for exˆ = (D ˆ x + α2 (x, y))(D ˆ y + α1 (x, y))u = 0, then u = ample Lu


R R R exp − α1 dy X(x) + Y (y) exp (α1 dy − α2 dx) dy , where X(x) and Y (y) are two arbitrary functions of the characteristic variables x, y. (L2 ) If h 6= 0, k 6= 0, transform the equation into one of the systems (7) (to fix the notations we choose the left system (S1 )) and then finding ˆ 1 v + α2 v)/h u = (X

tures and, using the differential substitutions (10), obtain the complete solution of the original equation. Alternatively ˆ 2 -transformations: rewrite the original one may perform X equation in the form of the right system (S2 ) in (7) and usˆ 2 w +α1 w)/k obtain the equation ing the substitution u = (X ˆ (−1) w = 0 with Laplace invariants L h(−1) = k, ˆ2 X ˆ 1 ln k − P X ˆ 1 ln k + X ˆ 2 (Q) k(−1) = 2k − h − X ˆ 1 (P ) + 2P Q. −X

(12)

ˆ 2 -transformation is a reverse of the X ˆ 1 -transIn fact this X formation up to a gauge transformation (see [1]). So we have (infinite in general) chain of second-order operators ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

X2 X2 X ˆ (2) X ˆ (1) X ˆX ˆ (−1) ← ˆ (−2) ← →1 . . . →1 L →1 L L L . . . ←2 L

(13)

and the corresponding chain of Laplace invariants . . . , h(−3) , h(−2) , h(−1) , h0 = h, h(1) , h(2) , h(3) , . . . (14) with recurrence formulas (11), (12). We do not need to keep the invariants k(i) in (14) since k(i) = h(i−1) . If on any step we have h(N ) = 0 then the chains (13) and (14) can not be continued: the differential substitution (10) is not defined; precisely on this step the corresponding LPDE is trivially factorable and we can find the complete solution for any of the operators of the chain (13). For simplicity let us choose ˆ1 = D ˆ x, X ˆ2 = D ˆy. characteristic variables (x, y), so that X The complete solution of the original equation in this case has the form

R R u = c0 (x, y) F + Gβ dy + c1 (x, y) F 0 + G ∂β dy + ∂x   (15) R N . . . + cn (x, y) F (N ) + G ∂∂xNβ dy . where F (x), G(y) are two arbitrary functions of the characteristic variables and ci (x, y), β(x, y) are some definite functions obtained in the process of Laplace transformations from the coefficients of the operator (3). As one may prove (see e.g. [8]) if the chain (13) is finite in both directions (i.e. we have h(N ) = 0, h(−K) = 0 for some N ≥ 0, K ≥ 0) one may obtain a quadrature-free expression of the general solution of the original equation: e e0 e (K−1) u =c0 F +c1 F 0 +. . .+cN F (N )+d0 G+d 1 G +. . .+dK−1 G (16) e with definite ci (x, y), di (x, y) and F (x), G(y) — two arbitrary functions of the characteristic variables and vice versa: existence of (a priori not complete) solution of the form (16) with arbitrary functions F , G of characteristic variables implies h(s) = 0, h(−r) = 0 for some s ≤ N , r ≤ K − 1. So minimal differential complexity of the answer (16) (number of terms in it) is equal to the number of steps necessary to obtain vanishing Laplace invariants in the chains (13), (14) and consequently naively-factorable operators. Complete proofs of these statement may be found in [8, t. 2], ˆ1 = D ˆ x, X ˆ2 = D ˆ y , for the general case [10, 11] for the case X cf. [11, p. 30] and [1]. Example 1. As a straightforward computation shows, for u = 0 the chain (14) is symmetric the equation uxy − n(n+1) (x+y)2 (h(i) = h(−i−1) ) and has length n in either direction. So the complexity of the answer (16) may be very high and depends on some arithmetic properties of the coefficients of c the operator (3); for the equation uxy − (x+y) 2 u = 0 the

(10)

substitute this expressions into the first equation of the left system (S1 ) in (7), obtaining a X1 -transformed equation ˆ (1) v = 0. It has Laplace invariants (cf. [1]) L ˆ 1 (2α1 −P )− X ˆ 2 (α2 )− X ˆ1 X ˆ 2 ln h+ QX ˆ 2 ln h h(1) = X − α3 + (α1 − P )(α2 − Q) = 2h − k ˆ1 X ˆ 2 ln h + QX ˆ 2 ln h + X ˆ 2 (Q) − X ˆ 1 (P ) + 2P Q, (11) −X k(1) = h. If h(1) = 0, we solve this new equation in quadratures and using the same differential substitution (10) we obtain the ˆ = 0. complete solution of the original equation Lu (L3 ) If again h(1) 6= 0, apply this X1 -transformation several times, obtaining a sequence of second-order operators ˆ (2) , L ˆ (3) , . . . of the form (4). If on any step we get h(k) = 0, L ˆ (k) u(k) = 0 in quadrawe solve the corresponding equation L

327

chains (13), (14) will be infinite unless the constant c = n(n + 1). Example 2. For a stochastic ODE x˙ = p(x) + α(t)q(x) with binary (dichotomic) noise α(t) = ±1 and switching fref (x, t)i and W1 (x, t) = quency ν > 0 the averages W (x, t) = hW f (x, t)i for the probability density W f (x, t) in the space hα(t)W of possible trajectories x(t) of the ODE satisfy a system of the form (9) (see [16]): ( Wt + (p(x)W )x + (q(x)W1 )x = 0, (17) (W1 )t + 2νW1 + (p(x)W1 )x + (q(x)W )x = 0.

Proof. The principal (nth-order) part of (19) decomˆi = poses into the product of the characteristic operators X ˆ y modulo lower-order terms. Other lower-order ˆ x − λi D D terms of any given order s may be also written as sums of ˆ i (modulo terms of order < s) in the following products of X unique way:

+

n−1 X

X

ˆ is + a0 (x, y). ˆ i1 · · · X as,i1 ···is (x, y)X

(21)

s=1 i1 <...
This is easily proved using induction over s. For s = 1 the ˆ x +p0,1 (x, y)D ˆ y = a1 (x, y)X ˆ1 + terms of order 1 are p1,0 (x, y)D ˆ a2 (x, y)X2 since the eigenvalues λ1 , λ2 are distinct. If this is proved for terms of order ≤ s1 we may suppose X (after ai local ˆ yj = ˆ xD ˆ s1 +2 = D ˆ x so change of variables) that X pi,j D

The Laplace invariants are h = ν 2 − [pxx q 2 (p + q) + p2x q 2 − px qx q(3p + q) − qxx pq(p + q) − qx2 p(2p + q)]/q 2 , k = ν 2 , so if ν, p(x) and q(x) satisfy a second-order differential relation h = 0, one can solve (17) in quadratures. Especially simple formulas may be obtained for polynomial p(x) = p1 x + p2 x2 , q(x) = q2 x2 , p1 > 0, p2 < 0: in this case k = ν 2 , h = h(−2) = ν 2 − p21 so if ν = p1 after the necessary transformations we obtain the following quadrature-free expression for the complete solution of the system (17):  q2  W = 2 G0 (y) − p1 G(y) + p1 F 0 (x) − p21 F (x) , x 1  W1 = 3 −q2 xG0 (y) + p1 (p2 x + p1 )G(y) + p1 q2 xF 0 (x) x  + p21 (p2 x + p1 )F (x) ,

i+j=s1 +1

! ˆ ys1+1+ p0,s1 +1 D

X

ˆ xi D ˆ yj D ˆ x = q(x, y)X ˆ1 X ˆ2 · · · X ˆ s1+1 pi+1,j D

i+j=s1

! X

+

ˆ xi D ˆ yj pi+1,j D

ˆ x = q(x, y)X ˆ1 X ˆ2 · · · X ˆ s1 +1 D

i+j=s1



 +

X

ˆ s1 +2 (moduˆ i1 · · · X ˆ is  X as1 ,i1 ···is1 (x, y)X 1

i1 <...
lo terms of lower order). ˆ inNow, provided we have the form (21) of the operator L, ˆ troduce the characteristic functions un ≡ u, un−1 = Xn un + ˆ n−1 un−1 + αn−1,n−1 un−1 + αn−1,n un , . . . , αnn un , un−2 = X ˆ 2 u2 +Pn α2s us with so far undefined αpq = αpq (x, y). u1 = X s=2 We have to define them inductively so that the expression P ˆ = X ˆ 1 u1 + n α1s us (again with α1s (x, y) to be deM s=1 ˆ In fact, the termined) would coincide with the operator L. ˆ is just X ˆ1 X ˆ2 · · · X ˆ n . The terms of highest-order term in M ˆ are just n products X ˆ i1 · · · X ˆi order n − 1 in M n−1 , i1 < ˆi, . . . < in−1 ≤ n (in every such product exactly one of X 1 ≤ i ≤ n, is missing) with coefficients αkk , 1 ≤ k ≤ n, so we just fix αkk to be equal to the respective an−1,i1 ···in−1 in ˆ −L ˆ to be of order n−2. Each coefficient (21). This makes M ˆi ˆ ˆ ˆ i1 · · · X at the terms X n−2 of order n − 2 in M − L includes derivatives of the already defined αkk and precisely one of αk−1,k , 2 ≤ k ≤ n, so choosing the latter appropriately we ˆ −L ˆ again. This step-down process will lower the order of M give us all αpq so we obtain the following system, equivalent ˆ = 0: to Lu  ˆ 1 u1 = − Pn α1s us , X   Ps=1  ˆ X2 u2 = − n s=2 α2s us + u1 ,  ...   ˆ Xn un = −αnn un + un−1 .

y = −t+ p11 ln p1 +(px2 −q2 )x ˆ 2 x = 0, X ˆ 1 y = 0) and F , are the characteristic variables (X G are two arbitrary functions of the corresponding characteristic variables. For the case ν 2 6= p21 we can compute other Laplace invariants of the chain (14): h(1) = h(−3) = ν 2 − 4p21 , h(2) = h(−4) = ν 2 − 9p21 , h(3) = h(−5) = ν 2 − 16p21 , . . . so for the fixed p(x) = p1 x + p2 x2 , q(x) = q2 x2 and ν = ±p1 , ν = ±2p1 , ν = ±3p1 , . . . one can obtain closedform quadrature-free complete solution of the system (17), with increasing complexity of the answer (16). Remark. The forms (15), (16) for the complete solution are local: in the general case due to nontrivial topological ˆ1 , X ˆ 2 in the picture of the trajectories of the vector fields X plane we are unable to guarantee existence of the global coordinate change (x, y) 7→ (x, y) (cf. Example 2 above).

3.

k

ˆ=X ˆ1 X ˆ2 · · · X ˆn L

The characteristic operators and left eigenvectors are simple: ˆ x , λ1,2 = −p(x) ± q(x), p11 = p21 = p22 = ˆi = D ˆ t − λi D X 1, p12 = −1. The characteristic system (8) for the new characteristic functions u1 = W − W1 , u2 = W + W1 is  ˆ 1 u1 = −(px − qx + ν) u1 + ν u2 , X (18) ˆ 2 u2 = ν u1 − (px + qx + ν) u2 . X

where x = −t+ p11 ln

Proposition 3. Any strictly hyperbolic LPDE (19) is equivalent to a n × n first-order system in characteristic form X ˆ i ui = X αik (x, y)uk . (20)

x , p1 +(p2 +q2 )x

GENERALIZED LAPLACE TRANSFORMATIONS OF n × n HYPERBOLIC SYSTEMS Hereafter we suppose that the LPDE of order n ≥ 2 X ˆ = ˆ xi D ˆ yj u = 0 Lu pi,j (x, y)D (19) i+j≤n

Remark 1. There are many ways of transforming a given higher-order LPDE into a first-order system. But for our purposes we need a precise equivalence, guaranteeing a oneto-one correspondence between the solution space of the

isX strictly hyperbolic, i.e. the characteristic equation pi,j λi = 0 has n simple real roots λk (x, y). i+j=n

328

ˆ 1 u1 − α11 u1 )/α12 . using the first equation of (8): u2 = (X Substitute it into the other equation of the system obtaining ˆ 1 u1 − α11 u1 )) − ˆ 2 ( 1 (X a second-order equation for u1 : X α12 α22 ˆ α21 u1 − (X1 u1 − α11 u1 ) = 0.

original equation and the solution space of the system. If for example we take the equation uxx − uyy − a(x, y)u = 0 and (after [7, Ch. I]) form the following first-order system   ux = p, qx = py , (22)  px = qy + a(x, y)u,

α12

ˆ 1 and X ˆ 2 in the second-order (L2 ) Swap the operators X term of this equation using the appropriate commutation laws and then rearrange the first-order terms   obtaining an ˆ 1 1 (X ˆ 2 u1 ) +p1 (x, y)X ˆ 1 u1 + equal expression of the form X α12   ˆ 2 u1 + p1 α12 u1 ) + ˆ 2 u1 + p3 (x, y)u1 = X ˆ 1 1 (X p2 (x, y)X α12 p2 ˆ 2 u1 + p1 α12 u1 ) + p (x, y)u1 = 0. (X

then we shall fix for it limited Cauchy data q|x=0 = uy |x=0 (so taking some subset of solutions of (22)) in order to have a one-to-one correspondence between the solution space of the original equation and some subset of the solution space of (22). Another straightforward possibility   ux = p, uy = q, (23)  p = q + a(x, y)u x y gives in fact an overdetermined degenerate system, it can not be put into the standard form (24) below. Remark 2. The converse of Proposition 3 is not true: already for n = 3 a generic system (20) gives for each ui an overdetermined involutive system of equations (see for example [7, Ch. I]). So hereafter we will construct our generalized Laplace transformations directly for systems in charˆ i 6= µ(x, y)X ˆ j for i 6= j. But acteristic form (20) with X for n × n first-order systems the equivalence can be proved. This is a well-known fact (see for example [7]), we give its simple proof for completeness of our exposition. Proposition 4. Any n × n first-order linear system (vi )x =

n X k=1

aik (x, y)(vk )y +

n X

bik (x, y)vk

(24)

k=1

with strictly hyperbolic matrix (aik ) (i.e. with real and distinct eigenvalues of this matrix) is equivalent to a system in characteristic form (20). Proof. A straightforward calculation similar to that in the proof of Proposition 2 shows that if P p ~i = (pi1 (x, y), . . . , pin (x, y)) are the left eigenvectors, k pik akj = ˆi = D ˆ x −λi D ˆ y are the characteristic operators, then λi pij , X P the new characteristic functions ui = k pik vk satisfy a system of the form (20). The characteristic form (20) of a given system (24) is deˆ i → γi (x, y)X ˆ i and gauge fined up to operator rescalings X transformations ui → gi (x, y)ui . The quantities hij = αij αji , hijk = αij αjk αki , i 6= j 6= k, and other similar cyclic products of the coefficients of (20) are invariant w.r.t. the gauge transformations so it is natural to call them Laplace invariants of (20). The complete set of such invariants can be found using the standard methods [18, 2]. We will not dwell upon this problem. Our generalization of the Laplace transformations is based on the following observation. For a given 2 × 2 system (8) ˆ 1 -transformation consists of the following the procedure of X steps:  ˆ 2 u = −α1 u + v X ˆ = 0 7→ ˆ Lu ˆ 1 v = hu − α2 v 7→ L(1) v = 0. X

3

α12

(L3 ) The last form of this second-order equation immediˆ 2 u1 + p1 α12 u1 )/α12 so we can ately suggests to set u2 = (X rewrite the equation in the form of the transformed 2 × 2 characteristic system  ˆ 2 u1 = −p1 α12 u1 + α12 u2 , X (25) ˆ 1 u2 = −p2 u1 − p (x, y)u2 . X 3 Remark. One should not be mistaken: if we apply this “swapping procedure” twice we do not necessarily return to the original system. This depends on the choice of the characteristic function to eliminate on step (L1 ) in (25): eliminating u2 we certainly return to the original system (8), but eliminating u1 from (25) we actually move one more step forward in the chain (13). This commutation trick gives the basis for our generalized Laplace transformations: (L1 ) For a given n × n characteristic system (20) choose one of its equations with a non-vanishing off-diagonal coˆ i ui − P efficient αik 6= 0, find uk = (X s6=k αis us )/αik and substitute this expression into all other equations of the system. We obtain one second-order equation X X
X αkk ˆ ˆ k 1 (X ˆ i ui − αis us ) − X αkp up − (Xi ui − αis us ) = 0 αik αik s6=k

p6=k

s6=k

and n − 2 first-order equations X X αjk ˆ ˆ j uj − X αjs us − (Xi ui − αis us ) = 0 αik s6=k

(26) (27)

s6=k

for j 6= i, k. (L2 ) The second step consists in rewriting the system (26), (27) in the following form with slightly modified unknown functions uj = uj + ρj (x, y)ui , j 6= i, k, ui ≡ ui , new coeffiˆp : cients βpq (x, y) but the same characteristic operators X X X X βkk ˆ ˆ i ( 1 (X ˆ k ui − βis us ))− X (Xk ui − βis us ) = 0, βkp up − αik αik s6=k

p6=k

s6=k

(28) ˆ j uj − X

X s6=k

X βjk ˆ βjs us − (Xk ui − βis us ) = 0, αik

j 6= i, k.

s6=k

(29) As we prove below this is always possible in a unique way. P ˆ k ui − (L3 ) Introducing uk = α1ik (X s6=k βis us ) rewrite (28), (29) as the transformed characteristic system  ˆ i uk = P βkp up , X  p   ˆ k ui = P (30) X s6=k βis us + αik uk ,   P  ˆ Xj uj = s βjs us , j 6= i, k.

If we will try to reformulate this in terms of 2 × 2 characteristic systems (8), we arrive at the following procedure: (L1 ) For a given system (8) choose one of the two variables ui and eliminate it, for example we may eliminate u2

Correctness of the step (L2 ). First of all we rewrite the l.h.s. of (26) (again using the appropriate commutation laws)

329


αis ˆ ˆ i 1 (X ˆ k ui ) +pi X ˆ i ui +pk X ˆ k ui −P as Ei = X s6=i,k αik Xk us + αik P P ˆ s6=k cis us and the l.h.s. of (27) as Ej = Xj uj − s6=k cjs us − ˆ k ui − pjj X ˆ j ui , where we have used the fact that any pjk X 3 characteristic operators are linearly dependent (we have only 2 independent variables!): ˆ i = φj (x, y)X ˆ k + ψj (x, y)X ˆj . X

(31)

Its equations can be consecutively solved: the first equations gives u3 = ex F (y). Using the standard method of the variation of constants we R solve the second equation obtaining u2 = ey (G(x) + 2 ex−y F (y) dy). In order to remove the quadrature we introduce a new arbitrary function Fe(y) instead of F (y): F (y) = ey Fe0 (y). Integrating by parts we obtain u2 = ey (G(x) + 2ex Fe(y)). The last equation gives u1 = C(x, y) exp x+y where C(x, y) satisfies the equation 2 −x−y ˆ (Fe0 (y) + 4Fe(y)) + X3 (C) = exp 2 (u3 + 2u2 ) = exp x+y 2 y−x 2 exp 2 G(x). Again introduce new arbitrary functions e 0 (x), Fe(y) = e−y F 0 (y), e F (y), such that G(x) = G G(x), x−y e then we can find C(x, y) = 2 exp y−x (3F (y)+ G(x)+exp 2 2 0 F (y)) + H(x − y). Finally we get the following complete solution of the transformed system (34):  0 e H(x − y), u1 = 2ey G(x) + ex (3F (y) + F (y)) + exp x+y   2  y e0 x 0 u2 = e G (x) + 2e F (y),    00 0 u3 = ex (F (y) − F (y)),

Introducing new uj = uj − pjj (x, y)ui , j 6= i, k, ui ≡ ui we ˆ j uj − P ˆ k ui have Ej = X cjs us − pjk X and
s6=k P αis ˆ 1 ˆ ˆ ˆ ˆ Ei = Xi αik (Xk ui ) +pi Xi ui +pk Xk ui − s6=i,k α Xk us + ik
P 1 ˆ ˆ ˆ ˆ cis us = Xi (Xk ui ) + pi Xi ui + pk Xk ui − Ps6=k αis ˆ αik ˆ s us ) + P ( X u − ψs X i s s6=i,k φs αik s6=k cis us . So adding to it a suitable combination of Ej we obtain P is ˆ i 1 (X ˆ k ui + αik pi ui − Es = X E i = Ei − s6=i,k φψssααik αik

P P αis αis k ˆ k ui + αik pi ui − (X us ) + αpik s6=i,k φs us ) + Ps6=i,k φs s6=k cis us , which coincides with (28), the basic coefficients βks (x, y), ρi (x, y) are therefore defined. Now (29) is in fact the equation Ej = 0 with slightly rearranged terms.

4.

AN EXAMPLE AND THE GENERAL FACTORIZATION PROCEDURE Let us consider the following 3 × 3 system  ˆ 1 u1 = u1 + 2u2 + u3 ,  X ˆ u = −6u1 + u2 + 2u3 , X  ˆ2 2 X3 u3 = 12u1 + 6u2 + u3 ,

(32)

e where F (y), G(x) and H(x − y) are three arbitrary functions of one variable each. The complete solution of the original system (32) can be easily found using the differential substitutions of the steps (L1 ) and (L2 ): u2 = u2 − 2u1 , ˆ 1 u1 − u1 − 2u2 = X ˆ 1 u1 + 3u1 − 2u2 (and certainly u3 = X u1 ≡ u1 !). Remark. As communicated to the author by Professor F. Schwarz, in this particular case the solution of (32) given above may be obtained computing for this systems its Janet base with respect to the lexicographic ordering of the functions ui and factorizing obtained equations which also have constant coefficients. One may conjecture that this is a general method suitable for constant coefficient systems of LPDEs.

ˆ1 = D ˆ x, X ˆ2 = D ˆy, X ˆ3 = D ˆx + D ˆ y . Let us eliminate with X u3 using the first equation: ˆ 1 u1 − u1 − 2u2 . u3 = X

ˆ 2 u2 − u2 − 2u3 = 0, E 3 = The resulting equations E2 = X ˆ 1 u3 − u3 = 0 and u3 = (X ˆ 3 u1 − u1 − 2u2 ) give us the X transformed system:  ˆ 1 u3 = u3 ,  X ˆ (34) X u = 2u3 + u2 ,  ˆ2 2 X3 u1 = u3 + 2u2 + u1 .

(33)

Substituting this expression into the other equations we obtain the following two equations: ˆ 2 u2 + 6u1 − u2 − 2(X ˆ 1 u1 − u1 − 2u2 ) = 0, E2 = X ˆ 3 (X ˆ 1 u1 −u1 −2u2 )−12u1 −6u2 −(X ˆ 1 u1 −u1 −2u2 ) = 0. E3 = X This completes the step (L1 ) of our generalized Laplace transformation. Step (L2 ) is done in 4 substeps: ˆ 1 u1 to X ˆ 3 u1 − X ˆ 2 u1 in E2 and 1) Using (31) change X ˆ 2 : E2 = X ˆ 2 u2 + 6u1 − u2 − 2X ˆ 1 u1 + collect the terms with X ˆ 2 u2 + 8u1 + 3u2 − 2X ˆ 3 u1 + 2X ˆ 2 u1 = X ˆ 2 (u2 + 2u1 + 4u2 = X ˆ 3 u1 = 0, so we now can introduce 2u1 ) + 8u1 + 3u2 − 2X the new function u2 = u2 + 2u1 . Change everywhere u2 to u2 − 2u1 : ˆ 2 u2 + 2u1 + 3u2 − 2X ˆ 3 u1 = 0, E2 = X ˆ3 X ˆ 1 u1 + 3X ˆ 3 u1 − 2X ˆ 3 u2 − 3u1 − 4u2 − X ˆ 1 u1 = 0. E3 = X ˆ 3 and X ˆ 1 in E3 , substitute X ˆ 3 u2 = X ˆ 1 u2 + X ˆ 2 u2 2) Swap X ˆ and collect the terms with X1 : ˆ 1 (X ˆ 3 u1 −u1 −2u2 )+3X ˆ 3 u1 −2X ˆ 2 u2 −3u1 −4u2 = 0. E3 = X ˆ 3 u1 − u1 − 2u2 ). This suggests to set u3 = (X ˆ 3 u1 to u3 + u1 + 2u2 : 3) Change everywhere X ˆ 2 u2 +2u1 +3u2 −2(u3 +u1 +2u2 ) = X ˆ 2 u2 −u2 −2u3 = E2 = X 0, ˆ 1 u3 + 3(u3 + u1 + 2u2 ) − 2X ˆ 2 u2 − 3u1 − 4u2 = E3 = X ˆ 1 u3 + 3u3 − 2X ˆ 2 u2 + 2u2 = 0. X ˆ 2 u2 in E3 : 4) Now get rid of the term −2X ˆ E 3 = E3 + 2E2 = X1 u3 − u3 = 0.

The general factorization procedure. ˆ = 0 or a system (24) I) If we have to solve an equation Lu transform it into the characteristic form (20). II) If the matrix (αij (x, y)) of the characteristic system is upper- or lower-triangular (similar to the matrix of (34)) solve the equations consecutively. III) If the matrix is block-triangular, the system factors into several lower-order systems; try for each subsystem the step IV. IV) In the general case of not-triangular matrix (αij (x, y)) perform several (consecutive) generalized Laplace transformations, using different choices of the pivot element αik 6= 0 on step (L1 ). The goal is to obtain a block-triangular matrix for one of the transformed systems. The main problem of this procedure is absence of upper bounds for the number of steps. As we have seen in Section 2 (Example 1) this bound depends on the coefficients of the equation (system) in a nontrivial way. On the other hand in this situation (the classical Laplace-factorization and our

330

generalization) the procedure of transformation does not require solution of differential equations and does not depend on the differential field of coefficients, contrary to the case of LODOs.

[14] N. Kamran and K. Tenenblat. Laplace transformations in higher dimensions. Duke Math. J., 84(1):237–266, 1996. [15] J. Le Roux. Extensions de la m´ethode de Laplace aux ´equations lin´eaires aux deriv´ees partielles d’ordre sup´erieur au second. Bull. Soc. Math. de France, 27:237–262, 1899. A digitized copy is obtainable from http://www.numdam.org/ [16] O.E. Leshakov, V.M. Loginov. Coagulation of particles in a stochastic environment. Siberian Journal of Industrial Mathematics, 3:159–171, 2000 (in Russian). [17] Ya. B. Lopatinskii. Linear diffrential operators. Diss. Doct. Sci., 71 p. Baku, 1946. Reprinted in: Ya. B. Lopatinskii. Teoria obschih granichnyh zadach. Kiev, 1984 (in Russian). [18] P.J. Olver. Equivalence, Invariance, and Symmetry. CUP, 1995. [19] L. Petr´en. Extension de la m´ethode de Laplace aux Pn P ∂iz ∂ i+1 z ´equations n−1 i=0 A0i ∂y i = 0. Lund i=0 A1i ∂x∂y i + Univ. Arsskrift. Bd. 7, Nr. 3, pages 1–166, 1911. [20] M. van der Put and M.F. Singer. Galois Theory of Linear Differential Equations. Grundlehren der Mathematischen Wissenschaften, v. 328, Springer, 2003. [21] V.V. Sokolov and A.V. Zhiber. On the Darboux integrable hyperbolic equations. Phys. Letters A, 208:303–308, 1995. [22] A.V. Zhiber and V.V. Sokolov. Exactly integrable hyperbolic equations of Liouville type. Russian Math. Surveys, 56(1):61–101, 2001. [23] A.V. Zhiber and S.Ya. Startsev. Integrals, Solutions, and Existence Problems for Laplace Transformations of Linear Hyperbolic Systems. Mathematical Notes, 74(5-6):803–811, 2003. [24] S.P. Tsarev. Factorization of linear partial differential operators and Darboux integrability of nonlinear PDEs. SIGSAM Bulletin, 32(4):21–28, 1998; also ”Computer Science” e-print cs.SC/9811002 at http://www.arxiv.org/. [25] S.P. Tsarev. Factorization of overdetermined systems of linear partial differential equations with finite dimensional solution space. In Proc. 4th Int. Workshop on Computer Algebra Scient. Comput. (CASC-2001), V. Ganzha, E. Mayr, V. Vorozhtsov (eds), pages 529–539. Springer-Verlag, 2001. [26] Z. Li, F. Schwarz and S.P. Tsarev. Factoring systems of linear PDEs with finite-dimensional solution spaces. J. Symbolic Computation, 36:443–471, 2003.

Acknowledgements We would like to thank Prof. V.M. Loginov who provided many interesting examples of hyperbolic systems originating in the theory of stochastic differential equations and Prof. D. Grigoriev, Prof. F. Schwarz for stimulating discussions in the long period of this research started with the paper [24]. My sincere thanks to Prof. F. Schwarz for his valuable remarks and a copy of [19].

5. REFERENCES [1] I.M. Anderson and N. Kamran. The Variational Bicomplex for Second Order Scalar Partial Differential Equations in the Plane. Duke Math. J., 87(2):265–319, 1997. [2] C. Athorne. A Z2 × R3 Toda system. Phys. Lett. A, 206:162–166, 1995. [3] C. Athorne. Darboux maps and D-modules. Theor. and Math. Physics, 122:135–139, 2000. ¨ [4] H. Blumberg. Uber algebraische Eigenschaften von linearen homogenen Differentialausdr¨ ucken. Diss. G¨ ottingen, 1912. [5] M. Bronstein. An improved algorithm for factoring linear ordinary differential operators. In Proc. ISSAC’94 , pages 336–340, ACM Press, 1994. [6] S.C. Coutinho. A primer of algebraic D-modules. London Math. Soc. Student Texts, v. 33, CUP, 1995. [7] R. Courant. Partial differential equations. N.Y.-London, 1962. [8] G. Darboux. Le¸cons sur la th´eorie g´en´erale des surfaces et les applications g´eom´etriques du calcul infinit´esimal. t. 2,4, Paris (1887-1896). [9] E.V. Ferapontov. Laplace transformations of hydrodynamic type systems in Riemann invariants: periodic sequences. J. Phys A: Math. Gen., 30:6861–6878, 1997. [10] A.R. Forsyth. Theory of differential equations. Part IV, vol. VI. Cambrudge, 1906. [11] E. Goursat. Le¸cons sur l’int´egration des ´equations aux d´eriv´ees partielles du seconde ordre a deux variables ind´ependants. t. 2, Paris, 1898. [12] E. Goursat. Sur les ´equations lin´eaires et la m´ethode de Laplace. Amer. J. Math., 18:347–385, 1896. [13] D. Grigoriev and F. Schwarz. Factoring and solving linear partial differential equations. Computing, 73:179–197, 2004.

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Preconditioners for Singular Black Box Matrices William J. Turner Department of Mathematics & Computer Science Wabash College Crawfordsville, IN 47933 USA

[email protected]

ABSTRACT

include a parameterized matrix based on rearrangeable permutation networks such as Beneˇs networks [12, §V] and unit upper and lower triangular Toeplitz matrices [8, Thm.2]. Other preconditioners, such as those described in Eberly and Kaltofen [4], also require the transpose of A. Chen et al. [2, §6] describe how to construct a PreCondInd preconditioner by generalizing a Beneˇs network to butterfly networks for arbitrary dimension n. Section 2.1 shows how to further generalize these switching networks to switches of arbitrary radix ρ, which decreases both the depth of the network and the number of switches required. These arbitrary radix switching networks cannot directly precondition the matrix. Instead of switching the rows or columns of a matrix, preconditioning matrices mix them with a generic exchange matrix to achieve the desired linear independence. Section 2.2 shows preconditioners using radix-ρ switching networks and random full Toeplitz matrices as generic exchange matrices solve the PreCondInd problem over a large field with a high probability. The generalized switching networks of arbitrary radix may offer only a limited increase in performance over previously known butterfly networks, but we believe they may provide other advantages. In particular, they provide us a way to search for preconditioning matrices to solve the PreCondInd problem. For example, we can generalize the result in Section 2.2 to Toeplitz matrices of arbitrary size. In particular, these results allow us to use a single random full Toeplitz matrix preconditioner to solve the PreCondInd problem. Turner [11; 10, Ch.3] uses determinantal divisors to relax slightly the generic rank profile condition of the KaltofenSaunders rank algorithm before applying a diagonal multiplier. Instead of requiring a nonzero leading principal minor of size r, any nonzero principal r × r minor suffices. This relaxation, along with Section 2.2, means one random full Toeplitz matrix can precondition the matrix so the KaltofenSaunders rank algorithm can then apply the random diagonal multiplier. Section 3 shows the diagonal multiplier is superfluous and a random full Toeplitz matrix alone fulfills the requirements of the Kaltofen-Saunders rank algorithm. This preconditioner requires fewer random variables and has an improved probability of success over butterfly network preconditioners, but at the cost of a slower matrix-vector product.

This paper develops preconditioners for singular black box matrix problems. We introduce networks of arbitrary radix switches for matrices of any square dimension, and we show random full Toeplitz matrices are adequate switches for these networks. We also show a random full Toeplitz matrix to satisfy all requirements of the Kaltofen-Saunders black box matrix rank algorithm without requiring a diagonal multiplier.

Categories and Subject Descriptors I.1.2 [Symbolic and Algebraic Manipulation]: Algorithms—algebraic algorithms, analysis of algorithms

General Terms Algorithms, Performance, Reliability, Theory.

Keywords Black box linear algebra, rank preconditioner, singular linear system, switching network, Toeplitz matrix, Wiedemann method.

1.

INTRODUCTION

Kaltofen and Saunders [8] provide algorithms based on the Wiedemann method to compute the rank of a black box matrix and to uniformly sample the solution space of a singular linear system. Both of these algorithms rely on preconditioning a matrix to place it into a generic rank profile as defined by Kaltofen and Lobo [7]. The Kaltofen-Saunders rank algorithm also requires multiplying the resulting matrix with a diagonal matrix. Chen et al. [2, Thm.3.1] show how to reduce the PreCondGen preconditioning problem of placing a matrix into a generic rank profile to the PreCondInd problem of localizing the linear independence of the system by preconditioning a matrix A to make the leading r = rank(A) columns linearly independent. Solutions to the PreCondInd problem

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. ISSAC’05, July 24–27, 2005, Beijing, China. Copyright 2005 ACM 1-59593-095-7/05/0007 ...$5.00.

2.

GENERALIZED NETWORKS

Preconditioners based on Beneˇs networks [1] solve the PreCondInd preconditioning problem. In this section, we

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Figure 1: We may represent radix-2 (left), radix-3 (middle), and radix-4 (right) switches by the complete bipartite graphs K2,2 , K3,3 , and K4,4 , respectively. We also call radix-2 a butterfly switch. 0

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LLLrrr LLLrrr LLLrrr rL rL rL xrrr LL& xrrr LL&  xrrr LL& 0 0YY 0YY 0 bbb1 1 0YY e1 YYYYYYYYYYYY eeeeYebb YeYbb bbbb bb Y Y Y bb e Y Y Y bb e Y Y Y bb YYYYYY YeYYYYY, YYYYY, eYe bb eebb qbbbbbbb   , reebb 

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Figure 3: An example of a radix-3 network switching desired indices into position. Notice we must wrap indices around the outside of the right subnetwork to allow them to switch into the correct position.

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Lemma 2.1. Let n = ρ` where ρ ≥ 2. The `-dimensional radix-ρ switching network can switch any r indices 1 ≤ i1 < · · · < ir ≤ n into any desired contiguous block of indices. For our purposes, wrapping the block around the outside preserves contiguity. (For example, we consider the 1s in the last row of Figure 2 contiguous.)

1

Figure 2: Radix-3 switching network. The solid arrows show how to set the switches to achieve the desired output. In this case, the network switches the four inputs designated with 1s to four contiguous outputs.

Proof. Let us prove the lemma by induction. The proof is trivial for n = 1 because no switches are required. Suppose the lemma holds for n/ρ. Let us divide the n nodes into ρ equal subnetworks. We can construct radixρ switching networks of dimension ` − 1 for each of these collections of n/ρ nodes. Let r0 = 0, rρ = r, and rk be such that irk is the last of the chosen indices before the end of the kth radix-ρ network: irk ≤ kn/ρ < irk +1 for k = 1, . . . , ρ − 1. Because the lemma holds for n/ρ, we know the kth subnetwork can arrange the indices irk−1 +1 , . . . , irk into any desired contiguous blocks for k = 1, . . . , ρ. The contiguous block desired from the network is either contained in the interior of the network in indices j, . . . , j + r − 1 or it wraps around the outside of the network and we can denote it by indices 1, . . . , j − 1 and n − r + j, . . . , n. We can convert this second situation into the first by instead thinking of switching the n − r indices not originally chosen into the contiguous block 1 ≤ j, . . . , j + n − r − 1 ≤ n. Thus, we need only consider the first situation. We can use the first subnetwork to place inputs i1 , . . . , ir1 so they switch into outputs j, . . . , j + r1 − 1. We can then use the second subnetwork to place inputs ir1 +1 , . . . , ir2 so they switch into outputs j + r1 , . . . , j + r2 − 1. We can continue in this manner until we use the ρth subnetwork to place inputs irρ−1 +1 , . . . , ir so they switch into outputs j + rρ−1 , . . . , j + r − 1. In each of these subnetworks, the indices may wrap around the outside of the subnetwork to allow them to switch into the correct position as Figure 3 demonstrates. Using the induction hypothesis, each of these ρ subnetworks has dimension ` − 1 and n(logρ (n) − 1)/ρ2 switches. With the addition of a level of n/ρ switches to combine these subnetworks, the final radix-ρ switching network has dimension ` requires a total of s = n logρ (n)/ρ switches.

improve on previous work by Parker [9] and Chen et al. [2] by generalizing their butterfly networks to networks of arbitrary radix switches. We then show a Toeplitz matrix satisfies the requirements of an arbitrary radix exchange matrix.

2.1

Arbitrary Radix Switching Networks

Consider an n × n matrix over a field F. We want to make the leading r rows or columns of the matrix linearly independent. A switching network can exchange rows or columns until the first r rows or columns are linearly independent. However, unlike Chen et al. [2], we want to exchange more than two rows or columns at a time to decrease the depth of the network and the number of switches required. A radix-ρ switch generalizes a butterfly switch to allow any permutation of ρ inputs. As Figure 1 shows, we can depict a radix-ρ switch as a complete bipartite graph of 2ρ nodes. We consider the nodes across the top of the graphs as the inputs to the switch and the nodes across the bottom as the outputs. This complete bipartite graph allows the network to route each input directly to any desired output. We want to use a radix-ρ switching network to switch any r inputs of an arbitrary number n inputs to the beginning of the network. However, as in Chen et al. [2, §6], we must first consider the case of switching any r inputs into any contiguous block when n is a power of ρ: n = ρ` . An `-dimensional radix-ρ switching network is a recursive network that uses ρ`−1 radix-ρ switches to merge the outputs of ρ radix-ρ subnetworks, each of dimension ` − 1, such that the network merges the ith outputs of each of the subnetworks, 1 ≤ i ≤ ρ`−1 . Figure 2 illustrates a twodimensional radix-3 switching network with nine nodes at each level. We begin by showing how a radix-ρ switching network can switch any r inputs into any contiguous block of outputs when n is a power of ρ. We will use this result when we consider arbitrary n.

Lemma 2.1 says we can switch any r rows or columns of an n × n matrix into any contiguous block when n = ρ` . Recall our original goal of not restricting n to a power of ρ. To do so, we must describe a generalized radix-ρ switching network for an arbitrary n.

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any desired contiguous block of indices; for our purposes, wrapping the block around the outside preserves contiguity. Furthermore, the network has a depth of dlogρ (n)e and a total of no more than ρdlogρ (n)e dlogρ (n)e/ρ switches of radix at most ρ. The network attains this bound only when c = 1 or ` = 0. Proof. The proof follows directly from Lemma 2.1 when c = 1. Otherwise, let r0 = 0, rc = r, and rk be such that irk is the last of the chosen indices before the end of the kth radix-ρ network: irk ≤ kn/ρ < irk +1 for k = 1, . . . , c−1. As in the proof of Lemma 2.1, we need only consider the case when the desired contiguous block is contained within the interior of the network in indices j, . . . , j+r−1, and we can use the kth radix-ρ subnetwork to place inputs irk−1 +1 , . . . , irk so they switch into outputs j + rk−1 , . . . , j + rk − 1. In each of these radix-ρ networks, the indices may wrap around the outside of the network to allow them to switch into the correct position as was needed in the proof of Lemma 2.1. The number of switches needed when c 6= 1 is the number for the c radix-ρ networks plus ρ` switches to combine the networks:

Figure 4: Generalized radix-3 switching network. The dotted boxes represent the extra layer of switches.

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Figure 5: Generalized radix-3 switching network. The solid arrows show how to set the switches to achieve the desired output. In this case, the network switches the three inputs designated with 1s to the first three outputs.

s=

ρ ρ` ρdlogρ (n)e c ρ` ` + ρ` ≤ ` + ρ` = dlogρ (n)e. ρ ρ ρ

The network never attains this bound when c 6= 1 and ` 6= 0. Furthermore, the network has depth ` + 1 = dlogρ (n)e for c 6= 1.

When n is not a power of ρ, let us decompose n as a sum of powers of ρ: 8 `i > p :` < ` < · · · < ` i=1 1 2 p

This means the generalized radix-ρ switching network uses ρdlogρ (n)e dlogρ (n)e/ρ switches when n = ρ` or n = c; otherwise the network uses fewer switches. Using this result, we can show the generalized radix-ρ switching network can switch any r inputs into the first block of r inputs for any n.

The radix-ρ switching network is the case when p = c1 = 1. We construct radix-ρ switching networks for each of the ρ`i blocks. We then build a generalized radix-ρ switching network by connecting these networks with switches recursively Pk−1 such that for k = 2, . . . , p, the network merges the i=1 ni of a generalized subnetwork with the far right nodes of each of the ρ`k blocks with switches of radix ck + P1 ≤ ρ. If ck > 1, the network merges the remaining nk − k−1 i=1 ni nodes of each of the ck blocks of size ρ`k with switches of radix ck < ρ. These two sets of extra switches do not interact, so they constitute only one extra layer of switches after the last level of radix-ρ switching networks. We may think of this level of connecting switches as one set of switches of radix ck +1 ≤ ρ that switch the corresponding outputs of the ck radix-ρ networks with the combined Pk−1 outputs of the generalized radix-ρ network for i=1 ni and P ρk − k−1 n “phantom” outputs to the left of the generali i=1 ized subnetwork. In this way, we require only one additional level of switches to connect all of the subnetworks. Figure 4 shows the general organization of a generalized radix-3 network, and Figure 5 shows a specific generalized radix-3 network acting on seven inputs. Before we consider an arbitrary network of size n, let us first expand Lemma 2.1 to include the case when p = 1 but c1 6= 1; in other words, n = c ρ` where 1 < c < ρ. This case does not arise when ρ = 2.

Theorem 2.1. Suppose ρ ≥ 2. The generalized radix-ρ switching network described above can switch any r indices 1 ≤ i1 < · · · < ir ≤ n into the contiguous block 1, 2, . . . , r. Furthermore, it has a depth of dlogρ (n)e and a total of no more than ρdlogρ (n)e dlogρ (n)e/ρ switches of radix at most ρ. The network attains this bound only when n = ρdlogρ (n)e or n < ρ. Proof. The proof follows from Lemma 2.2 if n = cρ` where 1 ≤ c ≤ ρ − 1. Otherwise, let us decompose n into P powers of ρ (2.1). We know p > 1 and np > p−1 i=1 ni . We prove the first part of the theorem P by induction. Suppose the theorem is true for p−1 i=1 ni . Let r1 be such that ir1 is the last index in the first subnetwork: ir1 ≤

r−1 X

ni < ir1 +1 .

i=1

We can switch the indices i1 , . . . , ir1 into the contiguous block 1, . . . , r1 using a generalized butterfly network. In addition, let r0 = 0, rcp +1 = r, and rk be such that irk is the last of the chosen indices before the end of the (k − 1)st radix-ρ network: irk ≤ (k − 1) ρ` +

`

r−1 X

ni < irk +1 ,

k = 2, . . . , cp + 1.

i=1

Lemma 2.2. Let n = c ρ where ρ ≥ 2 and 1 ≤ c ≤ ρ − 1. The generalized radix-ρ switching network described above can switch any r indices 1 ≤ i1 < · · · < ir ≤ n into

By Lemma 2.1, the kth radix-ρ subnetwork can arrange inputs irk +1 , . . . , irk+1 into any desired contiguous block of

334

outputs, k = 1, . . . , cp . In particular, we use the kth subnetwork to place these inputs so they switch into outputs rk + 1, . . . , rk+1 . In each of these radix-ρ networks, the indices may wrap around the outside of the network to allow them to switch into the correct position. The total number of butterfly switches is the number of switches for each of the subnetworks plus the number of switches combine them. When cp = 1, the network rePto p−1 `p switches for the combining. When quires i=1 ni < ρ cp 6= 1, it requires ρ` switches. We may also count the switches as the sum of the number of switches for each of the ni blocks plus the number of switches to connect these blocks: ! ! p−1 p p i−1 i X X X ci ρ`i `i X X `j `i `j + cj ρ + ρ − cj ρ . s= ρ i=1 j=1 i=2 j=1 i=1

product for the preconditioning matrix L where the symbols have been evaluated at fixed random values as a black box matrix. We can use an arbitrary random ρ × ρ matrix where each entry contains a different symbol, but it requires O(ρ2 ) field multiplications. We now describe a Toeplitz exchange matrix that will accomplish the task while requiring only O(ρ log(ρ) loglog(ρ)) field operations. As in Chen et al. [2, §6.2], we will prove the result by induction on the levels of the generalized butterfly network. Consider a field F and the symbolic Toeplitz matrix 2 3 αρ αρ+1 . . . α2ρ−1 6αρ−1 αρ . . . α2ρ−2 7 6 7 T =6 . (2.2) .. .. 7 , .. . 4 . . . . 5 α1 α2 ... αρ where α1 , . . . , α2ρ−1 are distinct indeterminates over F. The (i, j)th entry of T is T[i,j] = αρ+j−i . Let T[i1 ,...,im ;j1 ,...,jm ] denote the submatrix of T formed by the intersection of rows i1 , . . . , im and columns j1 , . . . , jm , so det(T[i1 ,...,im ;j1 ,...,jm ] ) denotes the corresponding minor. When employed as a generic exchange matrix embedded in an n × n identity matrix to obtain the symbolic preconditioner Tb , this symbolic Toeplitz matrix will interact with only ρ of the n columns of A. Let us denote the submatrix of A formed by these ρ columns and all n rows as A0 . Then, the symbolic preconditioning ATb only changes these columns, and the matrix equation A0 T captures the entire preconditioning. The submatrix A0 affected by T includes a subset of r0 of the r linearly independent columns of A where 0 ≤ r0 ≤ r. We must show these r0 columns remain linearly independent as they pass through this level of the network. Then, if we assign unique indeterminates to each symbolic Toeplitz exchange matrix, then the network must preserve the linear independence of the full r columns of A in each level. Q Thus, the first r columns of A( sk=1 Tbk (α1,k , . . . , α2ρ−1,k )) are linearly independent over F(α1,k , . . . , α2ρ−1,s ). To show we can move the linear independence of the desired r0 columns through a single exchange matrix we examine the leading term of the resulting preconditioned matrix A0 T . Using the lexicographic monomial ordering with α1 ≺ α2 ≺ · · · ≺ α2n−1 [3, Def. 2.2.3], the leading term of the minor the intersection of rows i1 , . . . , im and columns j1 , . . . , jm of T is ` ` ´´ lt det T[i1 ,...,im ;j1 ,...,jm ] m Y = (−1)bm/2c αn+jm+1−k −ik . (2.3)

ci >1

From the decomposition of n into powers of ρ (2.1), we know `i ≥ i − 1 for 1 ≤ i ≤ p, so the number of switches required when p > 1 is s≤

p p X X ρdlogρ (n)e dlogρ (n)e ni ρ `i < ρ` p ` p + ρ ` p = `i + . ρ ρ i=1 i=2

The network never attains this bound if p > 1. However, Lemma 2.2 tells us the network does attain the bound when p = 1 and c = 1 or ` = 0. These correspond to the cases n = ρdlogρ (n)e and n < ρ. The depth of the network when p > 1 is the depth of the largest subnetworks of a power of three plus one level to combine the rest of the network with these subnetworks. Thus, the complete network has a depth of `p + 1 = dlogρ (n)e.

2.2

Generic Exchange Matrices

Following Wiedemann [12], we can use switching networks to construct left and right PreCondInd preconditioners for an n × n matrix by embedding ρ × ρ exchange matrices into an n × n identity matrix. Each switch in the network implements a directed acyclic arithmetic circuit, which we can represent by a linear equation. For example, 2 3 ˆ ˜ ˆ ˜ a1,1,k a1,2,k a1,3,k x ˜1 x ˜2 x ˜3 = x1 x2 x3 4a2,1,k a2,2,k a2,3,k 5 a3,1,k a3,2,k a3,3,k represents the circuit for the kth radix-3 switch of a network shown in Figure 1. We embed the ρ × ρ matrix in an n × n identity matrix in the fashion of an elementary matrix that can exchange the ρ columns by replacing the principal— not necessarily leading principal—submatrix of the identity matrix composed of the desired rows and columns with the exchange matrix. We then consider the symbolic preconQ ditioner L = sk=1 Sk (α1,1,k , . . . , αρ,ρ,k ) where Sk symbolically implements the kth embedded switch in the network of s switches and αi,j,k is a symbol. We denote symbols and symbolic preconditioners with Greek and calligraphic letters, and we denote their evaluated counterparts with corresponding roman letters. Let A be a fixed n × n matrix of rank r. The first r columns of the symbolically preconditioned matrix AL are linearly independent over the field F(α1,1,k , . . . , αρ,ρ,s ) because we may evaluate the symbols in such a manner that the network switches r linearly independent columns to the left. We need to minimize the cost of the matrix-vector

k=1

The kth largest index of α in this leading term, which corresponds to the kth most significant α in the leading term, is n + jm+1−k − ik . Given the row indices i1 , . . . , im , the leading term of the minor uniquely determines the column indices j1 , . . . , jm , and vice versa. The Binet-Cauchy formula tells us det((A0 T )[I ,J ] ) is a sum of products of minors of A0 and T : det((A0 T )[I ,J ] ) =

X K ={k1 ,...,km } 1≤k1 <···
335

det(A0 [I ,K ] ) det(T[K ,J ] ).

(See, for example, Gantmacher [5, §2.5].) Because the submatrices T[K ,J ] are all homogeneous of degree one, their determinants are all homogeneous of degree m. The minors det(T[K ,J ] ) are linearly independent because each has a unique leading term under the lexicographic monomial ordering with α1 ≺ α2 ≺ · · · ≺ α2ρ−1 . There exists at least one index set K such that det(A0 [I ,K ] ) 6= 0 because I is a set of indices of linearly independent columns of A. The minor det((A0 T )[I ,J ] ) is a nonzero sum of linearly independent polynomials, so it cannot be zero. This means every minor of the symbolically preconditioned matrix A0 T involving only linearly independent columns of A0 must be nonzero. If the matrix A0 has rank at least r0 , then every set of m ≤ r0 columns of the symbolically preconditioned matrix A0 T must be linearly independent. Thus we can move the linear independence around within one switch. By using different indeterminates αi for each symbolic Toeplitz matrix, we ensure linearly independent columns remain linearly independent at each level of the generalized radix-ρ switching network. For the kth switch in the generalized radix-ρ switching network, we can embed the ρ × ρ symbolic Toeplitz matrix Tk into an n × n identity matrix as was done by Chen et al. [2, §6] to obtain the matrix Tˆk . By randomly choosing a value ai for each indeterminate αi , we convert the symbolic Toeplitz matrix Tk into the Toeplitz matrix Tk and the symbolic exchange matrix Tˆk into the randomly chosen exchange matrix Tˆk . We prove the following counterpart to Theorem 6.3 in Chen et al. [2].

The logρ (n) term in Theorem 2.2 means a preconditioner based on a generalized radix-ρ switching network offers some improvement as ρ increases. However, the number of random values required—and thus the size of the random set S from which they are chosen—increases.

3.

In Turner [11; 10, Ch.3], we consider a preconditioned matrix AD, where D is a symbolic diagonal matrix. We show every invariant factor of the characteristic matrix λI − AD of size m such that rank(A) < m ≤ n is divisible by λ, which gives an upper bound for the degree of λ in the minimal polynomial f AD . To find a lower bound, we rely on the proof of Lemma 4.1 in Chen et al. [2] to show the characteristic polynomial of AD has no repeated factors except λ. The invariant factors other than the largest, which is the minimal polynomial f AD , are all powers of λ. We then use the existence of a nonzero principal minor to prove a lower bound for the λ-degree equal to the upper bound. We now use a similar technique of examining the invariant factors of the characteristic matrix to show a random full Toeplitz matrix preconditioner satisfies the requirements of the Kaltofen-Saunders rank algorithm without requiring a diagonal multiplier. We begin by again using the invariant factors to prove an upper bound on the λ-degree of the minimal polynomial of the symbolically preconditioned matrix. Lemma 3.1. Let F be a field, let A ∈ Fn×n have rank r, let s ∈ Z>0 , and let B1 , B2 ∈ F[α1 , α2 , . . . , αs ]n×n , where α1 , . . . , αs are distinct indeterminates over F. Let λ be an indeterminate distinct from α1 , . . . , αs . Then, for all k with r + 1 ≤ k ≤ n, λ divides the kth invariant factor of λI − B1 AB2 . Furthermore, the degree of λ in the minimal polynomial of B1 AB2 is bounded by degλ (f B1 AB2 ) ≤ r + 1. Proof. The first part follows from Turner [10, Lem.3.5] and the fact rank(B1 AB2 ) ≤ rank(A) by Sylvester’s inequality [5, p.66]. The characteristic polynomial of B1 AB2 has λ-degree n and is the product of the invariant factors of the characteristic matrix, sk (λI − B1 AB2 ) where 1 ≤ k ≤ n. Because the minimal polynomial of B1 AB2 is the largest of these invariant factors, we know it has λ-degree

Theorem 2.2. Let F be a field, let A ∈ Fn×n have r linearly independent columns, let s be the number of switches in the generalized radix-ρ switching network from Theorem 2.1, and let S be a finite subset of F. Let N be the number of random numbers required in this network. If a1 , . . . , aN are randomly chosen uniformly and independently from S, then Q the first r columns of A( sk=1 Tbk ) are linearly independent with probability at least 1−

RANK PRECONDITIONER

ndlogρ (n)e rdlogρ (n)e ≥1− . |S| |S|

Proof. The network requires N < 2 ρdlogρ (n)e dlogρ (n)e random numbers because the radix-ρ switching network uses s ≤ ρdlogρ (n)e dlogρ (n)e/ρ switches of radix at most ρ by Theorem 2.1 and each switch uses no more than 2ρ − 1 random numbers. Q Let A˜ = A( sk=1 Tbk ). Because each of the dlogρ (n)e levels in the generalized radix-ρ switching network increases the degree of the polynomials by one, the columns of A˜ are vectors of polynomials in α1 , . . . , αN of degree dlogρ (n)e, and the determinant of an r ×r submatrix of this preconditioned matrix is a polynomial of degree rdlogρ (n)e. Because A has r linearly independent columns, we can designate the generalized radix-ρ switching network of Theorem 2.1 to switch these columns to the first r columns of the ˜ There is an r × r submatrix of the preconditioned matrix A. first r columns of A˜ whose determinant is not identically zero. Because this determinant is a polynomial of degree rdlogρ (n)e, the Schwartz-Zippel lemma tells us (a1 , . . . , aN ) is a root of it with probability at most rdlogρ (n)e/|S|. ThereQ fore, the corresponding r×r submatrix of A( sk=1 Tbk ) whose determinant is not zero with probability no greater than 1 − rdlogρ (n)e/|S|.

degλ (f B1 AB2 ) = n −

n−1 X

degλ (sk (λI − B1 AB2 )).

k=1

However, because λ divides at least n−r−1 of these invariant factors, degλ (f B1 AB2 ) ≤ n − (n − r − 1) = r + 1 . ˜

We must show the lower bound degλ (f A ) ≥ r + 1 without relying on a diagonal multiplier and Chen et al. [2]. In order to use the leading term of the characteristic polynomial to ˜ has no repeated factors except λ, we shall show det(λI − A) again turn to the uniqueness of the leading terms of the minors of the symbolic Toeplitz matrix T . This time we shall use the elementary symmetric functions of the eigenvalues of AT to find the leading term of the characteristic polynomial of AT . First we must limit the row and column indices allowed for a nonzero minor of the original matrix A. Recall every matrix A ∈ Fm×m is row-equivalent to a row-reduced echelon matrix [6, Thm.5]. Let us denote the columns of A corresponding to the columns of the corresponding row-reduced echelon matrix containing the leading nonzero entries of the rows as the pivot columns of A.

336

for 1 ≤ k ≤ m. Thus, for every index set J such that A[I ,J ] 6= 0,

Lemma 3.2. Let F be a field, and let A ∈ Fn×n have rank r and pivot column indices P = {p1 , . . . , pr } where 1 ≤ p1 < · · · < pr ≤ n. Furthermore, let Q = {q1 , . . . , qr } where 1 ≤ q1 < · · · < qr ≤ n be the indices of the last r linearly independent rows of A. If I = {i1 , . . . , im } where 1 ≤ i1 < · · · < im ≤ n and J = {j1 , . . . , jm } where 1 ≤ j1 < · · · < jm ≤ n are sets of m ≤ r row and column indices, respectively, of A with det(A[I ,J ] ) 6= 0, then j` ≥ p` and im+1−` ≤ qr+1−` for all 1 ≤ ` ≤ m.

m m Y ` ´ Y lt det(T[J ,I ] )  αn+im+1−k −pk  αn+qr+1−k −pk k=1

by equation 2.3. The bound on the leading term of the principal minor then follows from the Binet-Cauchy formula (3.1).

Proof. The rows indexed by the set I must be linearly independent for the minor A[I ,J ] to be nonzero. The last m linearly independent rows of A are the rows index by {qr+1−m , . . . , qr }. Thus, im+1−` ≤ qr+1−` for all 1 ≤ ` ≤ m. Because every matrix A is row-equivalent to a row-reduced echelon matrix RA , there exists an invertible matrix BA such that A = BA RA . The Binet-Cauchy formula says the minor A[I ,J ] of the matrix A is a sum of products of minors of BA and RA : X det(A[I ,J ] ) = det(BA[I ,K ] ) det(RA[K ,J ] ).

We can use this bound on the leading term of the nonzero principal minors of AT to bound the leading term of the characteristic polynomial λI − AT . To find this leading term, we need to find the leading terms of the r × r principal minors of AT . We begin by using one particular nonzero minor of A to show the leading term attains its bound. Lemma 3.3. Let F be a field, and let A ∈ Fn×n have rank r and pivot column indices P = {p1 , . . . , pr } where 1 ≤ p1 < · · · < pr ≤ n. If I = {i1 , . . . , im } where 1 ≤ i1 < · · · < im ≤ n is any index set of r linearly independent rows of A, then det(A[I ,P] ) 6= 0.

K ={k1 ,...,km } 1≤k1 <···
Suppose there exists an index ` such that j` < p` , and let Jb and N be the sets of indices Jb = {j1 , . . . , j` } and N = {1, . . . , n}. At most, the first first ` − 1 entries of each column RA[j1 ] , . . . , RA[j` ] of the row reduced echelon form RA of A are nonzero. Thus, the submatrix RA [N ,Jb] is not of full rank. By adding more columns, the submatrix RA[N ,J ] must also not have full rank. Thus, det(RA[K ,J ] ) = 0 for every index set K . Therefore, det(A[I ,J ] ) = 0, which contradicts the original assumptions.

Proof. The columns A[p1 ] , . . . , A[pr ] of A form a basis for the column space of A, which means there exists a matrix B ∈ Fr×n such that A = A[N ,P] B where N is the set of indices N = {1, . . . , n}. Considering only the rows of A indexed by the set I , we obtain A[I ,N ] = A[I ,P] B. By Sylvester’s inequality [5, p.66], rank(A[I ,N ] ) ≤ rank(A[I ,P] ) ≤ r. However, the rows of A indexed by the set I are linearly independent, so rank(A[I ,N ] ) = r. Thus, rank(A[I ,P] ) = r and det(A[I ,P] ) 6= 0.

We can use this result to bound the leading term of any nonzero principal minor of the symbolically preconditioned matrix AT .

We can now determine the leading term of any nonzero r × r principal minor of AT .

Theorem 3.1. Let F be a field, let A ∈ Fn×n have rank r, and let T ∈ F[α1 , . . . , α2n−1 ]n×n be the symbolic Toeplitz matrix (2.2) where α1 , . . . , α2n−1 are distinct indeterminates over F. Furthermore, let P = {p1 , . . . , pr } where 1 ≤ p1 < · · · < pr ≤ n be the pivot column indices of A, and let Q = {q1 , . . . , qr } where 1 ≤ q1 < · · · < qr ≤ n be the indices of the last r linearly independent rows of A. If I = {i1 , . . . , im } where 1 ≤ i1 < · · · < im ≤ n is any set of m ≤ r indices, then the leading term of the principal minor det((AT )[I ,I ] ) under the lexicographic monomial ordering with α1 ≺ α2 ≺ · · · ≺ α2n−1 is bounded by

Theorem 3.2. Let F be a field, let A ∈ Fn×n have rank r and pivot column indices P = {p1 , . . . , pr } with 1 ≤ p1 < · · · < pr ≤ n, and let T ∈ F[α1 , . . . , α2n−1 ]n×n be the symbolic Toeplitz matrix (2.2) where α1 , . . . , α2n−1 are distinct indeterminates over F. If I where 1 ≤ i1 < · · · < im ≤ n is any index set of r rows of A, then either the principal minor det((AT )[I ,I ] ) is zero or its leading term under the lexicographic monomial ordering with α1 ≺ α2 ≺ · · · ≺ α2n−1 is “ ” lt det((AT )[I ,I ] )

m “ ” Y lt det((AT )[I ,I ] )  αn+ir+1−k −pk



k=1 m Y

k=1

= (−1)br/2c det(A[I ,P] )

r Y

αn+ir+1−k −pk .

k=1

αn+qr+1−k −pk .

Proof. If the rows of A indexed by I are not linearly independent, then det(A[I ,J ] ) = 0 for every set of r column indices J . The Binet-Cauchy formula (3.1) then says the principal minor det((AT )[I ,I ] ) is zero. Suppose the rows of A indexed by I are linearly independent and det(A[I ,J ] ) 6= 0 for the index set J = {j1 , . . . , jr } where 1 ≤ j1 < · · · < jr ≤ n. Lemma 3.2 says jk ≥ pk for 1 ≤ k ≤ r, and thus

k=1

Proof. The Binet-Cauchy formula says each minor of the matrix AT is a sum of products of minors of A and T : X det((AT )[I ,I ] ) = A[I ,J ] T[J ,I ] . (3.1) J ={j1 ,...,jm } 1≤j1 <···<jm ≤n

Lemma 3.2 says if A[I ,J ] 6= 0, then jk ≥ pk and im+1−k ≤ qr+1−k for 1 ≤ k ≤ m, which means

r ` ´ Y lt det(T[J ,I ]  αn+ir+1−k −pk .

αn+im+1−k −jk  αn+im+1−k −pk  αn+qr+1−k −pk

k=1

337

If J = P, then

We can finally state the leading term of det(λI −AT ). We will use this leading term to show the characteristic polynomial has no repeated factors other than λ.

r Y ` ´ lt det(T[P,I ] ) = (−1)br/2c αn+ir+1−k −pk k=1

Theorem 3.3. Let F be a field, let A ∈ Fn×n have rank r, and let T ∈ F[α1 , . . . , α2n−1 ]n×n be the symbolic Toeplitz matrix (2.2) where α1 , . . . , α2n−1 are distinct indeterminates over F. Furthermore, let P = {p1 , . . . , pr } where 1 ≤ p1 < · · · < pr ≤ n be the pivot column indices of A, and let Q = {q1 , . . . , qr } where 1 ≤ q1 < · · · < qr ≤ n be the indices of the last r linearly independent rows of A. If λ is an indeterminate distinct from α1 , . . . , α2n−1 , then λn−r divides det(λI −AT ) and the leading term of the characteristic polynomial det(λI − AT ) under the lexicographic monomial ordering with λ ≺ α1 ≺ α2 ≺ · · · ≺ α2n−1 is

by equation 2.3. On the other hand, if J 6= P, then jk > pk for some k, and lt(det(T[J ,I ] )) ≺ lt(det(T[P,I ] )) . The leading term of the principal minor then follows from the Binet-Cauchy formula (3.1). We use our previous results about the leading terms of the minors of AT to find the leading terms of the elementary symmetric functions of the eigenvalues of AT . Recall the mth elementary symmetric function of the eigenvalues of AT is the sum of the m × m principal minors of AT : X Em (AT ) = det((AT )[I ,I ] ). (3.2)

(−1)b3r/2c A[Q,P] λn−r

I ={i1 ,...,im } 1≤i1 <···
αn+pm+1−k −qk .

k=1

Lemma 3.4. Let F be a field, let A ∈ Fn×n have rank r, and let T ∈ F[α1 , . . . , α2n−1 ]n×n be the symbolic Toeplitz matrix (2.2) where α1 , . . . , α2n−1 are distinct indeterminates over F. Furthermore, let P = {p1 , . . . , pr } where 1 ≤ p1 < · · · < pr ≤ n be the pivot column indices of A, and let Q = {q1 , . . . , qr } where 1 ≤ q1 < · · · < qr ≤ n be the indices of the last r linearly independent rows of A. The leading term of the rth elementary symmetric function of the eigenvalues of AT under the lexicographic monomial ordering with α1 ≺ α2 ≺ · · · ≺ α2n−1 is lt (Er (AT )) = (−1)br/2c A[Q,P]

r Y

Proof. The matrix AT has characteristic polynomial det(λI − AT ) =

n X

(−1)m Em (AT )λn−m .

(3.3)

m=0

Lemma 3.4 says Em (AT ) = 0 for m > r and λn−r divides the characteristic polynomial. The lemma also says lt (Er (AT )) = (−1)br/2c A[Q,P]

r Y

αn+qr+1−k −pk ,

k=1

which means ` ´ lt (−1)r Er (AT )λn−r

αn+qr+1−k −pk .

k=1

Otherwise, lt(Em (AT ))  Er (AT ) = 0 if m > r.

r Y

Qm

k=1 αn+qr+1−k −pk if m < r and

= (−1)b3r/2c A[Q,P] λn−r

r Y

αn+qr+1−k −pk .

k=1

Proof. Because rank(AT ) ≤ rank(A) = r by Sylvester’s inequality [5, p.66], we know every minor of AT of size greater than r is zero. Thus, Em (AT ) = 0 for m > r. Otherwise, let I be the index set I = {i1 , . . . , im } where 1 ≤ i1 < · · · < im ≤ n. If m < r, Theorem 3.1 says m “ ” Y lt det((AT )[I ,I ] )  αn+qr+1−k −pk ,

If m < r, then λ ≺ αn+qr+1−k −pk for all k means m Y ` ´ lt (−1)m Em (AT )λn−m  λn−m αn+qr+1−k −pk k=1

` ´ ≺ lt (−1)r Er (AT )λn−r . and the proof follows from equation 3.3.

k=1

Notice this leading term contains the smallest degree of λ possible and is at most linear in each of the αk . This is the result we desired to show the characteristic polynomial has no repeated factors other than λ.

which means, by the definitionQ of the elementary symmetric function (3.2), lt(Em (AT ))  m k=1 αn+qr+1−k −pk . If m = r, Theorem 3.2 says “ ” lt det((AT )[I ,I ] ) = (−1)br/2c det(A[I ,P] )

r Y

Theorem 3.4. Let F be a field, let A ∈ Fn×n have rank r with r ≤ n − 1, and let T ∈ F[α1 , . . . , α2n−1 ]n×n be the symbolic Toeplitz matrix (2.2) where α1 , . . . , α2n−1 are distinct indeterminates over F. Then, the matrix AT has characteristic polynomial det(λI − AT ) = λn−r g and minimal polynomial f AT = λ g where g is squarefree, λ does not divide g, and degλ (f AT ) = r + 1.

αn+ir+1−k −pk .

k=1

If I = Q, then “ ” lt det((AT )[Q,Q] ) br/2c

= (−1)

A[Q,P]

r Y

Proof. Because λn−r divides the characteristic polynomial by Theorem 3.3, we know det(λI − AT ) = λn−r g. By the same theorem, λn−r+1 does not divide the leading term of the characteristic polynomial, so we know the leading term of g is constant in λ and at most linear in each αk . Therefore, λ does not divide g. λ. Suppose g has a repeated factor h. Then, h2 divides g, and (lt(h))2 = lt(h2 ) must divide the leading term of g. This means lt(h) = 1, and thus g is squarefree.

αn+qr+1−k −pk .

k=1

If I 6= Q, then ik > pk for some k by Lemma 3.2, and “ ” “ ” lt det((AT )[I ,I ] ) ≺ lt det((AT )[Q,Q] ) . Thus, the leading term of the rth elementary symmetric function is lt(Er (AT )) = lt(det((AT )[Q,Q] )).

338

Because det(λI − AT ) has no repeated factors other than λ, all the invariant factors of det(λI −AT ) except the largest are either 1 or a power of λ, so f AT = λp g where p ≥ 1. Because degλ (λI − AT ) = n, we know degλ (g) = r, which means degλ (f AT ) ≥ r + 1. The proof then follows from the upper bound on degλ (f AT ) given by Lemma 3.1 where B1 = I and B2 = T .

matrices and a diagonal multiplier have probabilities of success of at least «2 „ « „ r(r + 1) rdlog2 (n)e 1− 1− |S| 2|S| and 1−

This means we can use the symbolic preconditioner AT in the Kaltofen-Saunders rank algorithm. The algorithm would be deterministic, but it would also be exponential in the size of the input matrix. We can use the SchwartzZippel Lemma [13] to convert this symbolic preconditioner to a randomized preconditioner that will make the KaltofenSaunders rank algorithm probabilistic and polynomial time.

respectively. On the other hand, Toeplitz matrices have slower matrix-vector products than butterfly network preconditioners, requiring O(n log(n) loglog(n)) field operations instead of only O(n log(n))).

4.

REFERENCES

[1] Beneˇ s, V. E. Permutation groups, complexes, and rearrangeable connecting networks. Bell System Tech. J. 43 (1964), 1641–1656. [2] Chen, L., Eberly, W., Kaltofen, E., Saunders, B. D., Turner, W. J., and Villard, G. Efficient matrix preconditioners for black box linear algebra. Linear Algebra Appl. 343-344 (Mar. 2002), 119–146. Special issue on Infinite Systems of Linear Equations Finitely Specified, edited by P. Dewilde, V. Olshevsky and A. H. Sayed. [3] Cox, D., Little, J., and O’Shea, D. Ideals, Varieties, and Algorithms: An Introduction to Computational Geometry and Commutative Algebra. Springer Verlag, Heidelberg, Germany, 1996. [4] Eberly, W., and Kaltofen, E. On randomized Lanczos algorithms. In Proc. ISSAC ’97 (New York, 1997), W. W. K¨ uchlin, Ed., ACM Press, pp. 176–183. [5] Gantmacher, F. R. Matrix Theory, vol. I. AMS Chelsea Pub., Providence, 1977. [6] Hoffman, K., and Kunze, R. Linear Algebra, 2nd ed. Prentice Hall, Upper Saddle River, NJ, 1971. [7] Kaltofen, E., and Lobo, A. On rank properties of toeplitz matrices over finite fields. In Proc. ISSAC ’96 (New York, 1996), Y. N. Lakshman, Ed., ACM Press, pp. 241–249. [8] Kaltofen, E., and Saunders, B. D. On Wiedemann’s method of solving sparse linear systems. In Proc. AAECC-9 (Heidelberg, Germany, 1991), H. F. Mattson, T. Mora, and T. R. N. Rao, Eds., vol. 539 of Lect. Notes Comput. Sci., Springer Verlag, pp. 29–38. [9] Parker, D. S. Random butterfly transformations with applications in computational linear algebra. Technical Report CSD-950023, UCLA, 1995. [10] Turner, W. J. Black Box Linear Algebra with the LinBox Library. PhD thesis, N. Carolina State Univ., Raleigh, North Carolina, May 2002. [11] Turner, W. J. Determinantal divisors and matrix preconditioners. Submitted to J. Symbolic Comput., June 2003. [12] Wiedemann, D. H. Solving sparse linear equations over finite fields. IEEE Trans. Inform. Theory IT-32, 1 (1986), 54–62. [13] Zippel, R. Probabilistic algorithms for sparse polynomials. In Proc EUROSAM ’79 (Heidelberg, Germany, 1979), E. W. Ng, Ed., vol. 72 of Lect. Notes Comput. Sci., Springer Verlag, pp. 216–226.

n×n

Theorem 3.5. Let F be a field, A ∈ F have rank r with r ≤ n − 1, and S be a finite subset of F. If 3 2 an an+1 . . . a2n−1 6an−1 an . . . a2n−2 7 7 6 n×n , (3.4) T =6 . .. 7 ∈ F . .. .. 4 .. . . 5 a1 a2 ... an where a1 , . . . , a2n−1 are chosen uniformly and independently from S, then the matrix AT has characteristic polynomial det(λI − AT ) = λn−r g(λ) and minimal polynomial f AT = λ g(λ) where g(0) 6= 0 and deg(f AT ) = r + 1, all with probability at least 1−

3r(r + 1) , 2|S|

n(n − 1) r(r + 1) ≥1− . 2|S| 2|S|

Proof. Suppose |F| > r(r + 1)/2; otherwise the result is trivial. By Theorem 3.4, if T ∈ F[α1 , . . . , α2n−1 ]n×n is the symbolic Toeplitz matrix (2.2) where α1 , . . . , α2n−1 are distinct indeterminates over F, then det(AT ) = λn−r g(λ, α1 , . . . , α2n−1 ) and f AT = λ g where g is not divisible by λ and f AT has degree r + 1 in lambda. Let y ∈ Fn be a vector such that y, (AT )y, . . . , (AT )r y are linearly independent. There is a (r + 1) × (r + 1) submatrix of the matrix with these vectors as its columns whose determinant is a nonzero polynomial in α1 , . . . , α2n−1 . This polynomial has total degree at most r(r + 1)/2 in the indeterminates α1 , . . . , α2n−1 . If values a1 , . . . , a2n−1 are chosen uniformly and independently from S for the indeterminates α1 , . . . , α2n−1 , the determinant is a nonzero element of F with probability at least 1 − r(r + 1)/(2|S|) by the Schwartz-Zippel Lemma. The vectors y, (AT )y, . . . , (AT )r y are linearly independent for our randomly chosen Toeplitz matrix (3.4) and the kth invariant factor of λI − AT is obtained from the kth invariant factor of λI − AT by replacing the indeterminates α1 , . . . , α2n−1 with the values a1 , . . . , a2n−1 , respectively. Not only does AT satisfy the requirements of the KaltofenSaunders rank algorithm, but T A does as well. The proof follows from the uniqueness of the Smith form to show the matrices λI − AT T and λI − T A have the same Smith form. This is an improvement in probability of success over requiring a diagonal multiplier. For comparison, the KaltofenSaunders rank algorithm using two butterfly network preconditioners and a diagonal multiplier or two unit Toeplitz

339

Solving Second Order Linear Differential Equations with Klein’s Theorem M. van Hoeij∗& J.-A. Weil†

ABSTRACT

1. INTRODUCTION

Given a second order linear differential equations with coefficients in a field k = C(x), the Kovacic algorithm finds all Liouvillian solutions, that is, solutions that one can write in terms of exponentials, logarithms, integration symbols, algebraic extensions, and combinations thereof. A theorem of Klein states that, in the most interesting cases of the Kovacic algorithm (i.e when the projective differential Galois group is finite), the differential equation must be a pullback (a change of variable) of a standard hypergeometric equation. This provides a way to represent solutions of the differential equation in a more compact way than the format provided by the Kovacic algorithm. Formulas to make Klein’s theorem effective were given in [4, 2, 3]. In this paper we will give a simple algorithm based on such formulas. To make the algorithm more easy to implement for various differential fields k, we will give a variation on the earlier formulas, namely we will base the formulas on invariants of the differential Galois group instead of semi-invariants.

The Kovacic algorithm [19] computes closed form (Liouvillian) solutions of second order linear differential equations over k = C(x). Since the appearance of [19], many papers have studied and refined the method. The version given in [27] uses invariants instead of the semi-invariants, which is easier to implement especially for differential fields k more complicated than C(x). The paper [15] gives good formulas for computing algebraic solutions (after [25]). The common basis of these algorithms is to derive solutions from (semi)invariants of the differential Galois group (see section 3). Another approach is the Klein pullback method: Klein ([18], also [1, 5, 2]) showed that if the projective differential Galois group is finite, then the equation is a pullback of an equation in a finite list of well-known standard hypergeometric Requations. This means that the solutions are of the form e g H(f ) where f, g ∈ k and H is a standard hypergeometric function H(x) = 2 F1 ([a, b], [c], x) whose parameters a, b, c appear in a finite list. Interest in this method has recently been revived [5, 20, 21] for classifying work, but finding pullback functions still relied on skill. In [4, 2, 3] Berkenbosch and the authors of this paper give (surprisingly simple) formulas for computing the pullback function f (as well as the function g). In [2, 3] Berkenbosch generalizes Klein’s theorem to third order operators. Our formulas from [4, 2, 3] rely on computing semi-invariants of the differential Galois groups, which is well-mastered for differential equations with coefficients in C(x). For more general differential fields, however, it may be easier (as noted in [27]) to use algorithms that compute invariants of the differential Galois group instead of semi-invariants. In order to use invariants, we will need to give formulas that are slightly different from those given in [4, 2, 3]. The contribution in this paper is of algorithmic nature: we give an algorithm for solving second order differential by pullbacks for a general differential field k by constructing new formulas which rely on invariants only. A field k is admissible for our algorithm if: k is an effective (computable) field (this includes extracting square roots), one has an algorithm for computing rational solutions of linear differential equations with coefficients in k and an algorithm for computing exponential solutions of second order differential equations. Examples of admissible fields are Liouvillian extensions of C(x) ([24]). Implementations of the above assumed algorithms are available for fields such as C(x), C(x, exp(f )) ([8]), quadratic extensions of C(x) ([10]), etc. For those

Categories and Subject Descriptors G.4 [Mathematical Software]: Algorithm design and analysis

General Terms Algorithms

Keywords Linear Differential Equations, Liouvillian solutions, Klein’s Theorem ∗ Dept. of Mathematics, Florida State University, Tallahassee, FL, USA. [email protected] Supported by NSF grant 0098034 †

LACO, Universit´e de Limoges, 123 avenue Albert Thomas 87060 Limoges cedex [email protected]

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340

fields k, the algorithm proposed here for computing Liouvillian solutions will be easy to implement. Although we recall the main ideas in sections 3, we assume in this paper that the reader has an elementary knowledge of differential Galois theory ([23]) and of the Kovacic algorithm [19, 23]. The algorithm in section 2 below follows the lines of the rational version of the Kovacic algorithm given in [27]. Section 2 contains the algorithm. Most of the remainder of the paper is devoted to its correctness and optional improvements. Section 3 contains material and definitions from differential Galois theory and Kovacic’s algorithm; section 4 recalls the pullback formulas from [4, 2, 3] for the case k = C(x), Section 5 proves the pullback formulas for a general differential field and the correctness of the algorithm. Finally, we remark that some recent papers [7, 12] showed how to solve certain classes of second linear differential equations as pullbacks of differential equations corresponding to special functions (Airy, Whittaker, etc). The present work is complementary to those whenever the differential equation has more than 3 singularities and the projective differential Galois group is not PSL2 . Acknowledgments. The authors would like to thank M. van der Put, M. Bronstein, M. Loday-Richaud and specially M. Berkenbosch for fruitful discussions while preparing this work. We thank the referees for their comments.

2.

tions, or (if G is not A5 ) nested radicals. We propose the representation as the default choice because it is the most compact representation. Moreover, converting these 2 F1 ’s to algebraic functions or nested radicals is easier to implement (table lookup) than the reverse conversion. s m The m-th symmetric power L of L is the operator whose solutions are spanned by products of m solutions of L. Given differential operators L ∈ k[∂] and ∂ −b, b ∈ k, the notation L ⊗ (∂ − b) refers to the operator whoseR solutions are the solutions of L multiplied by the solution e b of ∂ − b. Given a differential operator L = ∂ 2 + a1 ∂ + a0 , we define a0 its g-invariant to be gL := 2a1 + a00 . We can now state the algorithm. The steps have to be performed in the given order, and the algorithm exits when a solution is found. Pullback Algorithm, general k: Input: L with G(L) ⊂ SL2 (C) Output: Liouvillian solutions, expressed via solutions of the above standard operators 1. Determine if L has a solution y such that y 0 /y ∈ k (an exponential solution). If so, return a basis of Liouvillian solutions of L [15, 2, 19, 27, 23] 2 F1

s 4 2. Let B4 be a basis of solutions in k of L

(a) If B4 contains one element i4 , let ∂ 2 + a1 ∂ + a0 := R √ √ i0 L ⊗ (∂ + 4i44 ). Return 4 i4 e± −a0 or use section 5.3.

THE ALGORITHM

In this section, we state the algorithm assuming the reader is familiar with notations and concepts from differential Galois theory and Kovacic’s algorithm; unfamiliar readers should proceed first to the next sections for explanations and come back to this section afterward. Let k denote a differential field of characteristic 0. We consider the differential operator 2

L = ∂ + A1 ∂ + A0 ∈ k[∂]

(b) (implementation of this step is optional). If B4 contains two elements then let m = 6 and take solutions as in step 3 below (B6 will have one element i6 ), or use section 5.4. 3. For m in 6, 8, 12, let Bm be a basis of solutions in s m k of L . If Bm contains one element im , then let i0 2 L = ∂ + a1 ∂ + a0 := L ⊗ (∂ + mimm ). Now return the following basis of solutions of L √ √ m im H1 (f ), m im H2 (f )

(2.1) 00

This corresponds to the differential equation y + A1 y 0 + A0 y = 0. We assume that there exists w ∈ k such that 0 A1 = − ww (this is not restrictive since after a simple transformation one may assume the stronger condition A1 = 0, see section 3). We define the following standard differential operators StsDn StsG

1 x ∂− 2 2 ,n ∈ N x2 − 1 4n (x − 1) (8x + 3) (6ν − 1)(6ν + 1) = ∂2 + ∂+ 6 (x + 1) x 144 (x + 1)2 x = ∂2 +

where H1 (x), H2 (x) is a basis of solutions of StiG and where G and f are determined as follows: q a0 . (a) If m = 6, then G := A4 and f := 1 + 64 5 g2

(2.2)

L

This f will be in k.

7 (b) If m = 8, G := S4 and f = − 144

(2.3)

(c) If m = 12, G := A5 and f =

for (G, ν) ∈ {(A4 , 1/3), (S4 , 1/4), (A5 , 1/5)}. 4 5 x x2 + 3 ∂− 2 3 (x − 1) 144 (x2 − 1)2

StiD2

= ∂2 +

StiDn

= StsDn , n > 2 ` ´ 2 3 x2 − 1 5 2 = ∂ + ∂+ 3x (x2 − 1) 144x2 (x2 − 1) 1 (5 x − 2) 1 7 = ∂2 + ∂− 4 (x − 1) x 576 (x − 1)2 x

StiA4 StiS4

2 gL . a0

2 11 gL . 400 a0

The name of the standard operators refers to the projective differential Galois group P G(L) (see section 3 below) of L.

(2.4) (2.5)

4. Otherwise the operator has no Liouvillian solutions.

(2.6)

The above algorithm is correct but improvements are possible. In step 2a where B4 has one element, we have P G(L) = Dn√for some n > 2. If an integration algorithm for the field k( −a0 ) is available, then we could use it to try to simplify R √ the expression e± −a0 . However, if n 6= ∞ then there is an alternative that is likely to be more efficient. To implement this alternative, one starts by running a subroutine of the integration algorithm ([6]) that determines n. When

(2.7)

and StiA5 = StsA5 . These are well studied hypergeometric operators and their solutions are well-known. There are various ways to express the solutions of the above operators, one can use the hypergeometric function 2 F1 , or algebraic func-

341

n is found, if n 6= ∞, then instead of running the remainder of the integration algorithm one proceeds by using the formulas in section 5.4. Implementation of step 2b is optional. In step 2b, the projective Galois group is D2 (this denotes C2 × C2 ). If step 2b is not implemented, then in the D2 case the algorithm will proceed to step 3a and compute solutions using formulas meant for A4 . Although these formulas give correct solutions for the D2 case (note that D2
A4 and that these two groups have the same invariants of degree 6) one can find better (more compact) solutions in this case by using equation (2.4) and the formula from section 5.4.

3.

under the action of the differential Galois group G(L) of L. In other words h ∈ k. This function h is then called the value of the invariant polynomial I. 2. A homogeneous polynomial I(Y1 , Y2 ) ∈ C[Y1 , Y2 ] is called a semi-invariant with respect to a differential h0 ∈ k where h := I(y1 , y2 ). operator L if h We will list a few well known facts, for more details see [25, 23, 19]. For second order operators, there is a one to one correspondence between the (semi)-invariants of degree m and their values (for higher order operators this need not be the case). The values of invariants of degree m are pres m cisely the rational solutions of L , i.e solutions in k. The values of the semi-invariants of degree m are the so-called s m exponential solutions of L , that is, those solutions h of

s m 0 L for which h /h ∈ k. s m The operator L can be easily computed from the recursion given in (1.14) in [11] (see also [9]): Let L0 = 1, L1 = ∂ and

DIFFERENTIAL GALOIS THEORY

For completeness and to set notations, we briefly recall the rational Kovacic algorithm from [27]. Let L = ∂ 2 +A1 ∂ +A0 where A0 , A1 ∈ k. We consider a second order ordinary linear differential equation Ly = 0, We assume that A1 =

y 00 + A1 y 0 + A0 y = 0. f0 f

(3.8)

for some f ∈ k; this can be R A1

achieved after a change of variable y 7→ ye 2 which turns the equation (3.8) into the reduced form y 00 − ry = 0 with A2

Li+1 = (∂ + iA1 )Li + i(m − (i − 1))A0 Li−1 s m for 0 < i ≤ m, then Lm+1 = L .

A0

r = 41 + 21 − A0 . Given two linearly independent solutions of (3.8), say y1 , y2 (either “formal” or “actual functions on some open set”), the field K := k(y1 , y2 , y10 , y20 ) is a differential field (a field closed under differentiation) and is generated, as a differential field, by y1 and y2 over k. This field K is called a Picard-Vessiot extension of (3.8). The solution space in K is the C vector space generated by y1 and y2 , denoted by V in all that follows. The group of differential automorphisms of K over k (i.e., automorphisms of K over k that commute with ∂) is called the differential Galois group of (3.8) over k. We 0 denote it by G(L) = GalK/k (L). The condition A1 = ff ensures that G(L) ⊂ SL2 (C). The projective Galois group is defined by

3.2 The Subgroups of SL2 (C) Invariants and semi-invariants are elements of C[Y1 , Y2 ]. In the algorithm we will not calculate the invariants themselves, but only their values. For each semi-invariant, we will only compute the logarithmic derivative h0 /h of the value h of a semi-invariant. So in the following, when we write that there are n semi-invariants of degree m, we are counting the number of distinct h0 /h ∈ k for which h is a solution s m of L . And when we write that there are n invariants of s m degree m, we mean that the set of solutions of L in k has a basis with n elements. We recall the classification of subgroups of SL( C) (see e.g [19, 25, 27, 23]) and the invariants and semi-invariants of lowest degree. The group is reducible if there is at least one invariant line in V . A non-zero element of that line is an exponential solution, i.e., a solution whose logarithmic derivative is in k (see [23, 27, 15, 2] for more on this case). The rest of the classification (irreducible cases) is in the above references:

P G(L) := G(L)/(G(L) ∩ C ∗ ),

where G(L) ∩ C ∗ denotes the subgroup of those g ∈ G that act on V as scalar multiplication. R Multiplying the solutions by e b for b in k changes the Galois group G(L) but not the projective Galois group P G(L). The R operator whose solutions are y·e b , with y solution of L(y) = 0, is denoted L ⊗ (∂ − b). We will say that two operators L1 and L2 are projectively equivalent when there exists b ∈ k such that L1 = L2 ⊗ (∂ − b). It is easy to see that L1 , L2 are projectively equivalent if and only if they have the same reduced form. If L1 , L2 are projectively equivalent then P G(L1 ) = P G(L2 ).

Lemma 3.2 (Imprimitive groups). Assume that G(L) ⊂ SL2 (C) and that G(L) is imprimitive, i.e. irreducible and there exist two lines l1 , l2 ⊂ V such that G(L) acts on {l1 , l2 } by permutation. Then P G(L) ⊂ D∞ (infinite dihedral group). Three cases are to be considered. 1. P G(L) = D2 . Three semi-invariants S2,a , S2,b , S2,c of 2 is invariant), two invariants I4,a , I4,b degree 2 (S2,x of degree 4. One invariant I6 of degree 6, with I6 = S2,a S2,b S2,c . Note that the notation D2 does not refer to the cyclic group C2 but to C2 × C2 .

3.1 Invariants and Semi-Invariants The key to Kovacic’s algorithm is that the existence of Liouvillian solutions is (for second order equations) equivalent with the existence of a semi-invariant of the differential Galois group. Definition 3.1. Fix a basis y1 , y2 of the solution space V of L.

2. P G(L) = Dn , n > 2. One semi-invariant S2 of degree 2, one invariant I4 = S22 of degree 4, and another invariant I2n of degree 2n.

1. A homogeneous polynomial I(Y1 , Y2 ) ∈ C[Y1 , Y2 ] is called an invariant with respect to the differential operator L if its evaluation h := I(y1 , y2 ) is invariant

3. P G(L) = D∞ has only one semi-invariant S2 of degree 2 and one invariant I4 = S22 of degree 4.

342

A standard equation for each finite projective differential Galois group can be found among the hypergeometric equations a b c StP G = ∂ 2 + 2 + + x (x − 1)2 x(x − 1)

Lemma 3.3 (Primitive groups). Assume G is primitive, i.e neither reducible nor imprimitive, and G(L) ⊂ SL2 (C). Four cases are to be considered. 1. P G(L) = A4 ; two semi-invariant S4,a , S4,b of degree 4, one invariant I6 of degree 6, and one invariant I8 of degree 8, with I8 = S4,a S4,b

where the coefficients a, b, c are related to the differences λ, µ, ν of the exponents at 0, 1, and ∞ by the relations

2. P G(L) = S4 ; one semi-invariant S6 of degree 6, one invariant I8 of degree 8. 3. P G(L) = A5 ; one invariant I12 of degree 12. 4. G = SL2 (C); no semi-invariants and no Liouvillian solutions. The degrees for the (semi)-invariants of these groups als m low to give a list of possible symmetric powers L to investigate. This is the key to the Kovacic algorithm (semiinvariants) or its Ulmer-Weil rational variant [27] (invariants). Computing invariants (or semi-invariants), one can find the type of the differential Galois group (a little more needs to be done to discriminate Dn from D∞ , see section 4.4). We summarize this in the following immediate corollary

b=

1 − µ2 4

and

c=

4.2 Klein’s theorem hd i h i Definition 4.1. Let L1 ∈ C(z) and L2 ∈ k ∂ be dz linear differential operators. 1. L2 is a proper pullback of L1 by f ∈ k if the change of variable z 7→ f changes L1 into L2 .

Corollary 3.4. In the Pullback algorithm from section 2, in the case of step 1 the group is reducible, in case of step 2a the projective Galois group is D∞ or some Dn , n > 2. It is D2 in case of step 2b, A4 in step 3a, S4 in step 3b, A5 in step 3c, and PSL2 otherwise.

2. L2 is a (weak) pullback of L1 by f ∈ k if there exists v ∈ k such that L2 ⊗ (∂ + v) is a proper pullback of L1 by f . Theorem 4.2 (Klein, [18, 1, 2]). Let L be a second order irreducible linear differential operator over k with projective differential Galois group P G(L). Then, P G(L) ∈ {Dn , A4 , S4 , A5 } if and only if L is a (weak) pullback of StP G(L) .

For each possible finite projective group, pullback formulas can be computed; this is done in the next section.

4.

1 − λ2 4

1 − ν 2 + λ2 + µ2 . 4 `1 1 1 ´ for More precisely, one can choose (λ, µ, ν) = , , 2 2 n `1 1 1´ `1 1 1´ for P G = A4 , for P G = S4 , , , , P G = Dn , 3 2 3 3 2 4 `1 1 1´ and , , for P G = A5 . 3 2 5 The index P G refers to the projective differential Galois group of StP G corresponding to the chosen values of a, b, c. These equations and their solutions are well known. a=

PULLBACK FORMULAS, CASE K = C(X)

Let L have a projective differential Galois group P G(L) and suppose the standard equation with projective differential Galois group P G(L) has H1 , H2 as a C-basis of solutions. The theorem of Klein says that L is a pullback of StP G(L) . Suppose we know f and v as in definition 4.1, then a C-basis R R of solutions of Ly = 0 is given by H1 (f )e v and H2 (f )e v . H1 and H2 are known for all standard equations. To get the solutions in explicit form one should then determine the projective differential Galois group and, in case it is finite, determine f and v. It was remarked in [1, 5] (and somehow in [18]) that f can be expressed as a quotient of invariants of the differential Galois group, but this idea was not used algorithmically. We will build f (and v) using semi-invariants in section 4, and using invariants in section 5. The difficulty lies in the fact that L is a weak pullback of a standard equation, i.e it is only projectively equivalent to a proper pullback of the standard equation. The key to formulas is to compute a normal form such that the normal form of L will be a proper pullback of its standard form. Suppose that L has a differential Galois group G (and projective group P G) with semi-invariant S of degree m and value σ. And suppose the value of S with respect to the standard operator StP G equals σ0 (modulo C ∗ ). Then, the value of S w.r.t. both the differential operator SG = StP G ⊗ σ0 σ0 ) (∂z + 0 ) and the differential operator L = L ⊗ (∂x + mσ mσ0 is equal to 1 and the following property holds.

In this section, we recall our work with Maint Berkenbosch from [4, 2]. The next subsection is standard material [1, 2, 5, 20, 21]

4.1 Standard equations If y1 , y2 is a basis of solutions of L, then define CL := C( yy21 ), which is a subfield of the Picard-Vessiot extension K. The field CL does not depend on the choice of basis (replacing y1 , y2 by another basis corresponds to aR M¨ obius R transformation of yy12 ). Replacing y1 , y2 by e v y1 , e v y2 for some function v does not affect CL either. In fact, given two operators L1 and L2 , one has CL1 = CL2 if and only if L1 and L2 are projectively equivalent. The projective Galois group P G(L) acts faithfully on CL . P G(L) of invariants under this action can, by The field CL Luroth’s theorem, be written as C(f ) for some f ∈ k. We say that an operator St is a standard equation for P G(St) P G(St) equals C(z) for some z with z 0 = 1. if CSt Now assume that L has projective group P G and St is a standard equation with projective Galois group P G. If PG CLP G = C(f ), then z 7→ f maps CSt to CLP G . This, and the fact that CL determines L up to projective equivalence, are key ideas in Klein’s theorem below. Before stating this, we set a family of standard equations. All other standard equations can then be found using M¨ obius x 7→ (ax + b)/(cx + d) and projective equivalence L 7→ L ⊗ (∂ + v) transformations.

343

Lemma 4.3. L is a proper pullback of SG . Proof. The (semi)-invariant of SG corresponding to σ (in the above notations) has value 1 so it is mapped to 1 under any pullback transformation z 7→ f . L is a weak pullback by Klein’s theorem, so L ⊗ (∂ − v) will be a proper R pullback for some v; but its (semi)-invariant is e mv , which should be 1, so v must be 0 and hence L must be a proper pullback.

1. For m ∈ {4, 6, 12} check for a semi-invariant of degree m and call v its logarithmic derivative. 2. If yes, the projective group P G(L) is known. Let L = 1 L ⊗ (∂ + m v); this is a proper pullback of StsP G with invariant value 1. 3. Write L = ∂ 2 + a1 ∂ + a0 . Compute gL := 2a1 + g2

and the pullback mapping is f := 9s aL0

A direct examination (and relevant choices of standard equations) in each case will provide the pullback function f .

Remark 4.5. The change of variable z 7→ f changes gSt 1 to gSt (f ) · f 0 . Now, gSt = − 3(x+1) and the relation gL =

4.3 Formulas: the primitive case

0

− 3(ff+1) yields another method to find f . This approach will fail for imprimitive groups because then gL will be zero.

The projective Galois group is in {A4 , S4 , A5 } in this section. The standard equation in reference is StP G y = 0 1 where the differences of exponents are λ = at x = 0, 3 1 1 1 1 µ = at x = 1, and ν = for A4 , for S4 and for A5 2 3 4 5 at x = ∞.

4.4 Formulas: the imprimitive case In this case, the projective Galois group is P G(L) = Dn for n ∈ N. To simplify formulas, here, we choose the stan1 dard equation with exponent differences at +1 and −1 and 2 1 at infinity. It has a semi-invariant S2 = Y1 Y2 of degree 2 n and two semi-invariants Sn,a = Y1n +Y2n and Sn,b = Y1n −Y2n of degree n. The chosen standard equation

The differential Galois group of this equation has a semiinvariant S of degree m = 4 in the case of A4 , degree m = 6 in the case of S4 and m = 12 in the case of A5 with value σ0 (x) = x−m/3 (x−1)−m/4 . The new equation SG = StP G ⊗ 1 1 (∂ + + ) now has an invariant of degree m with 3x 4(x − 1) value 1. Rearranging it (via a M¨ obius transform, to obtain nicer formulas), we get the normalized standard equation: StsP G := ∂ 2 +

1 1 z ∂− 2 2 z2 − 1 4n z − 1 ` 1´ ` −1 1 ´ has exponents 0, , at +1 and −1 and at ∞; it 2 2n 2n has a semi-invariant of degree 2 and value 1. An operator L = ∂ 2 + a1 ∂ + a0 is a proper pullback of 2 1 1 a0 f0 SDn if a0 = − 2 2 and a1 = − 0 . The equation 4n f − 1 2 a0 Z √ Ly = 0 admits the solutions y1 , y2 = exp ± −a0 i.e. q p y1 = 2n f + f 2 − 1 and y2 = 1/y1 . The number n can thus be determined with (a subroutine √ of) the algorithm of elementary integration ([6]) applied to −a0 . For N ∈ N, the expressions y1N and y2N are permuted by the Galois group and are found to be a basis of solutions of LN := ∂ 2 + a1 p ∂ + N 2 a0 . In particular L2n has solutions f (rational) and f 2 − 1. Once n is known, we would like to compute f from a rational solution F of L2n . However, we would only know it up to a constant so we use its logarithmic derivative: StsDn = ∂ 2 −

1 (8 x + 3) s ∂+ 6 (x + 1) x (x + 1)2 x

with s = (6ν−1)(6ν+1) (recall that ν is 13 , 14 , 15 for cases 144 1 1 A4 , S4 , A5 respectively). It has exponents ( ν2 + 12 , − ν2 + 12 ) 1 1 at −1, (0, 2 ) at 0 and (0, 3 ) at ∞ where ν has the previous value in each case. Lemma 4.4. Let L = ∂ 2 + a1 ∂ + a0 be a normalized operator with P G(L) ∈ {A4 , S4 , A5 } (i.e it has an invariant of degree m with value 1 for the above values of m). Define a0 gL := 2a1 + a00 . Then L is a proper pullback of StsP G and the pullback mapping is 2 gL a0 Proof. Lemma 4.3 shows that L is a proper pullback z 7→ f of StsP G for some f . Computing this pullback and 0 00 f0 equating it to L gives the relations a1 = 2f + 6(f5f+1) − ff 0

f := 9s

02

a0

0

whence a00 = − f2f+1 − and a0 = (fsf +1)2 f formula follows by simple elimination.

f0 f

a00 , a0

+

2f 00 f0

Lemma 4.6. Let L = ∂ 2 + a1 ∂ + a0 be an irreducible operator with an invariant of degree 2 with value 1. Assume that P G(L) = Dn . Let F be a rational solution of 0 ∂ 2 + a1 ∂ + 4n2 a0 and let u := FF . Then the solutions of L q q p p are y1 = 2n f + f 2 − 1 and y2 = −2n f + f 2 − 1 with r 1 f= . u2

and the

In fact, the formula was not obtained that way: as we know that L is a proper pullback and that the solution f is unique (by Klein’s theorem and our normalization), we compute the expression of the image of StsP G under a generic pullback and perform differential elimination [13, 14] (there are other ways to find the formula but this way was the least amount of work). In the same way one can obtain formulas for other choices of standard equations but those turn out to be larger. So, given L = ∂ 2 + A1 ∂ + A0 with finite primitive projective group, the pullback function is found the following way: Pullback for A4 , S4 , A5 , semi-invariant version Input: L = ∂ 2 + A1 ∂ + A0 with P G(L) ∈ {A4 , S4 , A5 }. Output: Pullback function f .

1+

4n2 a0

Proof. By the above discussion, ∂ 2 + a1 ∂ + 4n2 a0 has a rational solution and F = cf for some constant f . Now we have f 02 = −4n2 a0 (f 2 − 1). Dividing out by f 2 yields the formula. Remark 4.7. Despite the square root in the expression of f , the function is rational. However, if the constant field of k is not algebraically closed, a quadratic extension of the constants may be needed in computing this square root (see also [2, 16] and references therein).

344

0

σ so its normal form has an invariant of degree 8 with ∂ + 8σ with value 1.

Pullback Formula for Dn , semi-invariant version Input: L = ∂ 2 + A1 ∂ + A0 with P G(L) = Dn (n unknown). Output: Pullback function f and the solutions. 1. Compute a semi-invariant of degree 2 and compute its logarithmic derivative v.

Lemma 5.1. Let L = ∂ 2 +a1 ∂ +a0 ∈ k[∂] be a normalized differential operator with projective Galois group P G(L) = S4 (L is normalized to have an invariant of degree 8 with a value 1). Define gL := 2a1 + a000 . Then L is a proper pullback of StiS4 and the pullback mapping is

2. If yes, let L = L ⊗ (∂ + 12 v); it is a proper pullback of SDn with invariant value 1. 3. Denote L = ∂ 2 + a1 ∂ + a0 . Determine a candidate for (a multiple of) n. (note: if there is more than one semi-invariant of degree 2, then n = 2)

f =−

Proof. That L is a proper pullback of StiS4 follows from lemma 4.3. Pick an unknown function f and form the change f 02 7 and of variable x = f in StiS4 . We obtain a0 = − 576 (f −1)2 f

4. Compute a rational solution F of Ln := ∂ 2 + a1 ∂ + 0 4n2 a0 and let u = FF . R v q p 5. Return the solutions y1 = e 2 2n f + f 2 − 1 and q r R v p 1 . y2 = e 2 −2n f + f 2 − 1 with f = u2 1+

5.

00

0

0

a1 = − ff 0 + 21 ff + 34 ff−1 Performing standard differential elimination on the latter, see [13, 14] and references therein, yields the above formula.

4n2 a0

With this formula, the algorithm in section 4.3 is straightforward to adapt (compute an invariant of degree 8 of L instead of a semi-invariant of degree 6).

PULLBACK FORMULAS, GENERAL K

5.1 Standard Equations

5.2.3 Tetrahedral case A4

The algorithm for general k uses only invariants (not semiinvariants). Hence, the relevant normal form for the standard and target equations will be the one for which an appropriate invariant (often one with the lowest degree) has value 1. For a projective group P G, a standard equation with semi-invariant of lowest degree with value 1 (resp. with invariants of lowest degree value 1) will be denoted StsP G (resp. StiP G ). A second idea that we will use is the fact that D2 ⊂ A4 ⊂ S4 . So, a standard equation for D2 (resp. A4 ) is a pullback of some StA4 (resp. StS4 ). Transformations between those equations can be found in [26] (or can be recomputed, as below). Like in the previous section, we will proceed in reverse order of the classification to give the pullback formulas

Let StsA4 denote the standard equation from section 4.3 with projective Galois group A4 . It has an invariant of degree 4 with value 1. As G(L) ⊂ SL2 (C), our L has only semi-invariants in degree 4, but it has an invariant in degree 6. So, proceeding as in section 5.2.2 (with lemma 3.3.1 in mind) yields a new standard operator StiA4 for A4 with an invariant of degree 6 having value 1: ` ´ 2 3 x2 − 1 5 1 StiA4 = ∂ 2 + ∂+ 3x (x2 − 1) 144 x2 (x2 − 1) 1 5 Its exponents are (0, 31 ) at 1 and -1, and (− 12 , 12 ) at 0 (the point ∞ is non-singular). We assume that the differential operator L has projective Galois group A4 and G(L) ⊂ SL2 (C). Thus L has an invariant of degree 6 with value σ. We normalize L by tensoring σ0 with ∂ + 6σ so the resulting normal form L has an invariant of degree 6 with value 1.

5.2 Primitive Cases 5.2.1 Icosaedral case A5 The group is determined by an invariant of degree 12, as in the C(x) case, so we use the formula from section 4.3.

Lemma 5.2. Let L = ∂ 2 +a1 ∂ +a0 ∈ k[∂] be a normalized differential operator with projective Galois group P G(L) = A4 , i.e L has an invariant of degree 6 with value 1. Then a L is a proper pullback of StiA4 . Let gL := 2a1 + a000 . Then the pullback mapping is s 64 a0 f =± 1+ 2 5 gL

5.2.2 Octaedral case S4 Let StsS4 denote the standard equation from section 4.3 with projective Galois group S4 . It has an invariant of degree 6 with value 1. However our target differential operator L has G(L) ⊂ SL2 . It only has a semi-invariant S6 of degree 6 and an invariant I8 of degree 8. Having computed the value of the (semi)-invariant of degree 8 of StsS4 , we tensor StsS4 1 with ∂ − 24(x+1) (and, via a M¨ obius transform, change the singularities to 0, 1 and ∞ to simplify the formula of lemma 5.1) to obtain the standard operator StiS4 = ∂ 2 +

2 7 gL 144 a0

Proof. One can use the same differential elimination argument as for lemma 5.1. Note that Klein’s theorem shows a0 that 1 + 64 must be the square of an element of k. 5 g2 Remark 5.3. The appearance of a square-root is no surprise because the standard equation for A4 has a symmetry (exchange 1 and −1) so there are two solutions to the pullback problem (see [16, 2] and references therein), each ”attached” to one of the two semi-invariants of degree 4. In the algorithm in section 4.3 we need to choose one of the two semi-invariants, hence the (apparent) uniqueness of the pullback formula there.

7 1 (5 x − 2) 1 ∂− 4 (x − 1) x 576 (x − 1)2 x

1 7 Its exponents are (0, 21 ) at 0, (− 24 , 4 ) at 1, and (0, 41 ) at ∞; it has an invariant of degree 8 with value 1. We assume that the differential operator L has projective Galois group S4 and G(L) ⊂ SL2 (C). Thus L has an invariant of degree 8 with value σ. We normalize L by tensoring

345

An alternative approach to find and prove the formula in the lemma 5.2 is the following. As L is a pullback of StiA4 , it is also a pullback of StiS4 because A4 ⊂ S4 . Now apply the S4 formula to the A4 standard equation, solve, and one obtains lemma 5.2. The same idea can also be used for D2 .

because the solution is not unique, factoring is inevitable in this process.

6. CONCLUSION Theorem 6.1. The algorithm of section 2 is correct.

5.3 Dihedral Groups Dn , n > 2

Proof. The steps compute the projective Galois group by [27] or corollary 3.4. Step 2a is sections 5.3 and 4.4; Step 2b is sections 5.4 and 4.4; Step 3a is section 5.2.3; Step 3b is section 5.2.2; and Step 3c is sections 5.2.1 and 4.3.

The case P G(L) ⊂ D∞ is characterized by the existence of an invariant I4 of degree 4. We assume that P G(L) 6= D2 so the space of invariants of degree 4 has dimension 1 (and I4 is the square of a semi-invariant of degree 2). Tensoring I0 L with ∂ + 4I44 , we obtain a normalized operator L which has an invariant of degree 2 with value 1. So we can use the algorithm from section 4.4 (start at step 3) and obtain the pullback function.

The algorithm presented here is very easy to implement for an admissible differential field. Further improvements and speedups can be provided in the case when k = C(x). The algorithm is implemented in Maple 9.5. A draft implementation (and a maple worksheet to check most formulas of this paper) can be consulted at http://www.unilim.fr/ pages_perso/jacques-arthur.weil/issac05/ Denote H(x) = 2 F1 ([−1/60, 11/60], [2/3], 1/(x+1)) which is one of the solutions of StsA5 . The Kovacic algorithm produces the minimal polynomial mK of y 0 /y for some solution y of StsA5 , whereas Fakler’s algorithm [15] produces the minimal polynomial mF of a solution y of StsA5 . Note that mF is preferable over mK . Now consider the following example: L = 48x(x − 1)(75x − 139)∂ 2 +(2520x2 −47712x/5+3336)∂−19x+36001/75 which has projective Galois group A5 . The pullback function f is rather large (the degree is 31). By default our algorithm uses hypergeometric functions to denote the answer. In essence this means that x in the expression H(x) above is being replaced by f . To get a solution of L in the same format as would have been produced by Kovacic’s resp. Fakler’s algorithm, one essentially has to substitute f for x in the solution that these algorithms provided for StsA5 . However, this substitution will lead to a large expression because x occurs many times in the expression mK resp. mF and all those occurrences are replaced by f . We compared the kovacicsols command in Maple 9.5 (which follows the usual Kovacic algorithm) with the algorithm presented here. The size of the output (measured with the command length) in Maple 9.5 was 236789 whereas for the new algorithm the size is only 1360. Note that this new algorithm is scheduled to appear in the kovacicsols command in the next version of Maple.

Remark 5.4. The difficulty in this subsection lies in deciding whether P G(L) is some Dn or D∞ . Computing n is achieved by computing the torsion of some divisor from the integration algorithm, which can be achieved under our assumptions on k, see [6] or [2, 3].

5.4 Quaternion Group D2 There is a problem to choose a relevant normalization because the space of invariants of degree 4 is two-dimensional and, in our normalizations, we would need to choose one among those that is a square of a semi-invariant of degree 2 in order to use the formulas from section 4.4. Although this is possible (e.g [27]), we propose a few simpler approaches (the reader is welcome to select whichever one she likes best). As G(L) ⊂ SL2 (C), the operator has a unique (up to constants) invariant of degree 6 with value σ (the product of the σ0 three semi-invariants of degree 2). Tensoring L with ∂ + 6σ , we obtain a normalized operator L whose invariant of degree 6 has value 1. Approach 1: We have D2 ⊂ A4 . Moreover, L has an invariant of degree 6 with value 1. So L is a proper pullback of StiA4 from section 5.2.3 and the pullback is computed directly with the algorithm from section 5.2.3. The good point is that no work is needed; the bad point is that the solutions will be given in terms of the solutions of StsA4 which is not very good if, for example, we want the minimal polynomial or an expression by radicals. Approach 2: In approach 1, we have computed a pullback e i (F ) with H e i solutions F from StiA4 so solutions of L are H of StiA4 . Now we precompute the pullback from D2i to Ai4 . First send singularities to 0, 1, ∞ by a M¨ obius transform; next, tensor by a first order operator so that the exponents are (0, 1/3) at 0 and ∞. Changing x to x3 , the preimages of 0 and ∞ will have exponents (0, 1) so they will be ordinary, while the preimages of 1 (i.e 1, j, j 2 ) will have exponent differences 1/2: the resulting equation is thus a standard D2 equation. Sending the singularities to −1, 1, ∞ and tensoring by a first order operator finally√sends us to the standard (x2 −1) e i ( 3 −3 operator StD2 . We find that H ) = Hi (x) with x3 −9 x Hi solutions of StiD2 . So the solutions of L will be Hi (f ) where f is a root of the third degree equation ´ ` ´ ` √ (5.9) 3 −3(f 2 − 1) − F f 3 − 9 f = 0

7. REFERENCES [1] Baldassarri, F; Dwork, B: Differential Equations with Algebraic Solutions, American Journal of Mathematics 101, 1979, p42-76. [2] Berkenbosch, M: Algorithms and Moduli Spaces for Differential Equations PhD dissertation, Rijksuniversiteit Groningen, 2004. [3] Berkenbosch, M: Pullbacks for Differential Equations, submitted [4] Berkenbosch, M; van Hoeij, M; Weil, J-A: Recent Algorithms for Solving Second-Order Differential Equations summary by Michele Loday-Richaud. INRIA research report ]5003. Algorithms seminar, 2001-2002 http: //algo.inria.fr/seminars/sem01-02/weil.pdf [5] Beukers, F; van der Waall, A: Lam´e equations with algebraic solutions. J. Differential Equations 197 (2004), no. 1, 1-25

By Klein’s theorem, the latter has three roots f in k which can be computed, e.g by factoring the above. We note that,

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[15] Fakler, W: On second order homogeneous linear differential equations with Liouvillian solutions. Computer algebra (Saint-Louis, 1996). Theoret. Comput. Sci. 187 (1997), no. 1-2, 27–48. [16] Hendriks, P.A.; van der Put, M: Galois action on solutions of a differential equation. J. Symbolic Comput. 19 (1995), no. 6, 559–576. [17] van Hoeij, Mark: The Minimum Polynomial of an Algebraic Solution of Abel’s problem. Preprint FSU00-02. [18] Klein, F: Ueber lineare Differentialgleichungen, Math. Ann. 12 (1877) 167–179. [19] Kovacic, J.J: An Algorithm for Solving Second Order Linear Differential Equations, Journal of Symbolic Computation 2, 1986, p3-43. [20] Lit¸canu, R: Lam´e operators with finite monodromy—a combinatorial approach. J. Differential Equations 207 (2004), no. 1, 93–11 [21] Maier, R.S: Algebraic solutions of the Lam equation, revisited. J. Differential Equations 198 (2004), no. 1, 16–34 [22] van der Put, M: Galois Theory of Differential Equations, Algebraic Groups and Lie Algebras, Journal of symbolic computation 28, 1999, p441-472. [23] van der Put, M: Singer, M.F: Galois Theory of linear Differential Equations, Grundlehren der mathematischen Wissenschaften, Springer 2003. [24] Singer, M.F: Liouvillian Solutions of Linear Differential Equations with Liouvillian Coefficients Journal of Symbolic Computation, 1991. [25] Singer, M.F ; Ulmer, F: Liouvillian and algebraic solutions of second and third order linear differential equations. J. Symbolic Comput. 16 (1993), no. 1, 37–73. [26] Vidunas, R: Algebraic transformations of Gauss hypergeometric functions, preprint (2004) http://arxiv.org/abs/math.CA/0408269 [27] Ulmer, F ; Weil, J-A Note on Kovacic’s algorithm. J. Symbolic Comput. 22 (1996), no. 2, 179–200.

[6] Bronstein, M: Integration of elementary functions. J. Symbolic Comput. 9 (1990), no. 2, 117–173. [7] Bronstein, M ; Lafaille, S: Solutions of linear ordinary differential equations in terms of special functions. Proceedings of the 2002 International Symposium on Symbolic and Algebraic Computation, 23–28 , ACM, New York, 2002. [8] Bronstein, M; Fredet, A: Solving Rlinear ordinary differential equations over C(x, e f (x)dx ). Proceedings of the 1999 International Symposium on Symbolic and Algebraic Computation (Vancouver, BC), 173–179 , ACM, New York, 1999. [9] Bronstein, M; Mulders, T; Weil, J-A: On symmetric powers of differential operators. Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation, 156–163, ACM, New York, 1997. [10] Burger R, van Hoeij M, and Labahn G: Closed Form Solutions of Linear Odes having Elliptic Function Coefficients. International Symposium on Symbolic and Algebraic Computation’04 Proceedings, 58-64, ACM, New York, 2004 [11] Chalkley, R: Relative invariants for homogeneous linear differential equations, J. Differential Equations 80, 107-153, 1989. [12] Chan, L; Cheb-Terrab, E.S: Non-liouvillian solutions for second order Linear ODEs. International Symposium on Symbolic and Algebraic Computation’04 Proceedings, 80-86, ACM, New York, 2004 [13] Hubert, E: Notes on triangular sets and triangulation-decomposition algorithms II: Differential Systems, in Symbolic and Numerical Scientific Computing 2630 p. 40-87 (Ed: Winkler, F. and Langer, U.), Springer Verlag Heidelberg, 2003 [14] Boulier, F; Hubert, E: diffalg: description, help pages and examples of use Symbolic Computation Group, University of Waterloo, Ontario, Canada, 1998. Now at http: //www.inria.fr/cafe/Evelyne.Hubert/diffalg

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Deterministic Equation Solving over Finite Fields Christiaan van de Woestijne Mathematisch Instituut, Universiteit Leiden P.O. Box 9512 2300 RA Leiden The Netherlands

[email protected] ABSTRACT

that taking square or higher roots cannot be done in deterministic polynomial time if the GRH is not assumed. In the course of my Ph.D. project with Hendrik W. Lenstra, Jr., I have developed an algorithm for solving general diagonal homogeneous equations in many variables over finite fields, that is both deterministic and runs in polynomial time. More specifically, the equation it solves is the following, for a given finite field F:

Deterministic algorithms are presented for the efficient solution of diagonal homogeneous equations in many variables over finite fields. As auxiliary algorithms, it is shown how to compute a field generator that is an nth power, and how to write elements as sums of nth powers, for a given integer n. All these algorithms take polynomial time in n and in the logarithm of the field size, and are practical as stated.

n X

Categories and Subject Descriptors

ai x n i = b

(1)

i=1

F.2.1 [Numerical Algorithms and Problems]: Computations in finite fields; G.4 [Mathematical Software]: Algorithm design and analysis; I.1.2 [Algorithms]: Algebraic algorithms; Analysis of algorithms

with ai and b all nonzero elements of F. The algorithm works for any characteristic and any n, although when n2 > |F| no solution may exist. Taking b = −1, we find an algorithm that computes a sequence of nth powers summing to zero, with at most n + 1 terms. The formulation of the algorithm is nontrivial and involves several steps that are interesting in their own right, all of which seem to be new. These include an extension of Shanks’s algorithm and a method for computing an element α of F whose nth power generates F over its prime field; whereas the solution of the general problem makes essential use of the ability to write elements of F as sums of nth powers. This last reduction is based on a group-theoretic proof of part of the Chevalley-Warning Theorem by Dem0 yanov ([5], also given in [7]). ˜ 3 (log |F|)2 ) operations in F to finThe algorithm uses O(n ish, and its practicality will be evident from the description. In Section 8 I compare its asymptotic complexity with that of a straightforward probabilistic algorithm. In the special subcase of finding a sequence of nth powers in F that sum to zero, the complexity drops to a mere ˜ 2 log |F|) operations in F, and I believe that this is optiO(n mal in regard to the dependence on |F|. Applications abound especially for the case of quadratic forms, since if char F is not 2, every quadratic form can be brought into diagonal form. Thus it follows that one can compute a rational point on a conic or higher-dimensional quadric over F by a fast deterministic algorithm. In another direction, one can use my algorithm to show that the problem of constructing an isomorphism between regular quadratic spaces (of dimension 2 or higher) over F has a deterministic reduction to taking a square root of the quotient of the discriminants. Full proofs of all results in this note will appear in my Ph.D. thesis “Deterministic equation solving over finite fields” (Universiteit Leiden, 2005). A preliminary version is available for download on my homepage.

General Terms Algorithms, Design, Performance

Keywords Deterministic algorithms, Equation solving, Finite fields

1. INTRODUCTION Currently known algorithms for solving equations over finite fields include: • brute force search • algorithms for factoring polynomials (see [2, 4, 16]) • Shanks’ algorithm for taking square (and higher) roots (see [12], [1], [4]) • methods for multivariate equations based on the above (see Section 8) • Schoof’s algorithm for taking square roots in prime fields (see [10]) However, all of these are either probabilistic (barring a proof of the Generalised Riemann Hypothesis for some) or take more than polynomial time. It should be especially noted

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348

2. PRELIMINARIES

independently to Kneser for the case n = 2 (see [8, Theorem XI.4.4]). I was able to prove the second statement by an extension of the same method. This statement was first proved by Schwarz [11] on the stronger assumption that d = (n, q − 1) < p, where p is the characteristic of F. The proof depends on the following Proposition.

I note that all complexity bounds given in this paper are “soft” bounds; this means that logarithmic factors have been ignored. See [16, Section 25.7] for a precise definition. I assume that finite fields are given by means of an irreducible polynomial of the right degree over the prime field. I will denote by F a finite field with characteristic p and having q = pe elements. For algorithms for the basic field operations, see [9]. I use the following facts about finite fields. First, the multiplicative group F∗ is cyclic of order q − 1. Hence for a positive integer n the quotient of F∗ by its subgroup of nth powers has d elements, where d equals the g.c.d. of n and the group size q − 1. It follows that every dth power is also an nth power, and conversely, and we might just as well replace n with d. If x and y are integers such that xn + y(q − 1) = d, then for every a ∈ F∗ , we have ad = (ax )n , hence conversion back and forth is easy. Therefore I will assume throughout the paper, when dealing with nth powers, that n divides q − 1. Next, I give some facts about the sums of nth powers in F.

Proposition 2.3 Let F be a finite field of q elements, and let f = a0 X0n + . . . + av Xvn be a diagonal form of degree n in v + 1 variables over F. Assume that ai 6= 0 for i = 0, . . . , v. Then: (i) If f is not isotropic, it represents at least v + 1 distinct classes of F∗ modulo nth powers. (ii) If every element of F is a sum of nth powers and f is not universal, then f represents at least v + 1 distinct classes of F∗ modulo nth powers. I do not prove the Proposition here, for space considerations and also because my algorithms for solving (1) provide algorithmic proofs.

3.

OUTLINE OF THE ALGORITHM

My algorithm for solving (1) is largely divided into three steps:

Proposition 2.1 Let F be a finite field of q elements, let n be a positive integer and let K be the subset of sums of nth powers of elements of F. Then: (i) K is a subfield of F. (ii) If K is a proper subfield of F, then we have n2 > q. (iii) K is equal to F if, and only if, F can be generated as a field by adjunction of an nth power to its prime field. (iv) Every nonzero element of K is a sum of at most n nth powers.

1. Given a finite field F and a positive integer n, compute α ∈ F such that αn generates F over its prime field. Thus, writing e for the degree of F over Fp , every element b of F can be written as b=

e−1 X

bi (αi )n ,

(2)

i=0

with bi ∈ Fp . I refer to [7, Section 4.2] for proofs of these statements, which are due in essence to Tornheim [15]. Finally, I comment on the solvability of (1); I recall a few definitions. A form is called isotropic if it has a nontrivial zero, anisotropic if it has only the trivial zero, and universal if it represents every nonzero element. A form is said to represent zero if and only if it has a nontrivial zero. Recall that if a form f represents a nonzero element b, then it also represents all elements of the form bxn for some x ∈ F. In this case also the coset of b modulo the nth powers is said to be represented by the form f .

2. Write elements of F as a sum of nth powers with at most n terms. Thus, interpreting the bi just given as integers between 0 and p − 1, inclusive, we have a sum of nth powers with ep terms; reduce this to at most n terms by a logarithmic decay of the number of terms. This solves (1) in the case where all coefficients are 1. 3. Writing a0 = −b and applying the second step to −1, −a0 /a1 , . . . , −a0 /an , we find a system of equations of the form 8 n n −a0 (y0,1 + . . . + y0,h )=0 > 0 > > n n > > −a1 (y1,1 + . . . + y1,h ) = a 0 xn 1,0 > 1 > < n n n −a2 (y2,1 + . . . + y2,h2 ) = a0 xn 2,0 + a1 x2,1 > > .. .. > > > . . > > : n n n ) = a 0 xn −an (yn,1 + . . . + yn,h n,0 + . . . + an−1 xn,n−1 n (3) Here initially, the right hand sides have only xj,0 nonzero, and we have hj ≤ n for 1 ≤ j ≤ n, and h0 ≤ n + 1. We now perform a reduction step that decreases the value of one of the hj by at least one, until one of them becomes 1, whereas in the meantime the xj,i in the right hand sides are filled in. The desired solution follows.

Theorem 2.2 Let F be a finite field of q elements, and let n be a positive integer. Then: (i) Every diagonal form of degree n over F in n + 1 variables is isotropic. (ii) Assume every element in F is a sum of nth powers in F. Then every diagonal form of degree n over F in n variables with nonzero coefficients is universal. The first statement of the Theorem is part of the ChevalleyWarning theorem. Note that the Theorem is not at all sharp; a straightforward application of Weil’s bounds on the number of solutions of diagonal equations over finite fields gives much stronger results (see [17], [13]). For example, if q > n4 , then every equation of the form axn + by n = c is solvable, if abc 6= 0. The interest, then, of the Theorem lies in its possessing a constructive proof. For the first statement, this proof is due to Dem0 yanov (see [5] and [7, Th´eor`eme 4.1]), and

4.

FINDING N TH POWER GENERATORS

I now give the first step of the algorithm in more detail. Let F be a finite field and n a positive integer. Proposition 2.1 tells us that if every element of F is an nth power, then

349

4.2

there exists some α in F such that αn generates F over its prime field. In this Section I show that it is possible to compute such an element α efficiently (and deterministically). I use the following auxiliary result, which could be considered as a multiplicative version of the primitive element theorem for separable field extensions.

Composing the right fields

I solve the problem of the computing the desired generator in three stages. A prime field Fp is generated over itself by 1n ; this gives the base step. Now suppose the prime field is small, by which I mean p ≤ n. Then we simply enumerate n2 + 1 elements of F and compute their nth powers. Among these powers, there are at least n + 1 distinct elements; hence if we adjoin all of them to Fp , we get a field with more than n elements. The compositum algorithm given above now serves to merge these n + 1 powers into just one nth power that generates this subfield over Fp . A problem arises if F does not have n2 + 1 elements. This is also the only case in which a solution to the current problem might not exist (cf. Proposition 2.1 above). However, because n is so large, we can simply try every element of F and still use polynomial time in n and log q. Next, suppose that there exists some subfield K of F with |K| > n, for which we know an nth power generator γ n . Let β be the given generator for F over Fp ; we now have the following result.

Theorem 4.1 There exists a deterministic algorithm that, given finite fields K and L, with K ⊆ L, and nonzero elements α1 , . . . , αt of L, computes integers x1 , . . . , xt such that α1x1 · · · αtxt generates the field K(α1 , . . . , αt ) over K, ˜ and uses O((log q) + te) operations in L, where e = [L : K]. The main result of this Section is as follows. Theorem 4.2 There exists a deterministic algorithm which, given a finite field F of q elements and a positive integer n dividing q −1, computes β ∈ F such that β n generates F over its prime field, or correctly asserts that no such β exists, us˜ 2 + n log q) operations in F. ing O(n The following sections outline the proofs of these results.

Lemma 4.4 Suppose |K| > n, and let c0 , . . . , cn be distinct elements of K. Then we have (β + ci )n 6∈ K for at least one i with 0 ≤ i ≤ n.

4.1 The compositum algorithm Let K be a finite field with q elements, let t be a positive integer, and for i = 1, . . . , t, let αi be algebraic over K of degree ei . I assume that all αi are contained in some finite overfield L of K, and hence the composite field M = K(α1 , . . . , αt ) is well defined. Its degree is equal to lcm(e1 , . . . , et ); I denote this degree by g. The notation M for the composite field, and g for its degree, is kept throughout this Section and the next. In the course of the algorithms of this Section, we will have many occasions to compute the degree of an element α of a given finite field L over a subfield K. The fastest way that I know for doing this is calling Algorithm 14.26 in [16], which the authors call the iterated Frobenius algorithm. ˜ This method computes degrees in O(e) operations in L. The compositum algorithm is based on the following key observation (cf. [3, Lemma 6.2]), in which φ denotes Euler’s totient function. I do not prove this result here.

Proof. Assume the contrary. Then for all i, we have (β + ci )n|K| = (β +ci )n , so (β +ci )|K|−1 is an nth root of unity in L, of which there are at most n. By the pigeonhole principle, there exist i and j with 0 ≤ i < j ≤ n such that (β + ci )|K|−1 = (β + cj )|K|−1 , which implies that β + ci ∈ K. β + cj But this is a contradiction, because β is not in K and ci 6= cj .

Corollary 4.5 With the same assumptions as in the Lemma, the elements (β + ci )n (for i = 0, . . . , n) together generate F over K.

Lemma 4.3 Let L/K be a finite cyclic extension of fields and let b1 , . . . , bn be a basis for L as a K-vector space. Then at least φ([L : K]) of the basis elements generate L over K as a field. There exists a basis for L over K with exactly φ([L : K]) field generators.

Proof. Retaining the same elements ci , apply the Lemma successively to all maximal subfields of F containing K. It follows that no such field contains all the elements (β + ci )n . Therefore these elements generate the whole field F over K.

φ(n) log log n exists and is posn itive (by Theorem 328 in [6]), it is to be expected that at least one in c log log e elements of a basis for L over K is a field generator, where c is a positive absolute constant.

With a second call to the compositum algorithm, we “compose” γ and the elements β + ci (for i = 0, . . . , n) to find a single element α whose nth power generates F over Fp . This solves our problem.

Remark. Actually, as lim inf n→∞

Algorithm. It is now easy to formulate an algorithm that

5.

satisfies the conditions of Theorem 4.1 above: given α1 , . . . , αt , compute a basis for K(α1 , . . . , αt ) over K; then test all the basis elements for field generators. This testing can be done very efficiently as we already know the degrees of the αi . We must find at least one generator by virtue of Lemma 4.3.

Theorem 5.1 There exists a deterministic algorithm which, given a finite field F with q elements, a positive integer n,

350

SELECTIVE ROOT EXTRACTION

In this Section I prove the following statement, which is needed for the second step of my main algorithm.

and n + 1 nonzero elements a0 , . . . , an of F, determines integers i and j and an element β ∈ F such that 0 ≤ i < j ≤ n and

The complexity would improve if we could replace the Tonelli-Shanks algorithm by a faster root taking algorithm. Now there do exist essentially quadratic probabilistic algorithms for root taking, which are mostly guises of Berlekamp’s polynomial factorisation algorithm (see [2] or [16, Section 14.5], for example); but these do not seem to suit the present deterministic application.

ai /aj = β n , ˜ using O(n(log q) + (log q)2 ) operations in F. The Theorem says actually that, given n + 1 elements in F, we can compute an nth root of the quotient of two among them; however, without close analysis of the elements ai one cannot predict which two. Because of this selection feature, the method was called Selective Root Extraction. My algorithm is an extension of the Tonelli-Shanks algorithm for taking roots in cyclic groups, which we first discuss.

6.

SUMS OF LIKE POWERS

I now detail the second step of the main algorithm. We consider a finite field F. Given a positive integer n, can we write any given element of F as a sum of nth powers of elements of F? And if so, how many such powers are needed? This problem, known as Waring’s problem for finite fields by analogy to its classical formulation with respect to the integers, has known active research in the 20th century. Some elementary results are recalled in Proposition 2.1 (especially (iv)), but these are not optimal in the cases where n is small with respect to |F|. A survey of recent results can be found in [18]. What concerns us here is the question of actually computing representations of given elements as sums of nth powers. My result is the following.

5.1 The Tonelli-Shanks algorithm The essence of the Tonelli-Shanks algorithm was already given by Tonelli in 1891 [14] for the purpose of extracting square roots modulo primes of the form 4k + 1. It has subsequently been rediscovered by Shanks [12] and by Adleman, Manders, and Miller [1] in the 1970’s, all of whom generalise the method to finding roots of arbitrary exponent, while Shanks also notes that the method can be applied to arbitrary cyclic groups. The discussion in [4, Section 1.5.1] is limited to the square root case. A notable property of the Tonelli-Shanks algorithm is that, given an element whose order is large enough, it proceeds deterministically to compute roots of a. Generally, to use it we must first guess such an element (for example, a nonsquare if we want square roots). In the present application, we are able to determine elements of sufficiently high order without having to search for them. A complexity analysis shows that, given an element of high enough order, the Tonelli-Shanks algorithm uses O (n+ (log q)2 log n) operations in F to compute an nth root.

Theorem 6.1 There exists a deterministic algorithm which, given a finite field F with q elements, a positive integer n and a nonzero element b of F, determines elements x1 , . . . , xn of F such that n X b= (4) xn i , i=1

or correctly asserts that b is not a sum of nth powers in F, ˜ 2 (log q) + n(log q)2 ) operations in F. using O(n

As far as I know, this is the first efficient deterministic algorithm to write finite field elements as sums of powers. The result below treats the special case of writing 0 as a sum of powers, in which case I have obtained an algorithm that uses only a linear amount of field operations.

5.2 The Selective Root Algorithm The Selective Root Algorithm is, in fact, the closest I can come to a deterministic root taking algorithm in finite fields, given the current state of knowledge. The main idea is simple. Let F be a finite field, with q elements, let n be a positive integer dividing q − 1, and let a0 , a1 , . . . , an be n + 1 nonzero elements of F. Denote by G the subgroup of F∗ generated by the ai . The group G is cyclic; therefore, the index [G : Gn ] is at most n. It follows that there exist i and j such that ai Gn = aj Gn ; in other words, for this particular i and j, there exists β ∈ G with β n = ai /aj . Now ai and aj are such that their quotient has not too large order in G; on the other hand, the order of β is rather large. Thus the real task of the algorithm is to look both for elements of large and of small order in the group G. The actual algorithm that comes out of this first factors n into primes and works with one prime at a time. This has the additional advantage, besides being simpler to understand, that the number of ai s to be examined is at least halved with every processed prime. The number of field operations performed by the algorithm is quadratic in log q, and thus the algorithm has essentially cubic bit complexity. I have been unable to obtain an essentially quadratic bound, except in situations where the orders of the arguments ai are bounded (see Section 6.2 for an example).

Theorem 6.2 There exists a deterministic algorithm which, given a finite field F with q elements and a positive integer n, determines elements x0 , . . . , xn of F such that n X

xn i = 0,

(5)

i=0

˜ 2 log q) operations in F. using O(n As already shown in the outline of the main algorithm, representing a nonzero element a of F as a sum of nth powers is easy, given an nth power generator of F over its prime field. (If such a generator does not exist, then by Proposition 2.1 we must have n2 > q, and we can simply enumerate all sums of nth powers in F to see if a is among them.) We must, however, reduce the number of terms in the sum to at most n, as claimed in the Theorem.

6.1

The reduction algorithm

Thus, suppose that we have a representation as follows: a=

M X i=1

351

yin

(6)

for some positive integer M and some nonzero elements yi of F. If this representation is obtained from the form (2) by simply writing all the coefficients as sums of ones, we have M ≤ ep, where e = [F : Fp ]. Now divide the sequence M terms yi into n + 1 subsequences, each having roughly the same number of components. Next, form the n+1 sums S0 up to Sn , where Sj is the sum of yin for all yi contained in the first j + 1 subsequences. If any of the Sj is zero, we immediately discard all the corresponding terms from the sum (6). If not, we apply Selective Root Extraction (see above) to the sequence (S0 , . . . , Sn ) and obtain

we can use r = 2; any element of even order is enough to apply our side exit; but we know such an element already, namely, −1. We thus write 1n + (−1)n = 0. Now we can modify my algorithm to take advantage of this situation: before applying Selective Root Extraction, we examine the element we want to take a root of, and if its order contains enough factors of some prime below n + 1, we solve our problem by means of roots of unity. On the other hand, if this never happens (e.g., in the case when q −1 itself contains no high powers) we only apply the Tonelli-Shanks algorithm to elements of bounded order. This leads to a ˜ 2 log q) operations in F in all cases. complexity of O(n Unfortunately, it seems very difficult to extend this technique to the case where we have distinct coefficients.

Sl = β n Sk for some β ∈ F and some integers k and l with 0 ≤ k < l ≤ n. Therefore, if we multiply all terms in the first k + 1 subsequences by β and discard all terms in the (k + 2)th up to (l + 1)th sequences, the value of the sum (6) does not change. In both cases, the number of terms M will drop by a factor of about n+1 , or more if more than one subsequence n can be discarded. The trick is applicable as long as we have M ≥ n + 1; hence we will end up having M ≤ n, as desired, and the number of iterations will be logarithmic in q.

7.

THE TRAPEZIUM METHOD

I now give details for the third and final step of the main algorithm. The following theorem is the main result of this note. Theorem 7.1 There exists a deterministic algorithm which, given a finite field F with q elements, a positive integer n dividing q − 1, and nonzero elements a1 , . . . , an and b of F, computes elements x1 , . . . , xn of F, such that

I point out that attention is required for the internal representation of a sum of the form (6). Namely, the initial number of terms M is exponential in log p; hence at every iteration we would need to perform an exponential amount of exponentiations and summations. But taking advantage of the way in which this sum arises, we see that we can do much better: internally, we still remember the coefficients bi from (2) and keep track of how they are split up by the grouping into subsequences. However, to get the complexity bound given in Theorem 6.1, one first brings the size of the coefficients bi down to at most n at the cost of increasing their number somewhat, with the result that we need O (log log q) iterations instead of O (log q). For example, one can try to write the bi as sums of nth powers in Z by repeatedly subtracting the greatest possible nth power. This very quickly brings the bi down to about nn , which is sufficient.

n X

ai x n i = b,

(7)

i=1

˜ 3 (log q)+ or decides that such elements do not exist, using O(n 2 2 n (log q) ) operations in F to complete. The proof of this result was already embarked upon in the outline of the algorithm, Section 3. As in the previous two steps, we might encounter insolubility of our problem only if n2 > q, by Proposition 2.1. And again, this implies that we have enough enumerate P time to n all elements represented by the form n i=1 ai xi over F and see if b is among them. Therefore, we can assume henceforth that a solution exists. Write a0 = −b, and assume we built the system of equations (3) (repeated here for the convenience of the reader): 8 n n −a0 (y0,1 + . . . + y0,h )=0 > 0 > > n n > > −a (y + . . . + y ) = a 0 xn 1 1,1 1,h 1,0 > 1 > < n n n −a2 (y2,1 + . . . + y2,h2 ) = a0 xn 2,0 + a1 x2,1 > > .. .. > > > . . > > : n n n ) = a 0 xn −an (yn,1 + . . . + yn,h n,0 + . . . + an−1 xn,n−1 n

6.2 A special case I now treat the special case of writing 0 as a sum of nth powers. For this it is enough to represent −1 in this way and bringing it to the other side of the equation; however, by a more involved approach we can improve the complexity of the algorithm significantly. The reason that the algorithm of the last section has essentially cubic bit complexity is the use of the Tonelli-Shanks algorithm. A more precise consideration of this algorithm shows that the complexity is only cubic if the order of the element whose root we compute is divisible by a high power of some prime that also enters into n. However, if r is a prime, dividing n, that divides the order of a to a higher order than it divides n, then we can use this to construct an element η whose nth power is an rth root of unity. As is well known, if ζ is an rth root of unity, then 1 + ζ + . . . + ζ r−1 = 0; thus, if r ≤ n + 1, we solve our problem by summing powers of η n . In fact, we need not limit ourselves to primes dividing n; it is enough if r divides q − 1. As an example, if n is odd,

(The name “trapezium method” derives from the shape of this system of equations.) The first equation is formed by writing −1 as a sum of nth powers, and then bringing −1 to the left. The others result from writing −a0 /ai as a sum of nth powers, for i = 1, . . . , n. Hence initially we have h0 ≤ n = 1 and hj ≤ n for j = 1, . . . , n by Theorem 6.1, whereas all xj,i are zero except the xj,0 for j = 1, . . . , n, which are all nonzero and remain so throughout the algorithm (this is very important). The goal is to lower the hj until one of them becomes 1; then our problem is solved by moving the term a0 xn j,0 to the left and dividing through by xn j,0 .

352

9.

n We try to lower the hj by bringing the last term aj yj,h j to the other side. We get the sequence ` n n a0 y0,h , a0 x n 1,0 + a1 y1,h1 , . . . , 0 ´ n n n . a0 xn,0 + . . . + an−1 xn,n−1 + an yn,h n

[1] Leonard Adleman, Kenneth Manders, and Gary Miller. On taking roots in finite fields. In 18th Annual Symposium on Foundations of Computer Science (Providence, R.I., 1977), pages 175–178. IEEE Comput. Sci., Long Beach, Calif., 1977. [2] Eric Bach. A note on square roots in finite fields. IEEE Trans. Inform. Theory, 36(6):1494–1498, 1990. [3] Eric Bach, Joachim von zur Gathen, and Hendrik W. Lenstra, Jr. Factoring polynomials over special finite fields. Finite Fields Appl., 7(1):5–28, 2001. Dedicated to Professor Chao Ko on the occasion of his 90th birthday. [4] Henri Cohen. A course in computational algebraic number theory, volume 138 of Graduate Texts in Mathematics. Springer-Verlag, Berlin, 1993. [5] V. B. Dem0 yanov. On representation of a zero of forms P n of the form m i=1 ai xi . Dokl. Akad. Nauk SSSR (N.S.), 105:203–205, 1955. [6] G.H. Hardy and E.M. Wright. An introduction to the theory of numbers. Oxford, at the Clarendon Press, 1965. Fourth edition, 3rd corrected printing. ´ [7] Jean-Ren´e Joly. Equations et vari´et´es alg´ebriques sur un corps fini. Enseignement Math. (2), 19:1–117, 1973. [8] T. Y. Lam. The algebraic theory of quadratic forms. W. A. Benjamin, Inc., Reading, Mass., 1973. Mathematics Lecture Note Series. [9] H. W. Lenstra, Jr. Finding isomorphisms between finite fields. Math. Comp., 56(193):329–347, 1991. [10] Ren´e Schoof. Elliptic curves over finite fields and the computation of square roots mod p. Math. Comp., 44(170):483–494, 1985. ˇ [11] Stefan Schwarz. On universal forms in finite fields. ˇ Casopis Pˇest. Mat. Fys., 75:45–50, 1950. [12] Daniel Shanks. Five number-theoretic algorithms. In Proceedings of the Second Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1972), pages 51–70. Congressus Numerantium, No. VII, Winnipeg, Man., 1973. Utilitas Math. [13] Charles Small. Diagonal equations over large finite fields. Canad. J. Math., 36(2):249–262, 1984. [14] Alberto Tonelli. Bemerkung u ¨ber die Aufl¨ osung quadratischer Congruenzen. Nachr. G¨ ottingen, (10):344–346, 1891. Reported in Dickson’s History, Vol. 1, Ch. VII, item 193, p. 215. [15] Leonard Tornheim. Sums of n-th powers in fields of prime characteristic. Duke Math. J., 4:359–362, 1938. [16] Joachim von zur Gathen and J¨ urgen Gerhard. Modern computer algebra. Cambridge University Press, Cambridge, second edition, 2003. [17] Andr´e Weil. Numbers of solutions of equations in finite fields. Bull. Amer. Math. Soc., 55:497–508, 1949. [18] Arne Winterhof. On Waring’s problem in finite fields. Acta Arith., 87(2):171–177, 1998.

This sequence has n + 1 elements, say c0 , . . . , cn . If one is zero, we are done anyway! Otherwise, use Selective Root Extraction to compute β ∈ F∗ with β n = ck /cl ,

i.e.

REFERENCES

c k = β n cl

(assume k > l). Replace now the kth term in the sequence by β n times the lth term, and we can reduce hk by one! Thus, in at most n2 steps, we will get one of the hj down to one. The complexity bound follows directly from the ones given above.

8. PROBABILISTIC METHODS It is interesting to compare the running time achievements of the algorithms given in this note with probabilistic methods for solving the same problems. Let us consider (7), as being the most difficult to solve; assume n divides q − 1. An obvious idea that comes to mind is the following: let x1 , . . . , xn−1 be random elements of F, test whether ! n−1 X n ai xi /b i=1

is an nth power in F, and if it is, take its nth root by means of a probabilistic root taking method. (It is possible that better methods exist, but I do not know of any.) Now for this to work, we must be sure that there are enough solutions, otherwise we are not likely to find one by guessing. Now to every n − 1-tuple (x1 , . . . , xn ) corresponds either zero, one, or n solutions to (7). A lower bound for the number of “lucky” elements of Fn−1 is thus obtained by dividing the number of solutions to (7) by n. From Weil’s estimates given in [17], on page 502, we can prove that if q > 3n2 , there are at least q n−1 /2 representations of b of the form (7). Thus if q > 3n2 , we may expect that every 2nth element of Fn−1 will give rise to at least one solution of (7). It is easy to estimate the expected running time of this algorithm as ˜ 2 + n log q) O(n operations in F; here the computation of nth roots is done ˜ log q) operations by polynomial factorisation, and takes O(n by [16, Corollary 14.16]. (One should not use the TonelliShanks algorithm.) I assume here that the generation of a random element of F is about as complex as multiplication. Thus, a probabilistic method will be faster than my method except in the special case given in Section 6.2. However, this running time rapidly deteriorates if q gets below 3n2 , whereas my bounds remain valid for q > n2 . For the case where q < n2 , Weil’s lower bounds on the number of solutions become negative and brute force, made polynomial time with the aid of dynamical programming, is the only remaining method.

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Stability Analysis of Biological Systems with Real Solution Classification Bican Xia

Dongming Wang

LMAM – School of Mathematical Sciences Peking University Beijing 100871, China

LMIB – School of Science, Beihang University Beijing 100083, China LIP6–UPMC–CNRS, 8 rue du Capitaine Scott F-75015 Paris, France

[email protected]

[email protected] ABSTRACT

local and global behaviors of such systems is crucial and challenging. It is extremely difficult to detect and analyze equilibria, stability, bifurcations, and chaos of biological dynamical systems using standard mathematical methods. Consider, for example, the class of plane autonomous polynomial differential systems, which are the simplest continuous dynamical systems. The problem of determining the number of limit cycles and their relative configurations is the second part of Hilbert’s 16th problem, which was posed 105 years ago and on which there is little progress. Therefore, the analysis of stability remains a challenge even for polynomial differential systems. In the research of biological systems, stability behaviors and bifurcations are often analyzed by means of numerical simulation and visualization. It is desirable that the results of such numerical analysis may be confirmed by formal mathematical reasoning. In this paper, we consider biological networks that may be modeled by autonomous systems of differential equations of the form 8 x˙ 1 = F1 (u, x1 , . . . , xn ), > > < x˙ 2 = F2 (u, x1 , . . . , xn ), (1) > ······ > : x˙ n = Fn (u, x1 , . . . , xn ),

This paper presents a new and general approach for analyzing the stability of a large class of biological networks, modeled as autonomous systems of differential equations, using real solving and solution classification. The proposed approach, based on the classical technique of linearization from the qualitative theory of ordinary differential equations yet with exact symbolic computation, is applied to analyzing the local stability of the Cdc2-cyclin B/Wee1 system and the Mos/MEK/p42 MAPK cascade, two well-known models for cell and protein signaling that have been studied extensively in the literature. We provide rigorous proofs and generalizations for some of the previous results established experimentally and report our new findings.

Categories and Subject Descriptors I.1 [Symbolic and Algebraic Manipulation]: Applications, Algorithms; J.3 [Life and Medical Sciences]: Biology and Genetics

General Terms Algorithms

where F1 , . . . , Fn are rational functions of u, x1 , . . . , xn with real coefficients and u is one or several real parameters. As usual, xi = xi (t), x˙ i = d xi /d t, and the parameters u are independent of the derivation variable t. For differential systems (1), we propose a new and general approach to detect their real equilibria by solving the rational-function equations F1 = 0, . . . , Fn = 0 symbolically and to analyze the stability of the equilibria by means of linearization and real solution classification. All the involved computations are performed symbolically and we are not aware of any existing work in this direction. The class of biological networks we consider is large enough to cover many complex systems, including biological positive-feedback loops for cell and protein signaling. We will use the well-known Cdc2-cyclin B/Wee1 system [11, 12] and the Mos/MEK/p42 MAPK cascade [1, 6], which have been studied extensively and experimentally in the literature, to illustrate the features, originality, and performance of our approach. We will also report on our computational results and new findings. The detection of bistability or multistability of such systems is an essential step for understanding how the systems function. Systems are bistable when they toggle between two discrete, alter-

Keywords Biological network, differential equations, equilibrium, stability, polynomial system, real root, solution classification, Cdc2-cyclin B/Wee1, Mos/MEK/p42 MAPK cascade

1.

INTRODUCTION

Computational studies of biological systems, also called bioinformatics, have emerged as a major area of research on the frontiers of mathematics, biology, and computer science in the last decade. Biological networks may be modeled mathematically by dynamical systems. The analysis of the

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do so, we use Lyapunov’s first method with the technique of linearization, that is, by considering the Jacobian matrix 0 1 ∂F1 ∂F1 ∂F1 ··· B ∂x1 ∂x2 ∂xn C B C B ∂F ∂F ∂F2 C 2 2 B C ··· B C ∂x1 ∂x2 ∂xn C . J=B B C .. C .. B .. B . C . . B C @ ∂Fn ∂Fn ∂Fn A ··· ∂x1 ∂x2 ∂xn

native stable steady states without being able to rest in intermediate states. We refer to [1, 12] for technical discussions on the behavior and importance of bistability and multistability in the biological context. The rest of the paper is structured as follows. In the next section, we shall recall some preliminary notions and results from the qualitative theory of ordinary differential equations and present the linearization technique and some stability criteria for our purpose. In Section 3 is provided a brief review on the existing methods dealing with systems of polynomial equations and inequalities. Emphasis will be placed on the techniques that will be used in later sections of this paper. In Section 4, we will present the details of the stability analysis for a simple yet nontrivial example: the Cdc2-cyclin B/Wee1 system. It will show how our method works step by step with symbolic computation. Our general approach to stability analysis for the considered class of biological systems will be described in Section 5. The practical value and potential of our approach will be demonstrated by several experimental results reported in Section 6: some of the previous results established with numerical simulation and visualization may be rigorously proved, refined, and generalized by our program using real solution classification. The paper will be concluded with a few remarks on the novelty, applicability, and limitation of our approach, the encountered computational difficulties, and our future work. A classical and widely used method for analyzing the stability of biological systems is based on phase plane or space diagrams, which plot the trajectories of the differential system around equilibria by numerical computation [1, 11]. This method is limited to plane and spatial differential systems. A more powerful and theoretical approach for analyzing stability behaviors together with a simple graphical method for deducing bifurcation diagrams for biological positive-feedback systems is described in [1]. The visualization technique is very useful in practice, but its theoretical rigor cannot be easily guaranteed. Our symbolic approach provides a mathematically rigorous framework for the stability analysis of a large class of biological systems of arbitrary dimension. The effectiveness of this approach comes from the novelty of using advanced techniques of symbolic real solving and real solution classification.

2.

Then system (1) may be written in the following matrix form: ¯)(x − x ¯)T + G, x˙ T = J(¯ u, x where the superscript T denotes matrix transpose and ¯ )(x − x ¯ )T G = [F1 (¯ u, x), . . . , Fn (¯ u, x)]T − J(¯ u, x ¯|) as x → x ¯. The following theorem serves to is o(|x − x ¯. determine the stability of the equilibrium x Theorem 1. (a) If all the eigenvalues of the matrix J(¯ u, ¯) have negative real parts, then x ¯ is asymptotically stable. x ¯) has at least one eigenvalue with (b) If the matrix J(¯ u, x ¯ is unstable. positive real part, then x ¯ when It is more difficult to determine the stability of x ¯ ) have zero real parts, but some of the eigenvalues of J(¯ u, x none of them has positive real part. In this case, if the eigenvalues with zero real parts correspond to a simple zero ¯), then x ¯ is stable; of the characteristic polynomial of J(¯ u, x otherwise, it may be unstable. In the case n = 2, we may have a more precise classification of the equilibria. Let the Jacobian matrix evaluated at ¯) for n = 2 be (¯ u, x „ « a b ¯) = J2 (¯ u, x c d ¯). More concretely, and λ1 , λ2 the two eigenvalues of J2 (¯ u, x λ1 , λ2 are the two roots of the characteristic polynomial ˛ ˛ ˛ a−λ b ˛˛ ˛ = λ2 + pλ + q, ˛ c d−λ ˛

EQUILIBRIA AND STABILITY OF DIFFERENTIAL EQUATIONS

where p = −(a + d) and q = ad − bc. Set ∆ = p2 − 4 q. We have the following criteria:

Let R denote the field of real numbers. For any real ¯ of u, a point x ¯ = (¯ parametric value u x1 , . . . , x ¯n ) in the n-dimensional real Euclidean space Rn is called an equilibrium (or a singular point, critical point, or steady state) of a system of differential equations of the form (1) if

C1. when q > 0, p > 0, and ∆ ≥ 0 (in this case, λ1 , λ2 are ¯ is a stable node; real and λ1 < 0, λ2 < 0), x C2. when q > 0, p < 0, and ∆ ≥ 0 (in this case, λ1 , λ2 are ¯ is an unstable node; real and λ1 > 0, λ2 > 0), x

¯ ) = · · · = Fn (¯ ¯ ) = 0. F1 (¯ u, x u, x C3. when q < 0 (in this case, λ1 , λ2 are real and λ1 λ2 < 0), ¯ is an (unstable) saddle; x

Therefore, the problem of computing equilibria amounts to finding the real solutions of the system of rational-function equations F1 = 0, . . . , Fn = 0 depending on the parameters u; the latter may be reduced to solving a system of n polynomial equations in n unknowns x1 , . . . , xn . We shall discuss how to solve such systems of equations as well as the involved computational difficulties in the following section. ¯ of u, let x ¯ be an For an arbitrary but fixed real value u ¯ . To equilibrium of (1). We want to analyze the stability of x

C4. when q > 0, p > 0, and ∆ < 0 (in this case, λ1 , λ2 are complex conjugates and Re λ1 = Re λ2 < 0, where Re ¯ is a stable focus (or spiral ); denotes the real part), x C5. when q > 0, p < 0, and ∆ < 0 (in this case, λ1 , λ2 are ¯ is an complex conjugates and Re λ1 = Re λ2 > 0), x unstable focus;

355

we need here is to find exact (number of) real solutions of polynomial systems and to establish conditions for parametric polynomial systems to have a prescribed number of real solutions, which are computationally difficult. There exist general methods such as the method of cylindrical algebraic decomposition (CAD) [4, 5] that may be applied to some of our problems in principle. However, it is well known that the CAD method and its variants have high computational complexity. Much of the recent research on real solving has focused on using the techniques of triangular sets and Gr¨ obner bases to transform the involved systems of polynomial equations and introducing specialized devices to handle the projection and lifting process for restricted classes of systems of polynomial equations and inequalities. Extensive work in this direction includes [2, 9, 18, 21]. The authors of this paper have been much involved in the design and implementation of triangular-set-based decomposition algorithms [13, 14, 15] and real solving based on discrimination systems [18, 19, 21], and we have the necessary software packages in hand for different kinds of experiments. Therefore, we base this initial analysis of stability of biological systems on the work using triangular sets and discrimination systems [14, 15, 19, 21]. It is apparent that other methods such as the CAD method [4, 5], quadratic quantifier elimination [16], and the method using Gr¨ obner bases and discriminant varieties [9] for real solving may also be applied to the same problem of stability analysis according to our general approach. We hope to explore such application and observe and compare the performances of different methods in the near future. In what follows, we present very briefly the method we will use for symbolic real solving and solution classification of (parametric) polynomial systems. Consider the following system of n equations

C6. when q > 0 and p = 0 (in this case, λ1 , λ2 are complex ¯ is a center of conjugates and Re λ1 = Re λ2 = 0), x ¯)(x − x ¯)T and the stability of the equilibx˙ T = J2 (¯ u, x ¯ of (1) depends on G (higher-order approximarium x tion); ¯) is singular C7. when q = 0, the Jacobian matrix J2 (¯ u, x ¯)(x − x ¯)T of and the first approximation x˙ T = J2 (¯ u, x (1) has infinitely many critical points. For n > 2, whether all the eigenvalues of the Jacobian ¯) have negative real parts can be determined matrix J(¯ u, x according to Routh–Hurwitz’s criterion. This will be explained at the end of Section 5. The method presented above is classical in the qualitative theory of ordinary differential equations and may be found in standard textbooks (e.g., [10, 23]). However, for practical application, there are several difficulties one must overcome. First of all, one has to compute the equilibria of the differential system (1), which are real solutions of a system of polynomial equations. Even in the simpler case when the parameters u are not present, in general the real solutions may not be given analytically. The components of the solutions are usually determined as real roots of some polynomials with rational coefficients. The approximate values of such real roots may be computed by numerical methods. Symbolically, the real roots may be isolated by rational intervals of arbitrarily small width, but real solution isolation for general systems of polynomial equations is still an outstanding problem of research in symbolic computation. In the presence of parameters u, one has to identify for ¯ of u the system of equations what parametric values u F1 (¯ u, x) = 0, . . . , Fn (¯ u, x) = 0

P1 (u, x) = 0, P2 (u, x) = 0, . . . , Pn (u, x) = 0,

has real solutions, and how many, for the unknowns x. How to describe or represent the real solutions in terms of the parameters u ? These are simple questions that cannot be easily answered. In order to decide the stability of the equilibria, one also has to determine rigorously the signs of the real parts of the eigenvalues of the Jacobian matrix at these equilibria. For instance, in the case n = 2, one needs to determine the signs of q, p, and ∆ in the above criteria. Even ¯ of u, as the equilibrium x ¯ may be defor a given value u termined only as real roots of polynomials and represented symbolically by means of isolating intervals, the sign determination is computationally nontrivial. It is clearly more difficult to derive the conditions on the parameters u for the real parts of the eigenvalues to have prescribed signs. The approach proposed in the present paper may solve most of these problems. We will present the details in the following sections.

3.

(2)

where P1 , . . . , Pn are polynomials in u and x = (x1 , . . . , xn ) with rational coefficients and u is one or several real parameters. Let P be any polynomial in u and x with rational coefficients. We are concerned with the following problems. P1. Assume that the parameters u are not present. Determine the number of real solutions of (2) for x and isolate all the isolated real solutions of (2) by rational intervals. P2. For any integer s ≥ 0, determine the conditions on u for system (2) to have exactly s distinct real solutions for x. P3. Assume that the parameters u are not present. Determine the sign of P at each isolated real solution of (2). P4. Determine the conditions on u for P to be 0, positive, or negative at the isolated real solutions of (2).

REAL SOLVING AND SOLUTION CLASSIFICATION OF POLYNOMIAL SYSTEMS

The method for solving these problems works by first transforming the set P = {P1 , . . . , Pn } of polynomials into finitely many regular sets T1 , . . . , Te [7, 14, 22] such that

Solving systems of polynomial equations has been one of the central topics in computer algebra. It is required and used in many scientific and engineering applications. There are several methods based on resultants, Gr¨ obner bases [3], and triangular sets [7, 8, 13, 17] which may be used to transform such systems into certain triangular form. What

Zero(P) =

e [

Zero(Ti /Ji ),

(3)

i=1

where Zero(P) denotes the set of all common zeros (in some extension of the field of rational numbers) of P1 , . . . , Pn ,

356

Zero(Ti /Ji ) = Zero(Ti ) \ Zero({Ji }), and Ji is the product of initials of the polynomials in Ti for each i (see [15] for the definition and computation of regular sets and other relevant concepts). Furthermore, we may assume that each Ti is simplicial [22] with respect to P (i.e., P is either zero at all the real zeros of Ti , or nonzero at every real zero of Ti ) and no two of the regular sets have common zeros [21]. Without loss of generality, we only consider one regular set T = Ti = [T1 (u, x1 ), . . . , Tn (u, x1 , . . . , xn )].

Figure 1: The Cdc2-cyclin B/Wee1 system and its phase plane diagram under unitary feedback (v = 1), reproduced from [1]

If the parameters u are not present, we first isolate the real roots of T1 for x1 by rational intervals. For each isolating interval I, substitute the two endpoints for x1 in T2 , respectively, and obtain two polynomials in x2 . From these two polynomials, one can obtain the isolating intervals for the real zeros of [T1 , T2 ] when x1 is on I, provided that I is sufficiently small. In this way, we can isolate all the real zeros of T. To determine the sign of P at each isolated real zero of T, one substitutes the intervals for the variables in P and determines the sign of P by interval arithmetic. If the algorithm proposed in [18] is used, we only need to take a point in the isolating interval arbitrarily and to determine the sign of P at this point. For algorithmic details, we refer to [18]. If the parameters u are present, we compute a so-called border polynomial B(u) of (2) according to the regular sets Ti [20, 21]. For each regular set Ti = [T1 , . . . , Tn ], the border polynomial contains the resultant of the leading coefficient of Tj and [T1 , . . . , Tj−1 ] as well as the resultant of the discriminant of Tj and [T1 , . . . , Tj−1 ] as its factors. If the sign of P is to be determined, then B(u) also contains the resultants of P and Ti . The border polynomial has the property that the number of distinct real solutions of the system (2) is invariant in each connected component (called cell ) of the complement of B(u) = 0 in the parametric space. It is closely related to the notion of discriminant variety introduced by Lazard and Rouillier [9]. Thus, to determine the number of distinct real solutions of (2) in each cell, it suffices to check the situation at one point of this cell. So one may take a sample point from each cell and isolate the real solutions of (2) at the sample point. Finally, the signs of the factors of B(u) together with the numbers of real solutions of (2) at the sample points form a real solution classification of (2), from which the conditions for problem P4 above may be easily obtained. The interested reader may consult [20, 21] for details.

4.

back loop is reduced to the following form 8 β1 x1 (vy1 )γ1 > > < x˙ 1 = α1 (1 − x1 ) − K + (vy )γ1 , 1

1

> β y x γ2 > : y˙ 1 = α2 (1 − y1 ) − 2 1 1γ2 , K2 + x 1

(4)

where α1 , α2 , β1 , β2 are rate constants, K1 , K2 are Michaelis (saturation) constants, γ1 , γ2 are Hill coefficients, and v is a coefficient (feedback) that reflects the strength of the influence of Wee1 on Cdc2-cyclin B. For easy reference and comparison, we take the same numerical values for the biological constants as in [1]: γ1 = γ2 = 4, α1 = α2 = 1, β1 = 200, β2 = 10, K1 = 30, K2 = 1. For simplicity of notation, let x = x1 and y = y1 . Then system (4) becomes x˙ =

P , 30 + v 4 y 4

y˙ =

Q , 1 + x4

(5)

where P = 30 − 30 x + v 4 (1 − 201 x) y 4 , Q = 1 + x4 − (1 + 11 x4 ) y, and v is a real parameter. Our problem is to detect the stability of (5). In particular, we want to know for what parametric value of v bistability may arise in this system, i.e., for what value of v system (5) may have two stable steady states. This amounts to determining the number of equilibria of (5), or equivalently, finding the number of real solutions of P = 0, Q = 0 for x, y in terms of v. In general, one can apply the methods of triangular sets, Gr¨ obner bases, and resultants to triangularize the polynomial system in equation. However, the system P = 0, Q = 0 for this example is too simple for any of these methods and the triangularization process is trivial: solving Q = 0 for y and substituting the solution to P , we obtain an irreducible polynomial of degree 17 in x with parameter v:

STABILITY ANALYSIS OF THE CDC2CYCLIN B/WEE1 SYSTEM

As an example to illustrate our general approach, in this section we analyze the stability of the Cdc2-cyclin B/Wee1 system. It is a bivariate system that describes the interplay between two proteins: the Cdc2-cyclin B complex and the Wee1 protein (see [1, 11, 12] and Figure 1). Its stability behavior may be determined numerically by the classical phase plane analysis [11] and another graphical method proposed by Angeli and others in [1]. We refer to [1] for the setting details of this example. Under certain assumptions, the system of differential equations that model the two-component, mutually inhibitory feed-

H = (439230 + 201 v 4 ) x17 − (439230 + v 4 ) x16 + (159720 + 804 v 4 ) x13 − (159720 + 4 v 4 ) x12 + (21780 + 1206 v 4 ) x9 − (21780 + 6 v 4 ) x8 + (1320 + 804 v 4 ) x5 − (1320 + 4 v 4 ) x4 + (30 + 201 v 4 ) x − 30 − v 4 . Now the problem is to derive the conditions on v for H to have 0, 1, 2, . . . real roots for x. This can be done by using the

357

method of real root classification sketched in the previous section. Our implementation of the method allows us to establish the conditions automatically. More concretely, the ¯ (which in fact is program may find a polynomial R = v R ¯ is of the discriminant of H with respect to x), where R degree 32 in v and has 9 terms and 4 real roots and is a little bit too large to be reproduced here. Denote the 5 real roots of R by v¯2 < v¯1 < v0 = 0 < v1 < v2 (where v1 = −¯ v1 ≈ 0.8315735076, v2 = −¯ v2 ≈ 1.796868764), which may be isolated as follows: » – » – » – » – 3 1 1 3 −2, − , −1, − , [0, 0], ,1 , ,2 . 2 2 2 2

S1. when 0 < v < v1 or v2 < v < +∞, q > 0 and ∆ > 0 hold at the only steady state, so this steady state is a stable node; S2. when v1 < v < v2 , one of the three steady states, at which q < 0, is an (unstable) saddle and the other two steady states (at which q > 0 and ∆ > 0) are stable nodes; S3. when v = 0, p > 0, q > 0, and ∆ > 0 hold at the unique steady state, so this steady state is a stable node; S4. when v = v1 or v = v2 , q > 0 and ∆ > 0 hold at one of the two steady states, so this steady state is a stable node, and q = 0, a < 0, d < 0, and bc > 0 hold at the other steady state. In the latter case, because q = 0 (i.e., the Jacobian matrix of (5) is singular), the method of linearization is inapplicable, but it is not difficult to see that the steady state in this case is unstable.

These intervals can be made arbitrarily small if we wish. As v is for the strength of the influence of Wee1 on Cdc2cyclin B (or in view of the symmetry), we may assume that v ≥ 0. From the output of the program, we get directly the following results: E1. when 0 < v < v1 or v2 < v < +∞, system (5) has only one steady state (or equilibrium);

Therefore, it is rigorously proved that the system exhibits bistability when v1 < v < v2 . This completes our analysis of the stability of (5).

E2. when v1 < v < v2 , system (5) has three steady states; E3. when v = 0, system (5) has a unique steady state;

5. DESCRIPTION OF THE GENERAL APPROACH

E4. when v = v1 or v = v2 , system (5) has two steady states.

In this section, we provide a short description of the algorithmic steps of our general approach for the stability analysis of biological systems. It is a formulation of the main steps shown in the above illustrative example. We are given an autonomous system of ordinary differential equations of the form (1) that models a biological network. In the case when the parameters u are not present, our problem is to compute the real steady states of the system symbolically, in the sense that the coordinates of each steady state may be given as rational intervals of arbitrarily small width, and to determine the stability of each steady state. In the presence of parameters u, our problem is to determine the conditions on u for the system to have prescribed numbers of stable and unstable steady states. The following steps provide solutions to these problems.

Finally, we need to determine the stability of the steady states, i.e., to determine whether the steady states are stable or not. To this end, we consider the Jacobian matrix of (5), whose entries are the partial derivatives of F =

P , 30 + v 4 y 4

G=

Q 1 + x4

with respect to x and y, i.e., a=

3 (10 + 67 v 4 y 4 ) 24000 v 4 xy 3 ∂F ∂F =− =− , b= , 4 4 ∂x 30 + v y ∂y (30 + v 4 y 4 )2

c=

40 x3 y ∂G =− , ∂x (1 + x4 )2

d=

∂G 1 + 11x4 =− . ∂y 1 + x4

Let

M1. Equate the numerators of the rational functions on the right-hand side of (1) to 0, yielding a system of polynomial equations of the form (2). In view of the background of the problem, there may be additional constraints on the system. For example, the denominators of the rational functions on the right-hand side of (1) should be nonzero and some variables may be positive.

2 p¯ p = −(a + d) = , (30 + v 4 y 4 ) (1 + x4 ) 3 q¯ , (30 + v 4 y 4 )2 (1 + x4 )2 ¯ 100 ∆ ∆ = p2 − 4 q = , (30 + v 4 y 4 )2 (1 + x4 )2

q = ad − bc =

where 4

4 4

M2. Decompose the polynomial set P = {P1 , . . . , Pn } into regular sets T1 , . . . , Te using any of the algorithms described in [7, 14, 22] (or more concretely, using the function RegSer available in the Epsilon library [15]), such that (3) holds. Make Ti and Tj have no common zero for all i 6= j and simplicial with respect to all constraints according to [21, 22].

4 4 4

p¯ = 30 + 180 x + 101 v y + 106 v x y , q¯ = 67 y 8 (1 + 11 x4 ) (1 + x4 ) v 8 + 20 y 4 (101 − 14788 x4 + 1111 x8 ) v 4 + 300 (1 + 11 x4 ) (1 + x4 ), ¯ = x8 (19 v 4 y 4 − 30)2 + 40 v 4 x4 (930 + 19 v 4 y 4 ) y 4 ∆

M3. If the parameters u are present, then go to step M4. Otherwise, isolate the real zeros of each Ti according to the algorithms presented in [18] (see the second paragraph from the end of Section 3). These real zeros are

+ 400 v 8 y 8 . It is easy to see that a < 0, d < 0, p > 0, ∆ ≥ 0 always hold. As output, our program gives the following results:

358

¯ (as u all the steady states of (1). Take any value for u are not present) and go to step M7.

defined to be the minors of P. According to the Routh– Hurwitz criterion, the real parts of all the roots of P are negative if and only if V(a0 , Γ1 , Γ3 , . . .) = V(1, Γ2 , Γ4 , . . .) = 0, where V(. . .) means the number of sign changes of the sequence.

M4. Compute the border polynomial B(u) of system (2) with all possible constraints (from T1 , . . . , Te ) according to the algorithm proposed in [21]. The real part of the hypersurface B(u) = 0 gives a topological decomposition of the parametric space. If, for example, u is a single parameter, then all the real numbers are decomposed into a finite number of open intervals and points by the real roots of B(u). Any two points in one such interval have the following property: the number and the stability of the steady states of (1) are invariant respectively at these two points.

6. EXPERIMENTS In this section, we report on some of our experiments with the proposed approach for stability analysis of biological systems. Our experiments have been made on the Cdc2-cyclin B/Wee1 model without taking values for some of the biological constants and on a more complicated modular, fivevariable example: the Mos/MEK/p42 MAPK cascade. As in our approach the obtained results are exact and rigorous, the involved symbolic computations are heavy in general. The occurring polynomials may be of very high degree with large integer coefficients. However, as shown by our experiments the symbolic methods underlying our approach are powerful enough as to be used for the stability analysis of reasonably complex biological systems. The condition on v derived in Section 4 for the Cdc2-cyclin B/Wee1 system to exhibit bistability is for the given values of the biological constants α1 , α2 , β1 , β2 , K1 , K2 , γ1 , γ2 . Estimation of the constant values are very difficult: some of the values may be determined experimentally and others may be chosen so that the model can simulate the type of biological behavior that is observed or expected. Our symbolic approach allows us to establish conditions on some constant parameters for the system to exhibit certain desired behavior such as bistability or multistability. To fix the idea, let us consider the Cdc2-cyclin B/Wee1 system as in Section 4, but without taking values for the Michaelis constants K1 , K2 . We want to know for what values of K1 , K2 and v the system exhibits bistability. From the meanings of the biological constants and variables, we know that K1 > 0, K2 > 0 and v, x1 , y1 are nonnegative. Our program may compute a polynomial R1 of degree 32 in v and degree 8 in either of K1 and K2 with 81 terms. Under the above assumption, we have a < 0, d < 0, p > 0, r ≥ 0, b ≤ 0, c ≤ 0. Then we can conclude that

M5. Take a sample point from each connected component (cell) of the complement of B(u) = 0 in the parametric space. This can be done by applying a partial CAD algorithm. If u is a single parameter, one only needs to take a sample point between ui and ui+1 for all i (0 ≤ i ≤ r) by isolating all the distinct real roots u1 , . . . , ur of B(u), where u0 = −∞ and ur+1 = ∞. ¯ , substitute u ¯ for u in Ti (notM6. For each sample point u ¯ does not make the vanishing of the initial of ing that u any polynomial in Ti ) and isolate the real solutions of the resulting regular sets, yielding the steady states of ¯ . Because of the property of B(u), system (1) at u = u ¯ is also the the number of steady states of (1) at u = u number of steady states of (1) when u is in the same cell. At the same time, one may obtain the signs of the ¯ . If the conditions on u for system factors of B(u) at u (1) to have a prescribed number of steady states are desired, we output the signs of the factors of B(u) at the sample points of those cells in which the system has exactly the prescribed number of steady states. ¯ is a steady state of system (1) at u = M7. Suppose that x ¯ . We want to determine the stability of (1) at x ¯. u By the technique of linearization explained in Section 2, we compute the Jacobian matrix J(u, x) of (1) and its characteristic polynomial H(u, x, λ). Now the problem is reduced to determining the stability of ¯, λ), which can be done by computing the signs H(¯ u, x ¯ , λ) according to of the Hurwitz determinants of H(¯ u, x the method described below. Finally, we output the signs of those cells in which system (1) has exactly the number of stable (or unstable) steady states as desired.

N1. when R1 < 0, the system has three steady states, of which two are stable (in this case q > 0) and the other is unstable (in this case q < 0); N2. when R1 > 0, the system has only one steady state which is stable. It follows that the system exhibits bistability if and only if R1 < 0. This generalizes the result given in [1] and Section 4. The computation in the case R1 = 0 is too heavy and could not be completed within three hours in Maple 9 on a notebook computer (Pentium 1.13 Ghz CPU with 256 M memory). From the results in the cases with specialized values of K1 , K2 , we guess that the system has two steady states, of which one is stable and the other is unstable, when R1 = 0. To determine the range of K1 , K2 for the system to exhibit bistability, we may compute a polynomial R2 of K2 :

A standard method to determine the stability of a polynomial is Routh–Hurwitz’s criterion [10, pp. 184–186]. Let P = a0 λm + b0 λm−1 + a1 λm−2 + b1 λm−3 + · · · be a real polynomial in 0 b0 B a0 B B 0 B P=B 0 B B 0 @ .. .

(a0 6= 0)

λ and consider the m × m matrix 1 b1 b2 · · · bm−1 a1 a2 · · · am−1 C C b0 b1 · · · bm−2 C C a0 a1 · · · am−2 C , C 0 b0 · · · bm−3 C A .. .. .. . . .

R2 = 1123963607439473175421875 K24 − 9244704652117591783090536 K23 − 5088828365064957511326382 K22 − 62301929415679096 K2 + 51046875.

where we take ai = 0 when i > m/2, and bj = 0 when j ≥ m/2. The Hurwitz determinants Γ1 , . . . , Γm of P are

359

Let the two positive real roots of R2 be k1 ≈ 0.77 · 10−9 and k2 ≈ 8.74. Our computation shows that the system exhibits bistability for some range of the feedback gain v when K1 > 0 and k1 < K2 < k2 , or no bistability otherwise. It follows that the system always exhibits bistability for some range of v, no matter what value K1 takes. This conclusion is related to a question in [1]. The computation of the above results took less than three minutes in Maple 9 on the abovementioned machine. Now we discuss our experiments on a three-tier MAPK cascade, based on the Mos/MEK/p42 MAPK cascade present in Xenopus oocytes, which has been studied extensively in the literature (see, e.g., [1, 6]). This system is modeled by the following five equations [1]: 8 xV2 > x˙ = − + vz3 V0 + V1 , > > K2 + x > > > > > xy1 V3 (1200 − y1 − y3 ) V6 > > > − , y˙ = > > 1 K K 6 + 1200 − y1 − y3 3 + y1 > > > < x(1200 − y1 − y3 ) V4 y 3 V5 (6) y˙3 = − , > K4 + 1200 − y1 − y3 K5 + y 3 > > > > > y 3 z 1 V7 (300 − z1 − z3 ) V10 > > − , > z˙1 = > K + 300 − z − z K > 10 1 3 7 + z1 > > > > > : z˙3 = y3 (300 − z1 − z3 ) V8 − z3 V9 . K8 + 300 − z1 − z3 K9 + z 3

Table 1: Steady states of the Mos/MEK/p42 MAPK cascade and their stability labels 1–10 11, 12 13 14–16 17 18 19, 20 21 steady 1 1 3 3 3 2 2 0 states stable 0 1 0 2 0 1 0 0 states Our program computed a polynomial B of degree 93 in v (the feedback strength) with 93 terms and 41 distinct real roots. Let vi (i = 1, . . . , 41) denote the 41 roots in ascending order, where v21 = 0. Because v is the feedback strength, we may assume that v ≥ 0; thus only 21 open intervals (v21 , v22 ), . . . , (v40 , v41 ), (v41 , +∞) need be considered. Our program has then isolated the real solutions of system (7) in these 21 intervals and returned the following results. In any of the first 12 open intervals, i.e., (v21 , v22 ), . . . , (v32 , v33 ), system (7) has only one steady state. In each of the next five intervals, i.e., (v33 , v34 ), . . . , (v37 , v38 ), the system has three steady states. The system has two steady states in (v38 , v39 ), (v39 , v40 ), or (v40 , v41 ) and no steady states in the interval (v41 , +∞). The characteristic polynomial of the Jacobian matrix of (6) is H = −λ5 + c4 λ4 + c3 λ3 + c2 λ2 + c1 λ + c0 ,

For this five-dimensional system, the graphical technique of phase plane or space analysis does not work and it is highly nontrivial to detect its multistability. According to [1], we take numerical values for the biological constants as follows: V0 = 3/2000, V1 = 1/500000, V2 = 6/5,

K2 = 200,

V3 V5 V7 V9

K4 = 1200, K6 = 1200, K8 = 300, K10 = 300.

= 8/125, = 5, = 3/50, = 5,

K3 K5 K7 K9

= 1200, = 1200, = 300, = 300,

V4 = 8/125, V6 = 5, V8 = 3/50, V10 = 5,

where the ci are rational functions in v, x, y1 , y3 , z1 , z3 , whose numerators are very large polynomials. For instance, the numerator of c1 is a polynomial of total degree 16 in v, x, y1 , y3 , z1 , z3 with 4165 terms. Let Γ1 , . . . , Γ5 be the Hurwitz determinants of H. For each of the 21 intervals, we need to check the signs of the Γi at each of the steady states. Let the 21 intervals be labeled with 1 through 21 in ascending order. The numbers of (stable) steady states of system (7) determined by our program are shown in Table 1. Moreover, when

Then equating the numerators of the rational functions on the right-hand side of (6) to 0, we obtain a system of five equations 8 P1 = 150000 vz3 + 750 vz3 x − 599999 x + 200 = 0, > > > 2 > > P 2 = 625 y1 + 750000 y3 + 625 y3 y1 + 19200 xy1 > > > > > − 8 xy12 − 8 xy1 y3 − 900000000 = 0, > > > > 2 > > < P3 = −11520000 x + 9600 xy1 + 8 xy1 y3 + 8 xy3 + 1500000 y3 − 625 y3 y1 − 625 y32 = 0, > > > P4 = 250 z12 + 75000 z3 + 250 z3 z1 + 1800 y3 z1 > > > > > − 3 y3 z12 − 3 y3 z1 z3 − 22500000 = 0, > > > > > > P5 = −270000 y3 + 900 y3 z1 + 3 y3 z1 z3 + 3 y3 z32 > > : + 150000 z3 − 250 z3 z1 − 250 z32 = 0. (7) Because of the background of the problem, the above system should satisfy the following constraints:

14999911 ≈ 0.749 or v = v37 ≈ 2.585, 20025000 is the unique positive root of

v = v34 = where v37

4251528000000000000000 v 3 +12802676662170000000000 v 2 −28309648634268638175000 v− 85804058978275037002757, system (7) has three steady states and none of them is stable. Thus, it is rigorously proved that system (7) exhibits bistability when v34 < v < v37 . This confirms the result for unitary feedback (i.e., v = 1) given in [1]. However, our result contradicts the conclusion in [1] that the system is bistable for any value of v between ≈ 0.7 and a very large real number. The upper bound of v we have obtained for the system to be bistable is only ≈ 2.585, which is not large. We hope that this upper bound can be confirmed by other theoretical and experimental studies.

7. CONCLUDING REMARKS

x ≥ 0, y1 ≥ 0, y3 ≥ 0, z1 ≥ 0, z3 ≥ 0, 1200 − y1 − y3 ≥ 0, 300 − z1 − z3 ≥ 0.

The general symbolic approach proposed in this paper has been applied to two well-known biological models, the Cdc2-cyclin B/Wee1 system and the Mos/MEK/p42 MAPK cascade. We have rigorously proved and generalized the result given in [1] for the Cdc2-cyclin B/Wee1 system and confirmed some of the results in [1] for the Mos/MEK/p42

Determining the steady states of (6) with the above-specified constant values is equivalent to finding the real solutions of the polynomial system (7) under these constraints.

360

MAPK cascade. Moreover, we have found an exact upper bound near 2.585 on the feedback gain v for the MAPK cas-

cade to exhibit bistability that is much smaller than an upper bound which is unknown but was claimed to be very large in the recently published paper [1]. Our approach uses exact symbolic computation and thus ensures that all the results obtained are mathematically rigorous. However, it is limited to the case when the Fi on the right-hand side of (1) are rational functions and it may be inapplicable when the Jacobian matrix is singular or some of its eigenvalues have zero real parts but none of them has positive real part (a much more complicated case in differential equations). The computations involved in our method may also be expensive and some of the polynomials representing the exact real values in our results are very large and cannot be reproduced in the paper. The interested reader may request those polynomials and the isolating intervals of their real roots from the authors. In this initial study, we have focused our attention mainly to the analysis of local stability for two concrete biological models. Our approach may be applied to many other models in biology and biochemistry. It may also be used and refined for the study of several relevant problems such as global stability and bifurcation of limit cycles. How to solve a large system of (parametric) polynomial equations and inequalities coming from biological networks, represent and classify their exact real solutions, and determine the sign of a rational expression evaluated at these solutions more efficiently, how to generalize our approach for other biological or nonbiological systems, and how to develop a practical software tool to automate the process of stability analysis are some examples of questions for our future research. We believe that investigations on these questions will make our symbolic approach a promising and powerful tool for the qualitative study of such biological networks that may be modeled by systems of ordinary differential equations. It may become a good alternative to the experimental approach based on numerical simulation and visualization.

[6]

[7]

[8]

[9]

[10] [11]

[12]

[13] [14] [15] [16]

[17]

[18]

Acknowledgments The authors wish to thank Eduardo D. Sontag for his communication concerning the equations (6) and the biological constants. This work is supported partially by the National Key Basic Research Project 2004CB318000 of China.

8.

[19]

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An Open Problem on Metric Invariants of Tetrahedra Lu Yang∗

Zhenbing Zeng

Software Engineering Institute East China Normal University Shanghai 200062, China

Software Engineering Institute East China Normal University Shanghai 200062, China

[email protected]

[email protected]

ABSTRACT

form:

In ISSAC 2000, P. Lisonˇek and R.B. Israel [3] asked whether, for any given positive real constants V, R, A1 , A2 , A3 , A4 , there are always finitely many tetrahedra, all having these values as their respective volume, circumradius and four face areas. In this paper we present a negative solution to this problem by constructing a family of tetrahedra T(x,y) where (x, y) varies over a component of a cubic curve such that all tetrahedra T(x,y) share the same volume, circumradius and face areas.

R2 = −

This means that the square of the volume of a tetrahedron is 2 with rational coefficients, and the square a polynomial in di,j 2 . According of the circumradius is a rational function in di,j to Heron formula of triangles, the squares of the areas of four faces of the tetrahedron are also rational polynomials 2 in di,j : 1 Ml , l = 1, 2, 3, 4. 16 Call di,j , V, R, Al the metric invariants of a tetrahedron. Since Ml can be regarded as polynomials in di,j , we have 6 polynomials connecting these metric invariants: A2l = −

Categories and Subject Descriptors: G.0 [General] General Terms: Algorithms Keywords: metric invariant, tetrahedron, distance geometry, manifold solution.

1.

M5 . 2M0

f1 (d1,2 , d1,3 , · · · f2 (d1,2 , d1,3 , · · · f3 (d1,2 , d1,3 , · · · f4 (d1,2 , d1,3 , · · · f5 (d1,2 , d1,3 , · · · f6 (d1,2 , d1,3 , · · ·

INTRODUCTION

Consider a tetrahedron T = P1 P2 P3 P4 in R3 . Let di,j = d(Pi , Pj ) be the distance between vertices Pi and Pj . It is well known that the volume V of tetrahedron T can be expressed by the Cayley-Menger determinant associated with the points Pl (1 ≤ l ≤ 4) as follows:

2 2 2
0 d1,3 d1,4 1

d1,2
2 2 2
d 0 d2,3 d2,4 1

1

2,1 2 2 2 V2 = d 0 d 1

. d 3,2 3,4 288

3,1 2 2 2 d d 0 1

d 4,1 4,2 4,3

1 1 1 1 0


, d3,4 , V ) = 288V 2 − M0 , , d3,4 , R) = 2M0 R2 + M5 , , d3,4 , A1 ) = 16A21 + M1 , , d3,4 , A2 ) = 16A22 + M2 , , d3,4 , A3 ) = 16A23 + M3 , , d3,4 , A4 ) = 16A24 + M4 .

According to a theorem in [1] on embedding a simplex in Rn , for any given positive real constants V, R, A1 , A2 , A3 , A4 , there exists a tetrahedron T = P1 P2 P3 P4 in R3 such that Pi Pj = di,j , whenever there is a solution (d1,2 , d1,3 , · · · , d3,4 ) with di,j > 0 to the system {f1 = 0, f2 = 0, · · · , f6 = 0}. In [2], a question proposed by M. Mazur asked whether or not a tetrahedron is uniquely determined by its volume, circumradius and face areas. P. Lisonˇek and R.B. Israel [3] gave a negative answer to this question by constructing two or more tetrahedra that share the same volume, circumradius and face areas, and suggested to discuss whether, for any positive real constants V, R, A1 , A2 , A3 , A4 , there are finitely many tetrahedra, all having these values as their respective metric invariants. In this paper, we present a negative solution to Lisonˇek and Israel’s problem.

For convenience, let M0 denote the above determinant and Ml the principal minor determinant of M0 obtained by deleting the l-th row and l-th column of M0 for l = 1, · · · , 5. Then, the circumradius R, i.e., the radius of the sphere circumscribed to T , can be expressed by di,j in the following ∗

The work is supported in part by NKBRPC-2004CB318003 and NNSFC-10471044. The first author, Lu Yang, holds concurrently positions both in Guangzhou University and Chinese Academy of Sciences.

2. A MANIFOLD SOLUTION TO THE METRIC INVARIANT EQUATIONS

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Our main result is the following theorem. Theorem 1. Let G(x, y) = 3 (1 − x)(17 − 18y)(1 + 3x + 3y) − 9x2 − 3x − 37,

362

g1,2 = 324 (1 − y) (1 + y), g1,3 = 324 (1 − x) (1 + x), g1,4 g2,3 g2,4 g3,4

y

= (29 − 18x − 18y) (7 + 18x + 18y), = 36 (7 − 3x − 3y) (1 + 3x + 3y), = (17 − 18x) (31 + 18x), = (17 − 18y) (31 + 18y),

0.6 0.4 0.2

and x

G0 = { (x, y) ∈ R2 | G(x, y) = 0, |x| < 1, |y| < 1}.

–0.4

Then, for each (ξ, η) ∈ G0 , it holds that

One can see that the polynomial G(x, y) is symmetric in x and y by expanding it. The shape of the semi-algebraic set G0 looks like a UFO, as shown in Fig. 1. Theorem 1 means that there are a family of infinitely many tetrahedra which share the same volume, circumradius and face areas. This presents a negative answer to Lisonˇek and Israel’s question. To prove the theorem, we need verify the following lemmas.

0.4

0.6

–0.4

Fig. 1 The last two, g2,4 (ξ, η) = (17 − 18 ξ) (31 + 18 ξ) > 0, g3,4 (ξ, η) = (17 − 18 η) (31 + 18 η) > 0, 3 3 , |η| < , by Lemma 1. Next, 4 4 observe that −9x2 − 3x − 37, the sum of the last three terms of G(x, y), is always negative, whenever G(x, y) = 0, the following inequality holds: also hold because |ξ| <

3 (1 − x)(17 − 18y)(1 + 3x + 3y) > 0.

Lemma 1. It holds for all (ξ, η) ∈ G0 that

We have known (1 − ξ)(17 − 18η) > 0 for (ξ, η) ∈ G0 , hence 1 + 3ξ + 3η > 0, and then

3 1 3 1 <ξ< , − <η< , 2 4 2 4 where G0 is defined in Theorem 1. −

g2,3 (ξ, η) = 36 (7 − 3ξ − 3η)(1 + 3ξ + 3η) > 0. Furthermore, 1 + 3ξ + 3η > 0 implies 7 + 18ξ + 18η > 0 and 3 3 that |ξ| < and |η| < imply 29 − 18ξ − 18η > 0, so we 4 4 have

Proof of Lemma 1: The set G0 is not empty since √ √   2 1 2 1 + , − ∈ G0 . 4 3 4 3

g1,4 (ξ, η) = (29 − 18ξ − 18η)(7 + 18ξ + 18η) > 0.

The curve G(x, y) = 0 does not intersect either of the lines x = 1, x = −1, y = 1, y = −1 because none of G(1, y), G(−1, y), G(x, 1), G(x, −1) has a real zero. So G0 is compact. For all (ξ, η) ∈ G0 , the maximum and minimum of ξ both are real zeros of the polynomial obtained by eliminat∂ G(ξ, η), that is, ing η from G(ξ, η) and ∂η

This completes the proof of Lemma 2.

 Lemma 3. The assignment { di,j = gi,j (ξ, η), (1 ≤ i < √ √ √ 43 3 , A1 = 84 3, A2 = 63 3, A3 = j ≤ 4), V = 441, R = 6 √ √ 63 3, A4 = 63 3 } solves the system { f1 , f2 , f3 , f4 , f5 , f6 } for every (ξ, η) ∈ G0 .

324 ξ 3 + 576 ξ 2 − 275 ξ − 233, which has only 2 real zeros in (−1, 1) as follows,

This is simple, just recall G(ξ, η) = 0 on doing substitution.

0.73069 · · · .

Proof of Theorem 1: Denote the Cartesian coordinates to be determined of the vertices P1 , P2 , P3 , P4 by

1 3 1 3 So we have − < ξ < , analogously, − < η < . 2 4 2 4

( 0, 0, 0 ), (x1 , 0, 0 ), (x2 , x3 , 0 ), (x4 , x5 , x6 ),

Lemma 2. It holds for all (ξ, η) ∈ G0 that gi,j (ξ, η) > 0,

0.2 –0.2

g1,2 (ξ, η) > 0, g1,3 (ξ, η) > 0, · · · , g3,4( ξ, η) > 0,  and there exists a tetrahedron T(ξ,η) with edge-lengths gi,j (ξ, η) (1 ≤ i < j ≤ 4), whose volume, circumradius and four face areas are equal to √ √ √ √ √ 43 3 , 84 3, 63 3, 63 3, 63 3, 441, 6 respectively.

−0.48681 · · · ,

–0.2

2 = gi,j (ξ, η), we have respectively. Since di,j

(1 ≤ i < j ≤ 4)

x21 = 324 (1 − η)(1 + η) x22 + x23 = 324 (1 − ξ)(1 + ξ) x24 + x25 + x26 = (29 − 18ξ − 18η)(7 + 18ξ + 18η) (x1 − x2 )2 + x23 = 36 (7 − 3ξ − 3η)(1 + 3ξ + 3η)

where G0 and gi,j are defined in Theorem 1. Proof of Lemma 2: The first two inequalities g1,2 (ξ, η) = 324 (1 − η) (1 + η) > 0, g1,3 (ξ, η) = 324 (1 − ξ) (1 + ξ) > 0

(x1 − x4 )2 + x25 + x26 = (17 − 18ξ)(31 + 18ξ) (x2 − x4 )2 + (x3 − x5 )2 + x26 = (17 − 18η)(31 + 18η).

are trivial since |ξ| < 1 and |η| < 1.

363

Solve the equation system for {x1 , x2 , · · · , x6 } and receive a manifold solution:  x1 = 18 (1 − η)(1 + η), 11 − 18ξ − 18η + 18ξη  , x2 = (1 − η)(1 + η) √ 7 3 , x3 =  (1 − η)(1 + η)

4. ACKNOWLEDGEMENTS The authors would like to express their gratitude to the referees for valuable comments and suggestions. One of the referee investigated the conjecture raised in the conclusion using a numerical test at random and always gets nine solutions, albeit some might not have positive values. This constitutes a probabilistic “proof” that the generic finite case has at most nine positive solutions, whenever face areas are pairwise distinct. Furthermore, the referee made another observation. In the generic case of nine solutions, one of them is always observed to give negative values, then a stronger conjecture would be that there are at most eight tetrahedra for a given set of parameter values for which only finitely many exist. Another referee pointed out that the proofs of Lemma 2 and Theorem 1 could be replaced by alternate ones as follows. Since the set G0 is compact, Lemma 2 can be proved by showing that none of the six systems of equations {G = 0, gi,j = 0} ( for 1 ≤ i < j ≤ 4 ) has a real solution in the 1 3 range − ≤ x, y ≤ , and then observing that, there is a 2 4 point (x0 , y0 ) in G0 such that gi,j (x0 , y0 ) > 0 for all i, j, (1 ≤ i < j ≤ 4). Moreover, the final part of the proof of Theorem 1 could be simplified by appealing to a theorem in Section 40 of Blumenthal’s book [1]. That theorem implies that the tetrahedron exists if and only if all the squared volumes V 2 , A21 , A22 , A23 , A24 have positive values, which therefore is not only necessary but also sufficient condition for existence of the tetrahedron.

18ξ + 11η − 18ξη − 18η 2  , (1 − η)(1 + η) √ 7 3η , x5 =  (1 − η)(1 + η) √ x6 = 7 3, x4 =

where (ξ, η) ranges over G0 = { (x, y) ∈ R2 | G(x, y) = 0, |x| < 1, |y| < 1}. Thus, we obtain a family of tetrahedra T(ξ,η) with vertices: P1 = ( 0, 0, 0 ),  P2 = (18 (1 − η)(1 + η), 0, 0 ),   √ 7 3 11 − 18ξ − 18η + 18ξη  ,  ,0 , P3 = (1 − η)(1 + η) (1 − η)(1 + η)   √ √ 7 3η 18ξ + 11η − 18ξη − 18η 2  P4 = ,  ,7 3 , (1 − η)(1 + η) (1 − η)(1 + η) which share the same volume, circumradius and face areas, √ √ √ √ √ 43 3 , 84 3, 63 3, 63 3, 63 3, 441, 6 according to Lemma 3. Now, Theorem 1 is proven.

3.

5. REFERENCES [1] L.M. Blumenthal, Theory and Applications of Distance Geometry, Chelsea, New York, 1970. [2] Problem 10717 (proposed by M. Mazur), Amer. Math. Monthly 106 (February 1999), 167. [3] Petr Lisonˇek, Robert B. Israel, Metric invariants of tetrahedra via polynomial elimination, Proceedings of the 2000 International Symposium on Symbolic and Algebraic Computation, (C. Traverso, Ed.), pp. 217-219. ACM Press 2000. [4] Lu Yang, Distance coordinates used in geometric constraint solving, in Automated Deduction in Geometry, F. Winkler (Ed.), LNAI 2930, pp. 216-229, Springer-Verlag 2004.

CONCLUSION

A negative answer is presented to an open problem proposed in ISSAC 2000 [3]: for any given positive real constants V, R, A1 , A2 , A3 , A4 , whether or not there are at most finitely many tetrahedra, all having these values as their volume, circumradius and four face areas, respectively. We construct a family of infinitely many tetrahedra T(x,y) which all share the same volume, circumradius and four face areas, whenever (x, y) ranges in a one-dimensional manifold. Our example, however, is in the case of A2 = A3 = A4 . We conjecture that, for given six positive constants V, R, A1 , A2 , A3 , A4 where A1 , A2 , A3 , A4 are pairwise distinct, there are at most nine tetrahedra, all having these values as their volume, circumradius and four face areas, respectively. What we discussed in this paper can be regarded as a generalized problem on geometric constraint solving that involves not only lengths or angles, but also areas, volumes and circumradius. See [4] for a recent approach to geometric constraint solving with distance geometry.

364

Admissible Orderings and Finiteness Criteria for Differential Standard Bases Aleksey Zobnin

∗

Department of Mechanics and Mathematics Moscow State University Moscow, Russia, 119992

al [email protected] ABSTRACT

finitely generated differential ideals is still open. It is completely solved in several important cases [20]: for radicals of ideals [1], for isobaric ideals, and for ideals with known standard bases. However, the latter may be infinite. This fact froze their studies for a long time. It was supposed that the ideal [x2 ] has no finite standard basis. But recently the author proved [21, 20] that finite bases of [xp ] do exist w.r.t. some orderings. These orderings were not considered by Carr` a Ferro and Ollivier. Thus, before tackling the finiteness of differential standard bases, it is natural to describe all admissible orderings on differential monomials. To every such ordering we attach a canonical set of matrices or an infinite matrix. Then we define additional properties of the orderings and study the relations between them. This allows us to formulate necessary conditions and sufficient conditions for an ideal to have a finite differential standard basis. Unfortunately, we cannot propose any criterion for determining this finiteness. But, in contrast to the previous research [2, 3, 13], we present a process that computes a reduced differential standard basis and always terminates if this basis is finite.

We prove that any admissible ordering on ordinary differential monomials in one differential indeterminate can be specified by a canonical set of matrices. The relations between some classes of these orderings are studied. We give criteria of finiteness of differential standard bases and propose an algorithm that computes such bases if they are finite.

Categories and Subject Descriptors I.1.2 [Symbolic and Algebraic Manipulation]: Algebraic algorithms

General Terms Algorithms, Design

Keywords Differential algebra, admissible orderings, Gr¨ obner bases

1.

INTRODUCTION

Gr¨ obner bases of ideals in polynomial rings play an important role in computational commutative algebra. There are a lot of generalizations of Gr¨ obner bases to other types of algebraic structures. We consider such an extension to rings of differential polynomials. The first attempts to do this were made independently by G. Carr` a Ferro [3] in 1987 (differential Gr¨ obner bases1 ) and by F. Ollivier [13] in 1990 (standard bases). These two approaches coincide in general. As well as in the algebraic case, these bases, if they are finite, allow to test the membership to given differential ideals. Nowadays the membership problem posed for

2.

PRELIMINARIES

An ordinary differential ring R is a commutative ring with a derivative operator δ, i.e., a linear operator that satisfies the product rule: δ(ab) = δa b + a δb. We set Θ := {δ k : k ≥ 0}. An ideal I of R is called differential iff δI ⊂ I. If F ⊂ R then [F ] denotes the differential ideal generated by F , i.e., the smallest differential ideal containing F . Let F be a field of constants, i.e., a differential field such that δF = 0. We assume that char F = 0. Let x be a differential indeterminate. Consider the differential ring F{x} := F [x, δx, δ 2 x, . . .], which is a polynomial ring in an infinite family of differential variables. It is called a ring of differential polynomials over F . Following Ritt [14], we denote a differential variable δ i x by xi . Any differential polynomial is a finite sum of differential terms, i.e., expressions of the form c Q · M , where c ∈ F and M is a differential i monomial 2 , M = ki=0 xα i , αi ≥ 0. We shall often omit the word “differential” for short. As a rule, we use capital letters for monomials and the Q letters f, g, h for polynomials. i Consider a monomial M = ki=0 xα i . The degree and the Pk weight of M are defined as usual: deg M := i=0 αi and

∗The work was partially supported by the Russian Foundation for Basic Research, project No. 05-01-00671. 1 Since the notion “differential Gr¨ obner basis” was redefined by E. Mansfield in the other sense [12], we write “differential standard basis” instead.

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2 Some scientists use the converse terminology for monomials and terms. We follow the notation of the book [4].

365

P wt M := ki=0 iαi . A polynomial f is called isobaric if all monomials in f have the same weight. Let M denote the set of all differential monomials. An admissible ordering on M is a total ordering ≺ satisfying the following axioms3 : 1. M ≺ N =⇒ M P ≺ N P 2. 1  P

Proposition 1. If monomial matrices Mk−1 and Mk are concordant then the corresponding orderings ≺k−1 and ≺k are concordant too. ¯ = R ∪ {∞}. We assume that x < ∞ for all x ∈ R. Put R

∀ M, N, P ∈ M;

Definition 3. Let U ⊂ Rn , U 6= ∅. A lexicographic supremum of U is the vector  1 v  .  ¯n v =  ..  := lex sup u ∈ R u∈U vn

∀ P ∈ M;

3. xi ≺ xj ⇐⇒ i < j. The author proved [19] that these properties are sufficient to guarantee that any admissible ordering well orders M. These orderings are rather more general than Weispfenning’s orderings [18]. Fix an admissible ordering ≺. Denote the leading monomial of a polynomial f ∈ / F by lm≺ f . Consider a differential ideal I in F {x}. A set G ⊂ I is called a differential standard basis of I if ΘG is an algebraic Gr¨ obner basis of I in F [x0 , x1 , . . .] (possibly, infinite). A differential standard basis is reduced if every g ∈ G is reduced w.r.t. Θ (G \ {g}). Let Rt = F [x0 , x1 , . . . , xt ] and I(t) = I ∩ Rt . There exists an integer t0 such that the Krull dimension of the ideal I(t) in Rt is constant for all t ≥ t0 [8, 10]. We refer the reader to [14, 8, 10, 13, 11] for further information on differential algebra and Gr¨ obner bases.

3.

such that v1

=

sup u1 ; u∈U

v2

=

 0,

if v 1 > u1 for all u ∈ U, sup u2 , otherwise;



u∈U : v 1 =u1

...... vn

=

 0,        

if (v 1 , . . . , v n−1 ) lex (u1 , . . . , un−1 ) for all u ∈ U, sup un , otherwise.

u∈U : (v 1 ,...,v n−1 )=(u1 ,...,un−1 )

Clearly, vk ≺lex w implies lex sup vk lex w ∀ w, vk ∈ Rn .

REPRESENTATION OF ORDERINGS

k

Let V = {x0 , x1 , x2 , . . .} and Vk = {x0 , x1 , . . . , xk }. Denote by Mk the set of all monomials in variables from Vk . Let us fix an admissible ordering ≺k on Mk for all k ≥ 0.

Theorem 1. Any concordant set of admissible orderings {≺k } such that xk ≺k+1 xk+1 can be specified by a concordant set of matrices.

Definition 1. The set of orderings {≺k } is called concordant if the restriction of ≺k+1 to Mk coincides with ≺k for all k ≥ 0.

Proof. It suffices to prove that for any k ≥ 1 it is possible to construct a matrix Mk specifying ≺k from the matrix Mk−1 using only the following sequential operations:

It is well known [5] that any monomial ordering on Mk can be specified by an m×(k+1) matrix M with real entries and lexicographically positive columns such that KerQ M = {0}:     α0 β0  .   .  αk βk β0 0 . ..  . xα . . . x ≺ x . . . x ⇐⇒ M ≺ M    lex 0 0 k k . αk βk

1. insertion of some column on the right; 2. insertion of the rows (0, . . . , 0, ±1) in some positions. | {z } k zeroes

Since the ordering is admissible, it is sufficient to consider the comparison of monomials of the form P and Qxdk , where P, Q ∈ Mk−1 . Let

We shall call such M a monomial matrix. Conversely, any monomial matrix specifies a monomial ordering. For α 0 . . . xk k we denote by M · P the vector a monomial P = xα
T 0 M α0 . . . αk . Thus, if M specifies ≺ then

v=

lex sup P,Q∈Mk−1 ; d>0:

Mk−1 · P − Mk−1 · Q ¯ k ∈R . d

P ≺k Qxd k

P ≺ Q ⇐⇒ M · P ≺lex M · Q.

The set {P, Q ∈ Mk−1 ; d > 0 : P ≺k Qxdk } is non-empty, since xk−1 ≺k 1 x1k . The same example shows that v is lexicographically positive. Let us construct the matrix Mk as follows. We insert the column v into the matrix Mk−1 on the right. Further, if there is a row in Mk of the form4
ai0 ai1 . . . ai, k−1 ∞ ,

Certainly, the same ordering can be specified by different matrices. Our first goal is to show that for concordant orderings it is possible to choose these matrices also concordantly in some sense. Definition 2. We say that a set of monomial matrices {Mk } is concordant if the matrix Mk−1 can be obtained from Mk by deleting the rightmost column and then by deleting a row of zeroes, if such a row exists. The following fact follows immediately from the definitions.

we replace it by two rows  0 0 ... 0 ai0 ai1 . . . ai, k−1

3 Axioms 1 and 2 are similar to those of monomial orderings in the non-differential case [5, 15].

4 By the definition of lex sup, if such a row exists then it is unique.

366

 1 . 0

Assume that after these operations KerQ Mk 6= {0}. Find two monomials P, Q ∈ Mk−1 and an integer d > 0 such that Mk · P = Mk · Qxdk . Let ( 1, if P ≺k Qxdk , a= −1, otherwise.

Theorem 2. Any admissible ordering on differential monomials can be specified by a concordant set of monomial matrices or, equivalently, by an infinite monomial matrix. Remark 1. By an infinite monomial matrix we mean an ordered system of infinite rows with real elements such that for any k ≥ 0 the first k + 1 columns contain only finitely many non-zero rows. Moreover, this finite set of rows must form a monomial matrix Mk specifying ≺k . Then {Mk } is a concordant set of matrices. We emphasize that the ordinal type of the ordered system of rows in infinite monomial matrix may differ from the ordinal types of N and Z. Of course, any concordant set of monomial matrices, as well as the corresponding infinite matrix, specify an admissible ordering.

We claim that a is independent of the choice of P , Q and d. Indeed, assume that P ≺k Qxdk . Let R k Sxek and Mk ·R = Mk · Sxek for some R, S ∈ Mk−1 , e > 0. Since Mk · P − Mk · Q = d Mk · xk , Mk · R − Mk · S = e Mk · xk , and P, Q, R, S ∈ Mk−1 , we have e (Mk−1 · P − Mk−1 · Q) = d (Mk−1 · R − Mk−1 · S), whence P e S d = Qe Rd . But this contradicts to e

d

e

P S ≺k Q

S d xde k

e

Note that the matrix specifying ≺0 is trivial: we may assume that M0 = ( 1 ). The inductive process of construction of matrices Mk is determinate. Thus the concordant set of matrices Mk obtained in the proof of Theorem 1 can be called canonical. We also call the corresponding infinite monomial matrix the canonical matrix specifying the ordering ≺ on differential monomials.

d

≺k Q R .

Thus a is defined correctly. We insert the row
0 0 ... 0 a at the bottom of Mk . We claim that Mk has the required properties. In fact, it is concordant with Mk−1 by construction. All columns in Mk are lexicographically positive. If a vector w ∈ Q k+1 is such that w 6= 0 and Mk w = 0 then its last component is non-zero, because KerQ Mk−1 = {0}. Hence the last component of the product Mk w is non-zero by the construction of Mk . Therefore, KerQ Mk = {0}. The restriction of the ordering specified by Mk to Mk−1 coincides with ≺k−1 . Let us show that Mk specifies ≺k . Consider arbitrary monomials R and S in Mk−1 and an integer e > 0. If R ≺k Sxek then

The following examples can be treated as definitions. Example 1. The lexicographic ordering (lex) is specified by the infinite canonical matrix5   ............................  ... 1 . . .   ... 1 . . .    ... 1 . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . .    1 ... . . . 1 ... ...

Mk · R ≺lex Mk · Sxek

Example 2. The degree-lexicographic ordering (deglex) is specified by the canonical matrix   1 1 ... 1 1 1 1 1 ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   ... 1 . . .    . . . 1 . . .  .  ... 1 . . .   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 ... ...

by the construction of the vector v and the matrix Mk . Now let R k Sxek . Take any P, Q ∈ Mk−1 and d > 0 such that P ≺k Qxdk . We have as above e d P e S d ≺k Qe S d xde k ≺k Q R ,

whence e (Mk−1 · P − Mk−1 · Q) ≺lex d (Mk−1 · R − Mk−1 · S). Hence, e v := e

lex sup P,Q∈Mk−1 ; d>0: P

Mk−1 · P − Mk−1 · Q lex d

Example 3. The canonical matrix of the degrevlex (degree-reverse-lexicographic) is   1 1 1 ... 1 1 ...  1 1 . . . 1 1 . . .   1 . . . 1 1 . . .    . . . . . . . . . . . . . . . . . . . . . . . . .  . . . 1 1 . . .    ... 1 . . .

≺k Qxd k

lex (Mk−1 · R − Mk−1 · S). Therefore, by the construction of Mk , Mk · R = Mk−1 · R lex Mk−1 · S + ev = Mk · Sxek . This inequality is strict, since KerQ Mk = {0}. Thus we have shown that Mk specifies ≺k and is concordant with Mk−1 .

ordering

........................ To obtain the canonical matrix of wt-revlex (the weightreverse-lexicographic ordering), one needs to replace the first row by the row
1 2 3 4 ... .

We apply the obtained result to representation of admissible orderings on the set of differential monomials. The proof of Theorem 1 shows that every admissible ordering is uniquely determined by a system of vectors {vk }, where ¯ k .But it is more convenient to work with the k ≥ 1, vk ∈ R concordant set of monomial matrices {Mk } instead of {vk }.

5

367

Hereby empty entries stand for zeroes.

Definition 5. An admissible ordering ≺ satisfying these equivalent conditions is called δ-lexicographic.

Example 4. Consider the ordering ≺ defined as M ≺ N ⇐⇒ (wt M, deg M, degx0 M + degx1 M, M ) ≺lex

Example 6. The orderings lex and deglex are δlexicographic. No β-ordering (i.e., an ordering such that lm δM = lmdegrevlex δM for all monomials M ) is δlexicographic.

≺lex (wt N, deg N, degx0 N + degx1 N, N ). It can be specified by the infinite matrix   0 1 2 3 4 5 ... 1 1 1 1 1 1 . . .   . . . 1 1   . . . . . . . . . . . . . . . . . . . . . . ,   1 . . .    1 . . . 1 ...

Proposition 3. Any strictly δ-stable ordering is δlexicographic. Proof. Suppose that ≺ is strictly δ-stable and there exist i and j, i ≤ j, such that xi xj  xi−1 xj+1 . Let us differentiate the members of this inequality. From strict δ-stability it follows that

which is not canonical. The canonical matrix of ≺ is   0 1 2 3 4 5 ... 1 0 −1 −2 −3 −4 . . .   0 0 −1 −1 −1 −1 . . .   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   1 . . .    1 . . . 1 ...

lm≺ (xi+1 xj + xi xj+1 )  lm≺ (xi xj+1 + xi−1 xj+2 )  xi xj+1 , whence xi+1 xj  xi xj+1 . After repeating this operation j − i + 1 times we get a contradiction: xj+1 xj  xj xj+1 . Proposition 4. If ≺ is a δ-lexicographic ordering then lm≺ δ n M = lmlex δ n M for all monomials M 6= 1 and all n ≥ 1. Proof. It suffices to prove that if monomials P and Q occur in δM and both inequalities P  Q and P lex Q hold then lm≺ δP  lm≺ δQ. Let xa and xb be the maximal variables occurring in P and Q, respectively. From P lex Q we conclude that a ≥ b. Since ≺ is δ-lexicographic, x x lm≺ δP = P a+1 and lm≺ δQ = Q b+1 . By Proposition 2, xa xb xb xa+1  xa xb+1 . Hence Qxa xb+1  Qxb xa+1 ≺ P xb xa+1 , whence lm≺ δQ ≺ lm≺ δP .

The equivalent concordant set of matrices is     0 1 2 3   0 1 2  1 0 −1 −2
0 1  1 , ,  1 0 −1 ,   0 0 −1 −1 , . . . . 1 0 0 0 −1 0 0 0 1

4.

Proposition 5. Let {Mk } be any concordant set of matrices specifying ≺ and Mk,j−1 denote the jth column of Mk . For ≺ to be δ-lexicographic, it is necessary and sufficient that for any k the sequence {Mk,j+1 − Mk,j } lexicographically strictly increases w.r.t. j. Proof. Rewrite the statement of Proposition 2 in terms of concordant set of matrices:

SOME CLASSES OF ORDERINGS

Definition 4. (Zobnin [21, 20]). An admissible ordering ≺ is called δ-stable (respectively, strictly δ-stable6 ) if for all non-trivial monomials M and N it follows from M ≺ N that lm≺ δM  lm≺ δN (respectively, lm≺ δM ≺ lm≺ δN ). Example 5. The orderings lex and deglex are strictly δstable. The orderings degrevlex and wt-revlex are δ-stable, but not strictly δ-stable, since x2i  xi−1 xi+1 , but lm δx2i = lm δxi−1 xi+1 .

≺ is δ-lexicographic ⇐⇒ xi xj ≺ xi−1 xj+1 for all 1 ≤ i ≤ j ⇐⇒ Mk · xi − M· xi−1 ≺lex Mk · xj+1 − Mk · xj ⇐⇒

Proposition 2. For an admissible ordering ≺ the following are equivalent:

Mk,i − Mk,i−1 ≺lex Mk,j+1 − Mk,j for all 1 ≤ i ≤ j and for all corresponding k.

1. lm≺ δM = lmlex δM for all monomials M 6= 1;

Corollary 1. Let an ordering ≺ be specified by an infinite matrix with the explicit first row (a0 , a1 , . . . , ak , . . .). If the sequence {ak+1 − ak } strictly increases then ≺ is δlexicographic.

2. xi xj ≺ xi−1 xj+1 for all 0 < i ≤ j, i.e., ≺ is lexicographic on isobaric monomials of degree 2; 3. all monomials occurring in δM are compared lexicographically for any non-trivial monomial M . Proof. 1 ⇒ 2. Let 0 < i ≤ j. Consider the monomial xi−1 xj . Its derivative is equal to xi−1 xj+1 + xi xj . We have

In terms of the canonical matrix strict δ-stability means that the differences of the columns do not increase very fast. Let the condition of the previous corollary be satisfied. Then strict δ-stability implies that after some leading zeroes, the first row forms an arithmetic progression7 .

lm≺ δxi−1 xj = lmlex δxi−1 xj = xi−1 xj+1  xi xj . 2 ⇒ 3. All monomials occurring in δM are of the form x M a+1 , where xa occurs in M . Let us compare two such xa monomials. Assume that i ≤ j. Then xi+1 xj+1 M ≺M ⇐⇒ xi+1 xj ≺ xi xj+1 . xi xj

Example 7. Let ≺ be specified by the canonical matrix   1 2 4 8 . . . 2k . . .  . . . . . . . . . . . . . . . . . . . . . . . . . .     1   ...  .   1     1 1 7 See http: //shade.msu.ru/˜difalg/DSB for the proof.

The last inequality is true by the assumption. Thus, all monomials in δM are compared lexicographically. 3 ⇒ 1. Evident. 6

Strict δ-stability was actually required by Ollivier [13].

368

m Q

i=0

αi 2i =

n P

β

i < j, the stability order of Mi and Mj w.r.t. ≺. Let k0 = max kij and N = maxi lm≺ δ k0 Mi . We may assume

βj 2j or these sums are equal and M ≺lex N .

that N = lm≺ δ k0 M1 . From Proposition 6 it follows that all monomials lm≺ δ k0 Mi are different. Then lm≺ δ k M1  lm≺ δ k Mj for j > 1 and k ≥ k0 . Therefore lm≺ δ k M1 occurs in δ k f and, moreover, is the leading monomial in δ k f . Let 1 xr be the maximal variable in M1 and M = M . Since ≺ xr k k is δ-lexicographic, lm≺ δ M1 = lmlex δ M1 = M xr+k for all k ≥ k0 .

i xα ≺ N = i

i=0 m P

n Q

xj j iff either

In other words, M =

j=0

i,j

j=0

By Corollary 1, this ordering is δ-lexicographic, but it is not n strictly δ-stable. Indeed, for n ≥ 1 we have x20 +1 xn  xn+1 , n since 2n + 1 + 2n > 2n+1 . But x20 +1 xn+1 ≺ xn+2 , because n n+1 n+2 2 +1+2 <2 . Proposition 6. Let ≺ be δ-lexicographic and M ≺ N be non-trivial monomials. Then either

Example 8. One can prove8 that the ordering ≺ described in Example 4 is δ-fixed. But it is not δ-lexicographic, since x21  x0 x2 .

lm≺ δ k M ≺ lm≺ δ k N for all k ≥ 1, or

Definition 8. Let ≺ be an admissible ordering. A polynomial f ∈ F{x} is ≺-quasi-linear if either f ∈ F or deg lm≺ f = 1. If the derivative of any ≺-quasi-linear polynomial is quasi-linear too, we say that ≺ is concordant with quasi-linearity.

lm≺ δ k M ≺ lm≺ δ k N for all 1 ≤ k < k0 and lm≺ δ k M  lm≺ δ k N for all k ≥ k0 . In other words, δ-lexicographic orderings become strictly δ-stable after taking some derivations of monomials M and N . Moreover, the sign of inequality between lm≺ δ k M and lm≺ δ k N can change at most once.

Proposition 8. Any δ-stable and also any δlexicographic orderings are concordant with quasi-linearity. Proof. If ≺ is δ-stable then xn  M implies that xn+1  lm≺ δM . The equality may appear only if deg M = 1. Then M = xk , where n > k. Thus, xn+1  xk+1 . Assume that ≺ is δ-lexicographic and xn  M , but xn+1  lm≺ δM = x , where xa is the maximal variable in M . It is clear M a+1 xa that a < n. Then xa xn+1  M xa+1 ≺ xa+1 xn contradicts Proposition 2.

Proof. Assume the contrary. First of all, note that lm≺ δ k M = lm≺ δ k N implies M = N , since ≺ is δlexicographic. Moreover, lm≺ δ i (lm≺ δ j M ) = lm≺ δ i+j M . Redefining M and N if necessary, we may suppose that M ≺ N,

(1)

lm≺ δM  lm≺ δN,

(2)

Example 9. Let the canonical infinite matrix of ≺ be like   1 2 3 4 5 6 7 8 ...  1 3 5 7 . . . . .............................

..., lm≺ δ k−1 M  lm≺ δ k−1 N,

(3)

lm≺ δ k M ≺ lm≺ δ k N.

(4)

x

Then the stability order of x2 and x1 w.r.t. ≺ is ∞. In fact, one can check that lm≺ δ 2k x2 ≺ x2k+1 , but lm≺ δ 2k+1 x2  x2k+2 for all k ≥ 0. This ordering is neither δ-fixed, nor concordant with quasi-linearity.

x

and lm≺ δN = N j+1 . InequaliLet lm≺ δM = M i+1 xi xj ties (1) and (2) imply that xj xi+1  xj+1 xi . From Propox sition 2 it follows that i > j. Further, lm≺ δ k M = M i+k , xi and similarly for N . From (3) and (4) we conclude that

Proposition 9. For any admissible ordering ≺ every ≺-quasi-linear polynomial is lexicographically quasi-linear. Proof. In fact, let lm≺ f = xm . Suppose that f contains a monomial that is lexicographically greater than xm . Then this monomial is either of the form M xn , where n > m, or M xdm , where d > 1. In any case this leads to a contradiction, since xm ≺ xn and xm ≺ xdm .

M xi+k−1 xj  N xj+k−1 xi and M xi+k xj ≺ N xj+k xi , whence xj+k−1 xi+k ≺ xj+k xi+k−1 , i.e., i < j. Contradiction. Definition 6. The stability order of two monomials M and N w.r.t. ≺ is either the minimal number k0 ≥ 0 such that all inequalities between lm≺ δ k M and lm≺ δ k N have the same sign for all k ≥ k0 , or ∞, if there is no such number.

Proposition 10. Let ≺ be an admissible ordering. Every ≺-quasi-linear polynomial is linear iff xi ≺ x20 for all i. In terms of matrices this statement turns into the following: quasi-linearity coincides with linearity iff in all matrices in the canonical set every column is lexicographically less than the doubled first column. Note that in this case ≺ is trivially concordant with quasi-linearity. Proof. ⇒. Assume that every ≺-quasi-linear polynomial is linear. Then the polynomial xi + x20 is not quasilinear. Hence, xi ≺ x20 . ⇐. Consider a ≺-quasi-linear polynomial f . Let lm≺ f = xk . Suppose that f contains a non-linear monomial M . Then M is divisible by some product xi xj . Thus, xk  M  xi xj  x20 . Contradiction.

The last proposition shows that the stability order of any two monomials w.r.t. a δ-lexicographic ordering is finite. In Example 9 below we construct an ordering and a pair of monomials with infinite stability order. Definition 7. An admissible ordering ≺ is δ-fixed if for any polynomial f ∈ F {x} \ F there exist a monomial M and indices k0 and r such that lm≺ δ k f = M xr+k for all k ≥ k0 . Proposition 7. Any δ-lexicographic ordering is δ-fixed. Proof. Take any f ∈ F {x} \ F and let M1 , . . . , Mn be the different monomials occurring in f . Denote by kij ,

8

369

See http: //shade.msu.ru/˜difalg/DSB for the proof.

5.

FINITENESS CRITERIA FOR DIFFERENTIAL STANDARD BASES

Corollary 4 (generalization of [3, Theorem 1]). Let ≺ be strictly δ-stable. The reduced differential standard basis of [f ] w.r.t. ≺ consists of f itself iff f is ≺-quasi-linear.

Theorem 3 (necessary condition of finiteness). Let ≺ be a δ-fixed admissible ordering. If a proper differential ideal I in F {x} has a finite differential standard basis G w.r.t. ≺ then I contains a ≺-quasi-linear polynomial.

Proof. Necessity. By Remark 2, a ≺-quasi-linear polynomial is contained in any differential standard basis. Hence f is ≺-quasi-linear. Sufficiency. Let lm≺ f = xi . As strictly δ-stable orderings are concordant with quasi-linearity, we have lm≺ δ k f = xi+k . Therefore S-polynomials of all derivatives δ i f and δ j f can be trivially reduced to zero.

This theorem was first proved by G. Carr` a Ferro [3] in the case of pure lexicographic ordering (and also in the case of partial derivatives). Our goal is to extend it to δ-fixed orderings.

n

Remark 5. The polynomial f = x20 +1 xn +xn+1 forms the reduced differential standard basis of [f ] w.r.t. the ordering defined in Example 7. Nevertheless, f is not quasi-linear w.r.t. this ordering, but its derivatives are. This shows that the requirement of strict δ-stability is essential.

Proof. In Section 2 we denoted by Rt the nondifferential ring of polynomials F [x0 , x1 , . . . , xt ]. Recall that the differential dimensional polynomial of I is constant, i.e., there exists t0 > 0 such that for all t ≥ t0 we have dimF Rt /(I ∩ Rt ) = h = const. Then for any (h + 1)-tuple of different variables xi1 , . . . , xih , xih+1 there exists a nonzero polynomial f ∈ I ∩ F[xi1 , . . . , xih , xih+1 ]. Let G = {g1 , . . . , gs }. Since ≺ is δ-fixed, for each gi there exist a monomial Mi and indices ki and ri such that lm≺ δ k gi = Mi xri +k for all k ≥ ki , where 1 ≤ i ≤ s. Let a variable xt be lexicographically greater than each Mi . Take a non-zero polynomial f ∈ I in variables xt , . . . , xt+h . Its leading monomial depends on these variables too. Hence lm≺ f must be divisible by the leading monomial of some derivative of an element of G. Therefore it must be divisible by some Mi . As all variables in Mi are smaller than xt , we have Mi = 1. Thus δ ki gi is ≺-quasi-linear.

Corollary 5. Let ≺ be a δ-lexicographic ordering and I be a proper differential ideal of F{x}. Then I has a finite differential standard basis w.r.t. ≺ if and only if I contains a ≺-quasi-linear polynomial. Proof. Follows immediately from Theorems 3 and 4, since δ-lexicographic orderings are both δ-fixed and concordant with quasi-linearity. The following fact follows from Proposition 9 and Theorems 3 and 4. It shows the key role of the pure lexicographic ordering. Theorem 5. If an ideal has a finite differential standard basis w.r.t. a δ-fixed ordering then it also has a finite differential standard basis w.r.t. the lexicographic ordering.

k

Remark 2. If ≺ is δ-stable then lm≺ δ g = xr+k implies that lm≺ g = xr . In this case a ≺-quasi-linear polynomial is contained in G.

Theorem 4 does not give any practical method to construct a finite basis, because the basis of I ∩ Ri−1 is unknown. Fortunately, we may use the following criteria:

Corollary 2 (generalization of [3, Theorem 1]). Let ≺ be δ-fixed. If the degree of each monomial in f1 , . . . , fn is greater than one then [f1 , . . . , fn ] has no finite differential standard basis w.r.t. ≺.

Lemma 1. Let F ⊂ F{x} satisfy the following conditions: 1. F is an algebraic Gr¨ obner basis of the ideal (F ) in the non-differential ring F [x0 , x1 , x2 , . . .] w.r.t. an admissible ordering ≺;

Remark 3. The requirement that ≺ is δ-fixed is essential. In fact, the ideal [xp ] has the finite differential standard bases {xp } w.r.t. any β-ordering for all p ≥ 1. See [21] for the proof of this fact.

2. ∀ f ∈ F

δf ∈ (F ).

Then F is both differential standard basis and algebraic Gr¨ obner basis of [F ] w.r.t. ≺.

Theorem 4 (sufficient condition of finiteness). Let ≺ be concordant with quasi-linearity. If a proper differential ideal I in F {x} contains a ≺-quasi-linear polynomial g then I has a finite differential standard basis w.r.t. ≺.

Proof. Since δF ⊂ (F ), from h ∈ (F ) it follows that δh ∈ (F ∪ δF ) = (F ). Hence [F ] = (F ). Thus F is an algebraic Gr¨ obner basis of [F ], and, the more so, a differential standard basis.

Proof. Let lm≺ g = xi . Since ≺ is concordant with quasi-linearity, lm≺ δ k g = xi+k . This means that every polynomial in I can be differentially reduced to a polynomial in Ri−1 w.r.t. g. Therefore a differential standard basis of I consists of g and algebraic Gr¨ obner basis of the ideal I ∩ Ri−1 . Evidently, it is finite.

Proposition 11. Let ≺ be concordant with quasilinearity, g be a ≺-quasi-linear polynomial and lm≺ g = xi . Let G ⊂ Ri−1 be a finite set of polynomials such that G is an algebraic Gr¨ obner basis of (G) and δG ⊂ (G, g). Then G ∪ {g} is a differential standard basis of [G, g] w.r.t. ≺. Proof. It suffices to apply Lemma 1 to the set F = G ∪ {δ k g, k ≥ 0}. The conditions of the lemma are satisfied. Indeed, the S-polynomials involving δ k g reduce to zero w.r.t. F , since the leading monomial lm≺ δ k g = xi+k does not appear in other leading monomials of F . All the other S-polynomials reduce to zero, since G is a Gr¨ obner basis of (G). The second condition of the lemma can be verified similarly.

Remark 4. The proof of Theorem 4 is also valid for orderings that are concordant with quasi-linearity at least for a given element g ∈ I. ` Ferro [3], Ollivier [13]). Corollary 3 (Carra If an ideal contains a linear polynomial then it has a finite differential standard basis w.r.t. any admissible ordering.

370

Consider the following completion process. The algorithm Diff Complete (G, r) returns the set of all elements of G and their derivatives that are in Rr . The Algorithm DiffAutoreduce (G, ≺) reduces every element g in G w.r.t. the elements of G \ {g} and their derivatives until the set G is differentially autoreduced.

Carr` a Ferro [2] denoted this ideal (G) by D(Is ). Having thrown away all non-trivial derivatives of elements from D(Is ), she got the so-called s-∆-bases of I. They can be viewed as approximations of finite or infinite differential standard bases. However, this construction is not sufficient for satisfactory approximations. One can check that s-∆-bases of ideals in the following Example 10 do not become stationary. To overcome this disadvantage, we use differential autoreduction (Algorithm DiffAutoreduce) that allows to obtain either reduced s-∆-bases or the reduced differential standard basis if it is finite. We have implemented9 improved Ollivier process in the computer algebra system Maple. The source code of this implementation can be downloaded from http://shade.msu.ru/˜difalg/DSB/Algorithms.

Improved Ollivier Process Input: F ⊂ F {x}, a finite set of polynomials; ≺, a δ-fixed admissible ordering that is concordant with quasi-linearity. Output: Reduced differential standard basis of [F ] if it is finite. Otherwise the process does not stop. G := F ; H := ∅; s := maxf ∈F ord f ; k := 0; repeat OldG := ∅; while G 6= OldG do H := Diff Complete (G, s + k); OldG := G; G := ReducedGr¨ obnerBasis (H, ≺); end do; k := k + 1; until G ⊂ F or G contains a quasi-linear polynomial; return DiffAutoreduce (G, ≺);

Example 10. The reduced lexicographic differential standard bases of the ideals Jn = [xn 1 + x0 ], n ≥ 3, are finite and consist of the following elements (that may be computed explicitly by our process for every n ≥ 3): 1. f = xn 1 + x0 ; 2. ga = n x0 x2 − x21 ; 3. gb = n xn−2 x22 + x2 = x2 (n x1n−2 x2 + 1); 1 4. gc = n(n − 2) xn−3 x32 − x3 . 1 Proof. First of all, we prove that these polynomials are in Jn . One can check it directly for n = 3 (for example, using our process) and verify that for all n > 3

Theorem 6. The improved Ollivier process stops iff [F ] has a finite differential standard basis w.r.t. the specified concordant with quasi-linearity and δ-fixed ordering ≺. In this case it returns the reduced differential standard basis of [F ].

ga = n x2 f − x1 δf , gb =

Proof. The sets G in the inner loop of the process generate an ascending chain of ideals in the Noetherian ring Rs+k . This chain becomes stationary, hence the inner loop terminates. By Theorems 3 and 4, the outer loop terminates iff [F ] has a finite standard basis w.r.t. ≺. The correctness follows from Proposition 11. Indeed, let g be a ≺-quasi-linear polynomial with the leading monomial xi that appears at the kth iteration of the inner loop. The set G is the reduced Gr¨ obner basis, therefore it does not contain other polynomials that depend on xi , xi+1 , . . .. Hence, G \ {g} ⊂ Ri−1 . Moreover, after the termination of the inner loop all first order derivatives of the elements of G \ {g} are in (G). Thus G is the reduced differential standard basis of I.

n xn−3 1

(x1 x3 + (n − 1) x22 ) δf − (n xn−2 x2 + 1) δ 2 f , 1

and gc = + n(3n − 2)x1n−3 x2 x3 +
+3n (n − + n(n − 1)(n + 2)xn−4 x32 · δf − 1
− n(3n − 2)xn−3 x22 + 3n2 (n − 1)x2n−5 x32 · δ 2 f − 1 1

=

nxn−2 x4 1 2

2

+ 3n (n −

1)2 x2n−6 x42 1

1)x2n−5 x22 x3 1

−(nx1n−2 x2 + 1) · δ 3 f . By Theorem 4, every Jn has a finite differential standard basis, because Jn contains a lexicographically quasi-linear polynomial gc . Consider the set G = {f, δf, ga , gb }. It is easy to prove that G and gc satisfy the conditions of Proposition 11. Thus, the polynomials f, ga , gb and gc form a differential standard basis of [G, gc ] = Jn . Moreover, this basis is reduced.

This process synthesizes the ideas of Carr` a Ferro [2] and Ollivier [13]. The original Ollivier process may not stop even if an ideal has a finite differential standard basis. Our Proposition 11 that allows to construct finite bases algorithmically is the generalization of Ollivier’s Criterion 3 for discarding useless reductions. However, Ollivier formulated his criterion only theoretically, and he did not claim that his criteria are sufficient for determining a finite basis. Furthermore, as it follows from the previous theorem, there is no need to check it explicitly in our process. Let F be a finite set of polynomials and I = [F ]. Following Carr` a Ferro, we consider ideals I(s) = I ∩ Rs and

Remark 6. One can prove (see, for example, [7]), that the ideals Jn are radical for n ≥ 3. At the same time [7, 20] the ideal J2 is not radical and its lexicographic differential standard bases are infinite, but its radical contains quasi-linear polynomials. Unfortunately, nobody knows an algorithm to determine whether there is a quasi-linear polynomial in I. Finally, we turn to the finiteness of differential standard bases w.r.t. orderings like degrevlex and wt-revlex. One can prove10 the sufficiency part of the following conjecture. However, the necessity is still an open problem.

Is = (δ k fi : fi ∈ F, δ k fi ∈ Rs ).

9 We used standard tools of the diffalg package [1, 6]. See http: //www-sop.inria.fr/cafe/Evelyne.Hubert/diffalg. 10 Visit http: //shade.msu.ru/˜difalg/DSB.

Clearly, Is ⊆ I(s). It is easy to see that after the kth iteration of the inner loop in our process we have Is ⊆ (G) ⊆ I(s).

371

[13] F. Ollivier. Standard bases of differential ideals. Lecture Notes in Computer Science, 508 : 304–321, 1990. [14] J. F. Ritt. Differential Algebra, vol. XXXIII of Colloquium Publications. New York, 1950. [15] L. Robbiano. On the Theory of Graded Structures. The Journal of Symbolic Computation, 2 : 139-170, 1986. [16] C. J. Rust and G. J. Reid. Rankings of Partial Derivatives. In Proceedings of the ISSAC-1997, pages 9–16, 1997. [17] C. J. Rust. Rankings o Derivatives for Elimination Algorithms and Formal Solvability of Analytic Partial Differential Equations. Ph.D. Thesis, University of Chicago, 1998. [18] V. Weispfenning. Differential term-orders. In Proceedings of ISSAC, pages 245–253, 1993. [19] A. Zobnin. Essential properties of admissible orderings and rankings. In Contributions to General Algebra, volume 14, pages 205–221, 2004. Available at http://shade.msu.ru/˜al zobnin/publications.htm. [20] A. Zobnin. On testing the membership to differential ideals. In Proceedings of CASC-2004, pages 485–496. Technische Universit¨ at M¨ unchen, 2004. Available at http://shade.msu.ru/˜al zobnin/publications.htm. [21] A. Zobnin. On standard bases in differential polynomial rings. To be published in Journal of Mathematical Sciences, 2005.

Conjecture 1 (M. V. Kondratieva, A. Zobnin). A proper ideal I has a finite differential standard basis w.r.t. a concordant with quasi-linearity β-ordering ≺ iff either I contains a ≺-quasi-linear polynomial, or I = [f p ], where f is ≺-quasi-linear and p ≥ 1.

6.

CONCLUSIONS

We proved that any admissible ordering on the set of ordinary differential monomials in one differential indeterminate can be specified by the canonical set of monomial matrices, or, equivalently, by an infinite matrix of special form. We studied some classes of such orderings and established the dependencies between them (see Appendix). Using our theory we established general criteria of finiteness of differential standard bases. We concluded that the finiteness of δ-fixed bases implies the finiteness of lexicographic ones. Finally, we introduced improved Ollivier process that computes suitable approximations of differential standard bases and stops iff the reduced basis is finite.

7.

ACKNOWLEDGEMENTS

The author is greatly thankful to M. V. Kondratieva, E. V. Pankratiev, O. Golubitsky, and A. Ovchinnikov for their assistance and valuable comments, as well as to anonymous referees for their reviews.

8.

REFERENCES APPENDIX

[1] F. Boulier, D. Lazard, F. Ollivier, and M. Petitot. Computing Representations for Radicals of Finitely Generated Differential Ideals. Technical Report IT-306, LIFL. 1997. [2] G. Carr` a Ferro. Gr¨ obner bases and differential algebra. Lecture Notes in Computer Science, 356 : 129–140, 1989. [3] G. Carr` a Ferro. Differential Gr¨ obner bases in one variable and in the partial case. Mathematical and Computer Modelling, 25(8–9) : 1–10, 1997. [4] D. Cox, J. Little, D. O’Shea. Ideals, Varieties and Algorithms. Springer-Verlag, New York, 1998. [5] H. Hong and V. Weispfenning. Algorithmic theory of admissible term orders. Preprint, 1999. Available at http://www4.ncsu.edu:8030/˜hong/papers/. [6] E. Hubert. Notes on triangular sets and triangulation-decomposition algorithms. I: Polynomial systems. II. Differential Systems. Chapters of Symbolic and Numerical Scientific Computations. Lecture Notes in Computer Science, vol. 2630, 2003. [7] E. R. Kolchin. On the exponents of differential ideals. Annals of Mathematics, 42 : 740–777, 1941. [8] E. R. Kolchin. Differential Algebra and Algebraic Groups. Academic Press, New York, 1973. [9] M. V. Kondratieva. Private communications. 2004–05. [10] M. V. Kondratieva, A. B. Levin, A. V. Mikhalev, and E. V. Pankratiev. Differential and Difference Dimension Polynomials. Kluwer Academic Publisher, 1999. [11] M. Kreuzer and L. Robbiano. Computational Commutative Algebra I. Springer-Verlag, Berlin, 2000. [12] E. Mansfield. Differential Gr¨ obner Bases. 1993.

Relations between some classes of orderings lex, deglex, wt-lex | {z } Strict δ-stability

degrevlex, wt-revlex | {z }

=⇒

δ-stability

⇓ δ-lexicographic orderings

⇓ =⇒

Concordance with quasi-linearity

⇓

⇓

δ-fixed orderings

Sufficiency of criterion

⇓

⇑

Necessity of criterion

Ideals containing linear polynomials z

}|

{

any ordering

372

Author Index Abramov, S. A....................................................5 Adams, J. ..........................................................13 Anai, H. ............................................................21 Aroca, J. M. ......................................................29

Hara, S. .............................................................21 Hitz, M. A.......................................................171 Hovinen, B......................................................177 Huang, F. ........................................................185

Beaumont, J. C. ................................................37 Bostan, A. .........................................................45 Boucher, D........................................................53 Bradford, R. J. ..................................................37 Bretto, A. ..........................................................61 Bronstein, M.....................................................68 Brown, C. W.....................................................76 Buchberger, B.....................................................1

Jeffrey, D. J.....................................................193 Johnson, J. R...................................................200 Kaltofen, E..............................................208, 216 Kauers, M. ......................................................156 Koiran, P.........................................................208 Krandick, W. ..................................................200 Laget, B. ...........................................................61 Li, B................................................................224 Li, Z. .................................................................68 Lichtblau, D. ...................................................231

Cano, J..............................................................29 Carvajal, O. A...................................................84 Chapman, F. W.................................................84 Chen, F. ..........................................................132 Chen, S. ..........................................................185 Chen, Y...........................................................224 Chen, Z. ............................................................92 Cluzeau, T. .......................................................45 Costermans, C. ...............................................100

Mao, W...........................................................237 McCallum, S.....................................................76 Monagan, M. ..........................................124, 245 Moreno Maza, M. ...........................................108 Morozov, D.....................................................216 Mourrain, B. ...................................................253

Dahan, X.........................................................108 Davenport, J. H.................................................37 Dayton, B. H...................................................116 de Kleine, J. ....................................................124 Díaz, S. P........................................................301 Deng, J............................................................132 Dumas, J.-G............................................140, 277

Ngoc Minh, H. ................................................100 Oancea, C. E. ..................................................261 Pavone, J. P.....................................................148 Pérez-Díaz, S. .................................................301 Pernet, C. ........................................................140 Petitot, M. .......................................................100 Petkovšek, M. .....................................................5 Phisanbut, N. ....................................................37 Pratibha...........................................................193

Eberly, W........................................................177 Enjalbert, J. Y.................................................100 Feng, R. ............................................................29

Reid, G............................................................269 Roach, K. B. ...................................................193 Rondepierre, A. ..............................................277 Ruslanov, A. D. ..............................................200

Galligo, A. ......................................................148 Gao, X. S. .........................................................29 Geddes, K. O. ...................................................84 Gerhold, S.......................................................156 Gillibert, L. .......................................................61 Grigoriev, D....................................................163

373

Salvy, B. .......................................................2, 45 Saunders, B. D..................................................13 Schneider, C. ..................................................285 Schost, É.................................................108, 293 Schwarz, F. .....................................................163 Sendra, J. R.....................................................301 Shen, L. ..........................................................132 Storjohann, A............................................92, 309

Wang, Q..........................................................224 Watt, S. M.......................................................261 Weil, J.-A........................................................340 Wen-tsun, W.......................................................4 Wittkopf, A.............................................124, 269 Wu, J...............................................................237 Wu, M...............................................................68 Wu, W.....................................................108, 269

Tournier, L......................................................317 Tsarev, S. P.....................................................325 Turner, W. J....................................................332

Xia, B..............................................................354 Xie, Y. ............................................................108 Yang, L. ..........................................................362 Yokoyama, K....................................................21 Yuhasz, G. ......................................................216

van de Woestijne, C........................................348 van Hoeij, M...................................................340 Verschelde, J. .................................................269 Villard, G........................................................309

Zeng, Z. ..................................................116, 362 Zobnin, A........................................................365

Wan, Z. .....................................................13, 140 Wang, D. ........................................................354

374