Real Algebraic Geometry: Proceedings of the Conference held in Rennes, France, June 24-28, 1991 (Lecture Notes in Mathematics) (English and French Edition)

Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, Ztirich E Takens, Groningen

1524

M. Coste

L. Mah6

M.-F. Roy (Eds.)

Real Algebraic Geometry Proceedings of the Conference held in Rennes, France, June 24-28, 1991

Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo Hong Kong Barcelona Budapest

Editors Michel Coste Louis Mah6 Marie-Fran~oise Roy IRMAR (CNRS, U.R.A. 305) Universit6 de Rennes I Campus Beaulieu F-35402 Rennes Cedex, France

Mathematics Subject Classification (1991 ): Primary: 14P Secondary: 12D15, 32B, 32C, 58A35, 68Q40

ISBN 3-540-55992-2 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-55992-2 Springer-Verlag New York Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1992 Printed in Germany Typesetting: Camera-ready by author/editor 46/3140-543210 - Printed on acid-free paper

PREFACE

The meeting on "Real Algebraic Geometry" was held in La Turballe, on the seashore not far from Rennes, from June 24 to 28, 1991. It took place ten years after the first meeting on "G6omttrie Algtbrique Rtelle et Formes Quadratiques" (*). These Proceedings contain survey papers on some of the developements of real algebraic geometry in the last ten years, and also contributions by the participants. Every paper has been submitted to a referee, and we want to thank all of them for their collaboration. The meeting, and the collaboration between the european teams which made it possible, received support from the Universit6 de Rennes 1, the GDR MathtmatiquesInformatique (CNRS), and the programs Rtseau Europ6en de Laboratoires, Acces, Alliance, Actions Inttgrtes Franco-Espagnoles. We would like also to thank SpringerVerlag for publishing this volume, and to express our gratitude to Ms. Yvette Brunel, for her precious help for the secretary of the meeting. Michel Coste Louis Mah6 Marie-Fran~oise Roy

(*) Lecture Notes in Mathematics 959, Springer (1982)

TABLE OF CONTENTS

Survey papers Knebusch M.: Semialgebraic topology in the last ten years

. . . . . . .

Parimala R.: Algebraic and topological invariants of real adgebraic varieties Polotovskii G.M.: On the classification of decomposing" plaae algebradc curves

1

37 52

Schcidcrcr C.: Read adgebra and its applications to geometry in the last ten years: some m a j o r developments and results

. . . . . . . . . .

Shustin E.I.: Topology of real plane algebraic curves

75

. . . . . . . .

97

Silhol R.: Moduli problems in real dgebraic geometry . . . . . . . .

110

Other papers Akbulut S., King H.: Constructing strange read Mgebraic sets

. . . . .

120

Andradas C., Ruiz J.: More on basic semialgebraic sets . . . . . . . .

128

Borobia, A.: Mirror property for nonsingular m i x e d configurations o f one line and k points in R 3 . . . . . . . . . . . . . . . . . . . .

Br6cker L.: Families of semialgebraic sets and limits

. . . . . . . .

140

145

Brumfiel G.W.: A H o p f flxed point theorem for semi-algebraic m a p s .

163

Brumfiel G.W.: On regular open semi-algebraic sets . . . . . . . . .

170

Castilla, A.: S u m s of 2n - th-powers o f meromorphic functions with compact zero set . . . . . . . . . . . . . . . . . . . . . . . .

174

Charzynski Z., Skibinski P. : T h e pseudoorthogonality o f the coordinates o f a holomorphic m a p p i n g in two variables with constant dacobian

178

Costc M., Reguiat M.: Trivialit4s en famille . . . . . . . . . . . .

193

Degtyamv A.: Complex structure on a r e d dgebraic variety . . . . . .

205

Dcnkowska Z.: Subanaly~icity and Hilbert's 16th problem

221

. . . . . .

Franqois¢ J-P., Ronga F.: T h e decidability of real dgebraic sets by the index formula . . . . . . . . . . . . . . . . . . . . . . . .

Gamboa J.M., Ueno C.: Proper polynomial maps: the real case . . . . .

235 240

Gondard-Cozette D.: Sur les ordres de nivean 2" et sur une extension du 17~me probl~me de Hilbert . . . . . . . . . . . . . . . . .

257

Itenbcrg I.V.: Curves o f degree 6 with one non-degenerate double point and groups o f m o n o d r o m y of non-singular curves

. . . . . . . . . .

267

VIII Jaworski E: The 17th ttilbert problem for noncompact reaJ analytic manifolds

289

Korchagin A.: Construction of new M-curves of 9-th degree . . . . . .

296

Krasinski T., Spodzieja S.: On linear differentiaJ operators related to the ndimensional jacobian conjecture . . . . . . . . . . . . . . .

308

Kurdyka K.: On a subanMytic stratification satisfying a W h i t n e y property with exponent 1 . . . . . . . . . . . . . . . . . . . . .

316

Lombardi H.: Une borne sur les degr~s pour le th~orbme des z~ros r~els effectif 323 Marshall M., Walter L.: Minimal generation of basic sem_./aJgebra/c sets over an arbitrary ordered field . . . . . . . . . . . . . . . . . 346 Mazurovskii V.E: Configurations of at most 6 lines in R P 3

. . . . . .

354

Mikhalldn G.: Extensions of Rokhlin congruence for curves on surfaces .

372

Mostowski T., Rannou E. : Complexit~ de la construction des strafes A multiplicit~ constante d'un ensemble alg~brique de C n . . . . . . 378 Shustin E.I.: Real plane Mgebraic curves with many singularities

. . . .

389

Vorobjov N.: Effective stratification of regular real Mgebra/c varieties

402

List o f participants

416

. . . . . . . . . . . . . . . . . . . .

Semialgebraic

topology

in the last ten years

Manfred Knebusch

Contents §1 Brumfiels program §2 The two approaches §3 The state of art in 1981 §4 Sheaves and homology §5 Locally semiMgebraic spaces §6 Abstract semialgebraic functions and real closed spaces §7 Cohomology with supports §8 Borel-Moore homology §9 Base field extension and comparison theorems §10 Homotopy sets §11 Weakly semialgebraic spaces §12 Generalized homology §13 Novel features of semialgebraic topology: three examples §14 An outlook References

1

2 5 8

10 11 14 16

18 20 22 24 27 31

34

§1 Brumfiel's p r o g r a m Before discussing the subject named in the title it seems appropriate to outline the situation in semialgebraic topology in 1981, at the time of the first Rennes conference on real algebraic geometry. Already in the seventies, in the long introduction to his book "Partially ordered rings and semialgebraic geometry" [B], G.W. Brumfiel had laid down a program for what we now call "semialgebraic topology". Here Brumfiel advocated a new way of handling topological problems which is closer to the spirit of algebraic geometry than traditional topology. Let me just quote the following passage: "It thus seems to me that a true understanding of the relations between algebraic geometry and topology must stem from a deeper understanding of real algebraic geometry, or, actually, semi-algebraic geometry. Moreover, real algebraic geometry should not be studied by attempting to extend classical algebraic geometry to nonalgebraically closed ground fields, nor by regarding the real field as a field with an added structure of a topology. Instead, the abstract algebraic treatment of inequalities originated by Artin and Schreier should be extended from fields to (partially ordered)

algebras, with real closed fields replacing the algebraically closed fields as ground fields" [B, p.2]. In the main body of the book [B] Brumfiel develops a "real algebra" by studying partially ordered commutative rings and various sorts of convex ideals, with the perspective that this real algebra should perform a similar role in semialgebraic geometry as commutative algebra does in present day algebraic geometry. But the book does not go very far into semialgebraic topology. §2 T h e two a p p r o a c h e s Even today not much semialgebraic topology has been done using Brumfiel's rather intricate real algebra from the seventies. Around 1979 two other approaches to semialgebraic topology emerged independently which turned out to be successful. These are the "abstract" approach by M. Coste and M.F. Roy, and the "geometric" approach by H. Delfs and M. Knebusch. Before we get into this let me remind you of what are perhaps the two most serious difficulties which one encounters if one works over a real closed base field R different from R. a) R '~ is totally disconnected in the strong topology (i.e. the topology coming from the ordering of R). b) R '~ has very few reasonable (i.e. geometrically relevant) compact subsets. In particular, the closed unit ball in R n is not compact. Let M be a semialgebraic subset of some R". In the abstract approach one adds to M "ideal points" which turn M into an honest (albeit not Hausdorff) topological space. More precisely, one passes from M to the corresponding constructible subset .£/of the real spectrum Sper R[T1,..., T,~] of the polynomial ring R[T1,..., T,,I (cf. [BCR, Chap. 7]). -~/turns out to have only finitely many connected components, a n d / ~ / i s quasicompact. Thus in some sense the difficulties described above are overcome. The subspace topology of M in M is the strong topology we started with. One could also pass from M to the subspace/17/m~x of closed points of ~/, which still contains M as a dense subset and is a compact Hausdorff space with only finitely many connected components. But although this compactification .~:/m~x of M has its merits (cf. [B1]), the more interesting and more useful space is /~/ itself. The main reason for this is that .~/is a spectral space, as defined by Hochster [Ho], and that the constructible subsets Y of M correspond bijectively with the semialgebraic subsets N of M via the relation Y = N. A very nice consequence of this is that the "semialgebraic structure" of M is encoded to a large extent in the topology of 37/, since the lattice .q(/~/) of constructible subsets of M is by definition the boolean lattice generated by the lattice J~(~¢) of quasicompact open subsets of M, and thus

J~(37/) is completely determined by the topology of 37/ (cf. [Ho]). We call the space the abstraction of the semialgebraic set M. The wisdom of passing back and forth between the semialgebraic sets and their abstractions has been displayed well in the book [BCR] by Bochnak, Coste and Roy. Curiously another very important and fascinating aspect of the abstract approach is scarcely touched on in that book: One can study the constructible subsets of the real spectrum Sper A of any commutative ring A. Thus the abstract approach opens the door for an "abstract" semialgebraic topology where no base field (real closed or not) needs to be present. Coste and Roy were certainly well aware of this aspect at an early stage (cf. for example Roy's paper on abstract Nash functions [R]) but chose not to give much space to this in their book with Bochnak. The geometric approach (cf. [DK]) relies on the following two ideas, the first one being very simple. 1° Don't consider any subset of a semialgebraic set M C R n which is not semialgebraic or any map f: M --~ N between semialgebraic sets which is not semialgebraic! In this context a map f is called semialgebraic if the graph of f is a semialgebraic set and f is continuous with respect to the strong topologies of M and N. 2 ° Install on M a Grothendieck topology such that the semialgebraic functions, i.e., semialgebraic maps to R on the open semialgebraic subsets U of M (open with respect to the strong topology) form a sheaf OM of R-algebras! Instead of studying M as a semialgebraic subset of R n study the ringed space (M, OM)! Let me give some comments and explanations on these ideas. A d 1°: The reason that this idea makes sense is Tarski's principle. It guarantees that many of the usual constructions of new sets and maps from given ones give us semialgebraic sets and maps if we start with such sets and maps. In particular, if f: M --* N is a semialgebraic map between semialgebraic sets then the image f(A) of a semialgebraic subset A of M is semialgebraic and the preimage f - l ( B ) of a semialgebraic subset B of N is semialgebraic. Continuity of f is not necessary for this but is appropriate since we want to do "topology". A d 2°: The Grothendieck topology on M is defined as follows. The underlying category is the category ~ ( M ) of open semialgebraic subsets of M (i.e. semialgebraic subsets which are open in the strong topology), the morphisms being the inclusion mappings. An admissible open covering (Ui[i E I) of some U E ~ ( M ) is a family (Uili E I) in ~ ( M ) with U = U Ui, such that there exists a finite subset J of I with

iEl U = U Ui. {Thus a property similar to quasicompactness is forced to hold.} Then iEJ

4 the semialgebraic functions on the sets U E ~ ( M ) indeed form a sheaf OM. It turns out that a morphism from (M, OM) to (N, ON) is determined by the underlying map f from M to N, and that these maps f are just the semialgebraic maps from M to N as introduced above (of. [DK, §7], by definition the morphism has to respect the R-algebra structures of the structure sheaves). Replacing a semialgebraic set M C R n by the ringed space (M, OM) allows us to forget the embedding M ~ R n. We call any ringed space of R-algebras which is isomorphic to such a space (M, OM) an affine semialgebraic space over R. By abuse of notations we do not distinguish between a semialgebraic set M and the corresponding ringed space (M, OM).

A semialgebraic path in M is a semialgebraic map from the unit interval [0,1] (which is a semialgebraic subset of R 1) to M. Having this notion of paths at hand one defines the path components of M in the obvious way. It turns out, that M has only finitely many path components M 1 , . . . , M r and that these are semialgebraic in M and closed, hence also open in the strong topology, cf. [DK]. Every Mi is "semialgebraically connected", i.e. Mi is not the union of two disjoint non empty open semialgebraic subsets, since this holds for [0,1], as is easily seen. Thus we have dealt with the first difficulty mentioned above, exploiting only idea N ° 1. By the way, the abstractions 2~/1,..., ~/r are the connected components of the topological space In order to cope with the second difficulty one also needs idea N ° 2. The category of affine semialgebraic spaces over R has fiber products. Thus we can define proper morphisms as in algebraic geometry. We call a semialgebraic map f: M --* N closed, if the image f(A) of a closed semialgebraic subset A of M is again closed. We call f proper if f is universally closed, i.e. for any semialgebraic map g: N ~ --* N the cartesian square

M ×N N ~

/ ~

¢1 M

N~

lg f

.~N

gives us a closed semialgebraic map f~. We call an affine semialgebraic space M complete if the map from M to the one-point space is proper. Even more than in algebraic geometry over an algebraically closed field, it is true for many purposes, that complete spaces are the right substitute for compact spaces in topology. For example, a semialgebraic function on a complete space attains its maximum and minimum.

It turns out that there exist in abundance relevant complete affine semialgebraic spaces. Namely, the following analogue of the Heine-Borel theorem holds: A semialgebraic subset M of R n is a complete space iff M is closed and bounded in R n. §3 T h e s t a t e of a r t in 1981 I give a rough sketch of the technical progress up till 1981. This is just to give an impression of the state of art at the first Rennes conference. It is not meant, of course, as a complete account of everything done up to that time. In the geometric theory we have the following list. 1) 2) 3) 4) 5) 6)

Connected components Complete affine semialgebraic spaces and the semialgebraic Heine-Borel theorem Dimension theory Existence of triangulations Hardt's theorem Semialgebraic homology

Here are s~me comments on these. N ° 1 and N ° 2 have been described above. One may add to N ° 2 that in 1981 we also had a good insight into the nature of proper maps between affine semialgebraic spaces [DK, §9 and §12]. A d 3: The dimension d i m M of a semialgebraie set can be defined as the maximal integer d such that M contains a subspace N which is isomorphic to the unit ball in R a {[DK, §8], there a different but equivalent definition had been given}. This notion of dimension behaves very well, better than in classical topology. For example, if a partition of M into finitely many semialgebraic subsets A 1 , . . . , Ar is given, then dim M is the maximum of the numbers dim AI,. •., dim A t . A d 4: If M is an affine semialgebraic space and A 1 , . . . , Ar are finitely many semialgebraic subsets of M then there exists a finite simplicial complex X over/it and an isomorphism of spaces (p: X ~ , M such that, for every i E { 1 , . . . , r}, the set ¢p-l(Ai) is a subcomplex of X [DK, §2]. Here the word "simplicial complex" is used in a non classical meaning: X is the union of finitely many open simplices a l , . . . , ~rt in some R N such that the intersection ~i fl ~j of the closures of any two simplices ¢ri, aj is either a common face of them or empty. Thus the closure )~" of X is a classical finite simpIicial complex (,~ finite polyhedron), and X is obtained from )( by omitting some open faces. Also "subcomplex" means just the union of some of the sets ( r l , . . . , ~rt. Clearly X = 3~ iff M is complete. In the case R = R the triangulation theorem has been well known since the sixties, even for semianalytic sets [L, Gi].

A d 5: Hardt's theorem states that for every semialgebraic map f: M --~ N there exists a partition of N into finitely many semialgebraic subsets N 1 , . . . , Nr such that f is trivial over each Nj, i.e. f-l(Nj) is isomorphic over Nj to a direct (= cartesian) product Nj × Fj, cf [DK1, §6]. The theorem had been proved for R = R by R. Hardt around 1978 [Ha]. A d 6: In his thesis [D] Delfs constructed homology and cohomology groups with arbitrary constant coefficients for affine semialgebraic spaces over any real closed field R. In the case R = R these groups coincide with the singular groups known from classical topology. Certainly Dells' homology theory was the most profound achievement in semialgebraic topology up till 1981. But the proofs of the triangulation theorem and of Hardt's theorem also needed new ideas beyond the known proofs for R = I~. The triangulation theorem is the main technical tool in developing semialgebraic homology (and also semialgebraic homotopy theory, cf. §10 below). Hardt's theorem is very useful if one wants to profit from semialgebraic homology. For a good example, cf. [DK1, §7]. I will say more about semialgebraic homology in the next section §4. R e m a r k . Only recently (1989) I learned from Cert-Martin Greuel about the unpublished dissertation of Helmut Brakhage [Bra] (Heidelberg 1954, thesis advisor F.K. Schmidt). Here Brakhage studies semialgebraic topology over an arbitrary real closed field. He exploits idea N ° 1 of the geometric theory (cf. §2) to an enormous extent and obtains many of the results we had found up to 1981, in particular the triangulation theorem. The introduction to Brakhage's thesis reads very much like the talks Dells and I used to give around 1980. He would have saved us a lot of work if we only would have known about his thesis. Brakhage is now a professor at Kaiserslautern, working mostly in applied mathematics. It is difficult to give a good picture of the state of art in semialgebraic topology in 1981 on the abstract side, since in the abstract theory the main bias was towards algebraic problems. Topology seems to have been studied mainly as an aid for solving algebraic problems of current interest. I give the following list. 1) 2) 3) 4) 5) 6)

Connected components Compactness of constructible sets Specialization theory Dimension theory Abstract Nash functions Separation of connected components by global quadratic forms

Here only N ° 1 - 4 truly belong to semialgebraic topology, but N ° 5 and 6 use topology in an essential way, and have also turned out to be stimulating for semialgebraic topology since 1981.

N ° 1 has been discussed above, N ° 2 alludes to the easily accessible but extremely important fact, that the real spectrum Sper A of any commutative ring A is compact in the constructible topology. This means that, if X is a constructible subset and (Y~[i E I) is a family of constructible subsets of SperA with X C [,.J Y~, then there iEl exists a finite subset J of I with X C U Y~" The quasicompactness of J~/stated iEJ above is a rather special consequence of this. A d 3: If x and y are points of a topological space X then we say that y is a specialization of x (and x is generalization of y) if y lies in the closure of the set {x}. We write x ~- y for this. N ° 3 alludes to some - again simple but important facts about specializations in a real spectrum SperA, cf. [CR2], [BCR, 7.1], [KS, III §3 and §7]. In particular, the specializations of a given point x in Sper A form a chain, i.e. if x ~- y and x ~- z then y :,-- z or z ~- y. Moreover if neither x ~ y nor y ~- x then there exist disjoint open subsets U, V in Sper A with x E U and y E V.

-

A d 4: The dimension dim X of a constructible subset X of Sper A is defined as the supremum of the lengths of the specialization chains in X. {Up till now it has been adequate to put dim X = oo if the lengths do not have a finite bound.} The main result is that, if M is a semialgebraic set over some real closed field, then this "combinatorial" dimension dim J~/ of the abstraction /~¢ coincides with the semialgebraic dimension dim M from above, cf. [CR2], [BCR]. A d 5 a n d 6: One of the most important achievements in the early work of Coste and Roy is the construction of a sheaf of "abstract Nash functions" 91A on the real spectrum of an arbitrary commutative ring A JR], which generalizes the sheaf of classical Nash functions for algebraic manifolds over R. Indeed, right from the beginning they had the idea of constructing the real spectrum as a ringed space (Sper A, 92A) [CR], [CR1], thus bringing semialgebraic geometry close to the spirit of abstract algebraic geometry in the sense of Grothendieck. The sheaf 91A is more algebraic in nature than the sheaf of semialgebraic functions discussed in §2. It does not belong to semialgebraic topology, but nevertheless relies on the topological fact that every etale morphism A ~' ~ B induces a local homeomorphism Sper qo: Sper B Sper A. Building on this, Mah6 was able to solve one of the main open problems of quadratic form theory from the seventies [K1, Problem 16] affirmatively, namely the separation by global quadratic forms of the connected components of the set V(FI) of real points of an affine algebraic variety V, and later, together with Houdebine, also of a projective algebraic variety V over R [M], [HM]. In fact, they prove such a theorem over any real closed field R, and also for the real spectrum of any commutative ring. Mah~'s theorem in [M] is probably the first result which signaled to the outside world

that something new in principle had happened in real algebraic geometry around 1980. §4 S h e a v e s a n d h o m o l o g y After 1981 semialgebraic topology has been dominated by two major new trends: A strong interaction between the geometric and the abstract theory, and the employment of new spaces. An important instance of the first trend is sheaf theory. Let M be a semialgebraic set over some real closed field R. Then a (set valued) sheaf over M is essentially the same object as a sheaf over the abstraction _~/. Indeed, as was already known before 1981 [CR2], [D], [De], a semialgebraic subset U of the affine semialgebraic space M is open iff the abstraction/) is open in/~/. Moreover, a family (Ui[i E I) of open semialgebraic subsets of M is an admissible open covering of U iff (/)i[i E I) is an open covering o f / ) . The reason for this is the definition of the Grothendieck topology on M on the one hand, and the quasicompactness of U on the other. Since the quasicompact open subsets of/~/form a basis of the topology of/iT/, all of this gives us a canonic£1 isomorphism Y H .9~ from the category of sheaves on i to the category of sheaves on Jl~/, via the rule ~-(U) = 5g'(U). Henceforth we only consider sheaves of abelian groups. Recall that M is dense in/~/. For x E M the stalks ~'z and fi'z are equal. It may well happen that all stalks .T'x, x E M, are zero, but .T is not zero. {An example is given in [92, 1.1.7].} This is by no means astonishing: Of course, Y # 0 iff fi" # 0. Then, since ~/ is an honest topological space, there exists some a E M with ~a # 0. But it may happen that none of these points a lies in M. This discussion makes it clear that most often sheaf theoretic techniques work better in the abstract settir~g than the geometric one. Only there one can argue "stalk by stalk" without further justification. Now is a good moment to say something about the semialgebraic homology theory of Hans Dells, sinc e he has been able to simplify his theory greatly by using sheaves and the interplay back and forth between semialgebraic sets and their abstractions [D1]. I first describe the main problem in defining homology groups Hq(M,G) for a semialgebraic set M over some real closed field R and some abelian group of coefficients G. Let us assume for simplicity that M is complete. We choose a triangulation 90: X ~ * M. Here X is a finite simplicial complex in the classical sense but over R; X may be regarded as the realization [K[/~ over R of an abstract finite simplicial complex K, a purely combinatorial object (cf. [Spa, 3.1]; the realization is defined exactly as in the case R = R). It is intuitively clear that Hq(M,G) should coincide, up to isomorphism, with the combinatorial homology group Hq(K,G) from classical topology. To make an honest

definition out of this, one has to verify that (up to natural isomorphism) the group Hq(K, G) does not depend on the choice of the triangulation. The now traditional way to prove this is to define a complex C.(M, G) of singular chains and to verify the seven Eilenberg-Steenrod axioms for the homology groups [ES, I §3]. Then one obtains, in a well known manner, that tIq(C.(M, G)) ~- Hq(I(, G) for the triangulation ~ above. {One also has to consider noncomplete spaces M and the relative chain complex C.(M, A; G) for A a semialgebraic subset of M. I omit these technicalities.} We can indeed define singular chain groups Cq(M, G) along classical lines, decreeing that a singular simplex is a semialgebraic map from the q-dimensional standard simplex Aq to M. Six of the seven Eilenberg-Steenrod axioms can bc proved as in the classical theory, always using semialgebraic maps instead of continuous maps. But the excision axiom is difficult. The classical way to prove it is to make a given singular cycle Z "small" with respect to a given covering of M by two open (semialgebraic) sets U1, U2, by iterated barycentric subdivision of Z. This means that every singular simplex occuring in the subdivided cycle has its image either in U1 or U2. But if the base field R is not archimedian then this procedure completely breaks down, since then usually a given bounded semialgebraic set cannot be covered by finitely many semialgebraic sets all whose diameters are smaller than a given c > 0. In his thesis [D] Delfs found the following way out of this difficulty. He defined cohomology groups Hq(M, G) as the sheaf cohomoIogy groups of the constant sheaf GM on M, and similarly relative cohomology groups Hq(M, A; G) as the sheaf cohomology groups of a suitable sheaf GM,A on M. {Recall that M is equipped with a Grothendieck topology.} For these groups Hq(M, A; G) Delfs succeeded in verifying the Eilenberg-Steenrod axioms. Then he knew that Hq(M,G) is isomorphic to the combinatorial group Hq(K, G). Thus Hq(K, G) is independent of the choice of the triangulation, up to natural isomorphism. From this Delfs concluded that also Hq(K, G) is independent of the choice of triangulation [D]. The verification of six of the seven Eilenberg-Steenrod axioms for the groups Hq(M,A,G) is straightforward, but this time the homotopy axiom causes difficulties. Delfs surmounted these difficulties in [D] by a complicated geometric procedure. Later Delfs found an easier way [D1]. He realized that the homotopy axiom follows from the statement that, for any sheaf ~" on M, the adjunction homomorphism 5r --~ rr.r*~', with 7r the projection from M × [0, 1] to M, is an isomorphism and Rqr,(rr*.T") = 0 for q > 1. [D1, Prop. 4.2 and 4.4]. This then could be deduced via a stalk by stalk argument from the fact that Hq([0, 1], G) = 0 for q > 1 and any abelian group G, which in turn can be verified in an easy geometric way. The crucial point is that one needs the fact Hq([0, 1], G) = 0 not just over R but ovcr the residue class fields k(x) of all points x E M. Roughly one can summarize that Delfs reduced the verification of the homotopy axiom to an easy special case using sheaf theory, at the expense of enlarging the real dosed base field in many ways.

10 §5 L o c a l l y s e m i a l g e b r a i c spaces Dells and I had already introduced "scmialgcbraic spaces" over a real closed field R before 1981 by gluing together finitely many affine semialgebraic spaces over R along open subspaces [DK, §7]. What then was still missing was a handy criterion for a semialgebraic space M = (M, (-gM) to be again affine. Such a criterion would allow the building of semialgebraic spaces M from semialgebraic sets in an "abstract" manner, i.e. without explicitly looking at polynomials, such that M eventually turns out to be an affine space, in other words, a semialgebraic set. In 1982 R. Robson proved his imbedding theorem [Ro] which gives such a criterion. The theorem says that a semialgebraic space M over R is affine iff M is regular, i.e. a point x and a closed semialgebraic subset A of M with x ~/A can be separated by open semialgebraic neighbourhoods. {A subset A of M is called closed semialgebraic if the complement M - A is an open semialgebraic, i.e., an admissible open subset of

M.} Robson's theorem reMly paved the way for the trend of employing new spaces in the geometric theory. Before I go into details about this I should say some words about covering maps. Having semialgebraic paths at hand we may define the fundamental group 7rl(M, x0) for M a semialgebraic space over R and x0 E M, as in the classical theory, by considering homotopy classes of semialgebraic loops with base point x0. Of course, homotopies also have to be defined in the semialgebraic sense, starting from the unit interval [0,1] in R, cf. §10 below. It turns out that for affine M the group 7rl (M, x0) is very respectable. It is finitely presented and coincides with the topological fundamental group in the case R = R. {These are consequences of the two comparison theorems on homotopy sets [DK2, III §3 and §5], to be discussed in §10 below.} Assume since now that M is affine and path connected. The question arises whether the subgroups of 7rl(M, x0) classify "semialgebraic covering spaces" of M, as one might expect from classical topology. It seems clear what a semialgebraic covering map 7r: N ~ M has to be: N should be a semialgebraic space and 7r a semialgebraic map. Further there should exist an admissible open covering (Uili E [) of M such that ~ is trivial over each Ui with discrete fibers, i.e. rr-](Ui) ~ Ui × Fi over Ui for a discrete semialgebraic space Fi. But what does it mean for a semialgebraic space F to be discrete? Reasonable answers, one can think of, are: dim F = 0; the path components of F are one-point sets; the one-point sets in M are open in F; the one-point sets in M form an admissible open covering of M. - All of these properties mean the same thing, namely that the space F consists of finitely many points. We conclude that every semialgebraic covering map r: N ~ M has finite degree.

]1 Working with path lifting techniques one verifies that the semialgebraic coverings ~r:N --~ M of M are indeed classified by the conjugacy classcs of subgroups of ~rl(M, x0) of finite index [K3]. Using Robson's embedding theorem one also sees that N is again affine. Having verified this in 1982 [DK3], Dells and I realized that the category of semialgebraic spaces is not broad enough. There should exist some sort of covering space N of M corresponding to any given subgroup H of ~rl (M, x0), in particular a "universal covering", corresponding to H = {1}. This led us to introduce locally semialgebraic spaces. A locally semialgebraic space M over R is obtained by gluing together (maybe infinitely many) affinc semialgebraic spaces over R along open semialgebraic subspaces. Of course, the gluing is meant in the sense of ringed spaces with Grothendieck topologies, cf. [DK2, I §1]. The nice locally semiatgebraic spaces are those which are regular (defined in the same way as above) and paracompact, as defined in [DK2, I §4]. The category LSA(R) of regular paracompact locally semialgebraic spaces over R contains the category of affine semialgebraic spaces over R as a full subcategory. In LSA(R) we have a fully satisfactory theory of covering spaces. In particular ever), space M E LSA(R) has a universal covering (cf. [DK3, §5]; a full treatment of this topic still awaits publication [K3]). In LSA(R) there exist fibre products. There is also a good notion of subspaces. Namely, if M is a locally semialgehraic space and (Mill • I) is an admissible open covering of M, such that every M / i s an affine semialgebraic space, then a subset A of M is cMled a subspace if A N M/ is semialgebraic in Mi for every i • I. Indeed, collecting the affine semialgebraic spaces A N M / w e obtain on A the structure of a locally semialgebraic space over R, which is independent of the choice of the covering (Mi[i • I). This space A is regular and paracompact if M has these properties [DK2, I, §3 and §4]. Up to now LSA(R) has proved to be the appropriate basic category for all geometric studies over R, as long as one does not pass to abstract spaces. In particular the triangulation theorem for semialgebraic sets (eft §3 above) extends to a triangulation theorem of equal strength for these spaces (simultaneous triangulation of M and a locally finite family of subspaces of M, cf. [DK2, Chap.II]). Also the homology theory of Dells discussed above extends to these spaces [DK2, Chap.III]. And we have a fairly good homotopy theory in LSA(R) at hand, to be discussed below. §6 A b s t r a c t s e m i a l g e b r a i c f u n c t i o n s a n d real closed spaces We come back to the relationships between a semiatgebraic set M over R and its abstraction/~/. Recall from §4 that the sheaves on the affine semialgebralc space M correspond uniquely with the sheaves on 37/. In particular we have a sheaf of rings

12 (-')M on 5-I which corresponds to the sheM 0 M of semialgebraic functions on M. The question arises whether (~M generalizes in a natural way to a sheaf of rings Ox on any constructible subset X of any real spectrum Sper A, which then can be regarded as a sheaf of "abstract" semialgebraic functions on X. This is indeed the case. Around 1983 G. Brumfiel [B4] and N. Schwartz [S] gave two solutions of this problem. A (slightly "corrected", cf. [D, 1.7], [S, Example 58]) version of Brumfiel's definition runs as follows. Let p: Sper A[T] --* Sper A be the natural map from the real spectrum of the polynomial ring A[T] in one variable over A to Sper A, induced by the inclusion A ~ A[T]. For any quasicompact open subset U of the space X the elements of Ox(U) are the continuous sections s of plp-l(u): p-I (U) --~ U such that s(U) is a closed constructible subset of p-1 (U). What does this mean? For any x E A we may identify p-1(x) with the real spectrum Sper k(x)[T], where k(x) denotes the residue class field of Sper A at x, a real closed field. This real spectrum is the abstraction of the real affine line over k(x). Thus k(x) injects into p - l ( x ) as a dense subset (cf. §2). For a section s ms above, s(z) lies in this subset and hence corresponds to an element f(x) of k(z), which should be regarded as the value of the abstract semialgebraic function f given by s. The section s is completely determined by the values f(x) and should be regarded as the graph of f. N. Schwartz defined an abstract semialgebraic function f on U directly as a family (f(x)lx E U) E l-Leu k(x) with compatibility relations between the values f(x) coming from canonical valuations Ax,y: k(x) --~ n(x, y) U c~. For any pair (x, y) with x E U and y a specialization of x in U, ~(x, y) is an overfield of k(y), and A~,v has to map f(x) to f(y) E k(y) C t~(x, y). The definition of Schwartz has the advantage that here it is immediately clear that Ox(U) is a ring, while in Brumfiel's definition one has to work for this. Then Dells proved that the definitions of Brumfiel and Schwartz give the same sheaf Ox [D1, §1]. The stalks of Ox are local rings. In the geometric case, i.e., if A = R[T1,...,Tn] and X = /17/ with M C R n a semialgebraic set, we indeed have Ox = (gM. From now on we call the ringed space (_~/, OM) - instead of just the topological space .ltT/- the abstraction of the affine semialgebraic space (M, OM). In the paper [B4] cited above Brumfiel introduced abstract semialgebraic functions as a tool to prove a vast generalization of Mah6's theorem on the separation of connected components by global quadratic forms. For every commutative ring A there is a natural homomorphism from the Witt ring W(A) to the orthogonal Kgroup KOo(Sper A) of the real spectrum of A. Brumfiel proves that both the kernel and the cokernel of this homomorphism are 2-primary torsion groups. Thus, from our viewpoint, the localization 2 - ~ W ( A ) of W(A) at the prime 2 is a purely topological object.

]3 Brumfiel's paper is a bold step into the realm of abstract semialgcbraic topology. A full understanding of it is a challenge even today, since some arguments are only sketched. For a discussion cf. [K, §6], and for a treatment in the geometric case cfi [BCR, 15.3]. N. Schwartz studied in [S] the spaces (X, Ox), with X a constructible subset of some real spectrum Sper A, for their own sake. The ring O(X) of global sections of Ox is a sort of "real closure" of the ring A. Schwartz describes how to obtain O(X) from the ring A in a constructive way. He further makes the important discovery that the natural map from X to Sper O(X) is an embedding which makes X a dense subspace of Sper O(X). Even more is true: the closed points and also the minimal points of Sper O(X) all lie in X. In the special case that X is convex in Sper A with respect to specialization, it turns out that the ringed spaces (X, Ox) and Spec O(X) are equal. In the geometric case X = ~ / t h i s happens iff the semialgebraic set M C R n is locally closed in R n. Later Schwartz realized that all we have said above about (X, Ox) remains true if X is a proconstructible subset of Sper A, i.e., the intersection of an arbitrary family of constructible subsets of X [$1], [$2]. He called any ringed space isomorphic to such a space (X, Ox) an aO~ne real closed space. He then introduced the category 7"~ of real closed spaces as a full subcategory of the category of all locMly ringed spaces. The definition of a real closed space is simple: a ringed space (X, Ox) - always with X a genuine topological space, no Grothendieck topology - is called real closed if every point x E X has an open neighbourhood U such that (U, Ox[U) is an affine real closed space. The books [$1], [$2] are both versions of Schwartz's Habilitationsschrift [S]. For the insiders they constitute a sort of bible of abstract semialgebraic topology - an incomplete bible, I should add, since more can and should be written down with the methods developed there. The shorter version [S1] is easier to read, while [$2] is closer to the original Habilitationsschrift and contains much more material. In IS] Schwartz defined real closed spaces using as building blocks only constructible subsets of real spectra, instead of proconstructible ones. I will call these more special ringed spaces here "abstract locally semialgebraic spaces" and denote their category by 7~0. The analogy with locally semialgebraic spaces over a real closed field R is striking. But there is more than analogy. One can attach to any locally semialgebraic space (M, OM) over R an abstract locally semialgebraic space (/~, 0 M ) in a rather obvious way, starting from the abstractions of affine semiMgebraic spaces discussed above. Schwartz proves that this gives an embedding of LSA(R) into the category ~0, making LSA(R) a full subcategory of the category of abstract locally semialgebraic spaces over Sper R [S], [$1], [$2]. A good thing about ~ is that here more constructions - in particular more quotients - are possible than in LSA(R).

14 In view of what has been said above about affine real closed spaces, it is clear that real closed spaces are close to schemes and many of them are in fact schemes. This constitutes a rather thorough algebraization of semialgebraic topology, since the notion of schemes originates from polynomial equalities and non-equalities ( f = 0, f 0) instead of inequalities ( f > 0, f > 0). The books [$1], [$2] give ample evidence that scheme theoretic notions and techniques work well in the category 7~. The books are close in spirit to the foundations of Grothendieck's abstract algebraic geometry [EGA I], [EGA I*]. In particular, the transition from a locally semialgebraic space to its abstraction is fully analogous to the transition from an algebraic variety over an algebraically closed field to the associated scheme. Of course, in many respects [$1] and [$2] are simpler than Grothendieck's theory, since here only "topological" phenomena have to be captured. This is already reflected by the fact that no nilpotent elements occur in the structure sheaves. It also pays well to pass from the category 7-4o to 7~. For example, a finite subset of a real spectrum Sper A is always proconstructible but only rarely constructible. This implies that for a real closed space (X, Ox) every finite subset S of X gives us a "subspace" (S, Os) of (X, Oz) which is again real closed. Especially useful are the two-point spaces S = {~, x} coming from the real spectrum Sper 0 of a convex subring 0 of a real closed field K with ~ the minimal point ("general point") and x the closed point ("special point") of Sper 0. {Recall that 0 is a valuation ring.} These two-point spaces occur in valuative criteria for various properties of morphism between real closed spaces, cf. ~13 A below. By the way, Sper 0 = Spec 0 as a ringed space. My student Michael Prechtel has given an interesting classification of all real closed spaces which contain only finitely many points [P].

§7 Cohomology w i t h s u p p o r t s Starting with this section, a geometric space means a regular paracompact locally semialgebraic space over some real closed field R, and an abstract space means a regular paracompact abstract locally semialgebraic space. A word of explanation for these terms: A real closed space X is called regular if the specializations of a point x E X form a chain. X is called paracompact if X has a locally finite covering (Xi]i E I) by affine open subsets Xi. There are several reasons why this terminology makes sense. On the one hand the abstraction M of a locally semialgcbraic space M over some real closed field R is regular if M is regular and is paracompact if M is paracompact. Thus the abstraction of a geometric space is an abstract space. Oil the other hand, if X is an abstract space then the subspace X m~x of closed points of X is a paracompact topological space [D2, Chap. I].

15 Abstract spaces are very amenable to sheaf cohomology with supports, as defined in classical topology (cf. e.g. [Bre D. This has been amply demonstrated by Dells in Chapter II of his tIabilitationsschrift [D2]. Recall that if • is a family of supports on X (= antifilter of closed subsets), and ~- is a sheaf on X (as always, with values in abetian groups), then F~(X, 7 ) denotes the group of sections s E .T(X) with support supps E ~, and .T" ~ H{(X,F) is the q-th derived functor of the left exact functor

There is a close connection with the sheaf cohomology on the subspace X max, provided X max is dense in X and every point of X has a specialization in X max (which holds in many applications). Then we have a canonical continuous retraction r: X ~ X raax. The direct image functor r, from the category of sheaves on X to the category of sheaves on X max is easily seen to be equal to the inverse image functor i* for i the inclusion X max ~-~ X. For every sheaf ~" on X this gives us canonical isomorphisms tl~(X,~) ~= ~ u qv "~ v m ~ ,z. , ~. . j , and for every sheaf ~ on X, canonical isomorphisms H{(X ma*,~) ~ H{(X, i,G), with • denoting the set of intersections {A N xm~*Ia E ~}. In the case that X is affine (or, more generally, a "normal spectral space") this important observation goes back to Carral and Coste [CC]. If M is a geometric space then we have a canonical isomorphism fi" H ~" from the category of sheaves on M to the category of sheaves over/~'/, as explained above in the special case that M is affine (§4). A family of supports ¢b on M is by definition an antifilter of closed subspaces of M. We can define cohomology groups H~(M,U) in much the same way as in the classical theory. {Fo(M, U) is the group of all s E .T(M) such that sIM - A = 0 for some A E ¢.} Let ~ denote the antifilter of closed subsets of/~/generated by the abstractions A of all A C ¢. Then it is evident that

U (M, 7) = H (i4, for all q _> 0. There are many useful families of supports ~, even more than in classical topology. In particular we can choose for ~ the family of all complete sub@aces of M, "complete" being defined as in §2. We denote this family by c, suppressing its dependence on M in the notation. In the abstract setting we also have the notion of complete spaces. Here, for X an abstract space, we denote by c the antifilter of closed subsets generated by the complete subspaces of X. The groups IIq~(M,F) resp. H~(X, J:) are the analogues of the cohomology groups with compact support in classical topology. For M a geometric space we have H~(M, J:) = Hqc(~l,~). In Chapter II of [D4] Dells develops the formal theory of sheaf cohomology with supports for geometric and abstract spaces virtually to the same extent as the classical theory (e.g. [BreD . Some important topics are omitted, in particular cup products,

16 but it is evident that these topics can be dealt with by the same methods. Dells has often talked to me about such things in very clear terms. As a sample of his theory I will now say something about the cohomology of fibers [D2, II §8]. Let f : M ~ N be a morphism between geometric spaces and Y" a sheaf on M. Let f:/~/ --* N denote the abstraction of f . For every c~ C fir the fibre f-l(c~) is the abstraction of a geometric space over k(~). In particular, if y C N, then f - 1 (y) = I - 1 (y)~. Fix some q C No. For every c¢ E fir we consider the cohomology group which we denote more briefly by H ~ ( f - l ( c ~ ) , ~ ) . The question arises as to how these groups are related for varying ~. A first answer is that there exists a sheaf (5 on N, namely ~5 = Rqf!.T", such that, for every o~ C /V, H ~ ( f - l ( c ~ ) , ~ ) is the stalk of ~ at c~,

H~(f-~(oO, )elf-~(~))

(*)

(R f!.T) <~

Hqc(/-l(a),.T').

Here 9~ H Raf!.~ is the q-th derived functor of a left exact functor 5v H f.,9v defined as follows. For U an admissible open subset of N the group (fiY')(U) consists of all sections s of .T'(f-I(U)) such that there exists some closed subspace A of f - I (U) such that s vanishes on M - A and the restriction flA: A --~ N is proper. It follows from (*) that, for y E N, (**)

(Rq f!.T)y = Hq(f -1 (y),.T').

The equations (*) and (**) throw some light on the relationship between the geometric and the abstract setting. In the geometric theory we are interested in the groups H~(f-l(y),.T ) for y e N. But it is evident from (*) and (**) that the groups Hc~(f-l(c~), ~ ) with c~ e / 9 , not necessarily ~ e N, have an influence on these groups.

§8 B o r e l - M o o r e h o m o l o g y Again, let M be a geometric space and q) a family of supports on M. Let GM be the constant sheaf on M attached to some abelian group G. We denote the groups H$(M, GM) more briefly by tI$(M, G). If ¢ is the set of all closed subspaces of M, henceforth denoted by cld (= "closed"), then these groups are just the semialgebraic cohomology groups Ha(M, G) of Delfs discussed in §4. It would also be desirable to have at hand homology groups Hq'I'(M,G) with support in q). This is a problem even for q) = ctd. It is intuitively clear that the group Hq(M, G), as described in §5, should coincide with H,~(M, G), and not with I-I~ld(M,G), since a simplicial chain, as used there, clearly has support in a complete subspace of M. {By the way, every complete subspace of M is affine semialgebraic, cf. §13 A }. In classical topology homology groups with closed support have been defined by Borel and Moore for locally compact spaces [BM], [Bre, Chap.V]. They used a rather complicated sheaf theoretic procedure, but their groups turned out to be very useful.

]7 Dells found out that, for M a locally complete geometric space, such Borel-Moore groups BMIIq(M,G) can be defined in a down-to-earth combinatorial manner as follows [D2, Chap. III], IDa]. We choose a triangulation ~o:X ~, M. Then X is a locally finite and locally closed simplicial complex over the base field R. This means that every open simplex of X is the face of only finitely many open simplices of X, and that X is open in )(. {Recall that we use the notion "simplicial complex" in a non-classical sense (§3).)~" is a simplicial complex in the classical sense.} All the combinatorial technique is based on open simplices instead of closed ones. We choose a (say) total ordering of the vertices of )( and then can talk about oriented open simplices. For pairwise different vertices x 0 , . . . , x q of X let (x0,...,xq) denote the open oriented q-simplex with these vertices and the orientation given by the sequence x 0 , . . . , xq. We define a chain complex BMc.(x, G) as follows. BMCq(X, G) consists of all formal sums ~ meo" with ~r running through the set of positively oriented open o"

simplices of X and me E G. The boundary operator is given by the classical rule q =

i=0 but omitting on the right hand side all open simplices which do not lie in X. The fact that X is locally closed guarantees that 0 o 0 = 0. BMHq(M, G) is defined as the q-th homology group of the chain complex BMG.(X, G). Dells verifies that these "semialgebraic Borel-Moore homology groups" do not depend on the choice of the triangulation [D2, III.2.1]. They have the formal properties one is used to from topological Borel-Moore homology, and in the case R = R they coincide with the classical Borel-Moore groups. From now on we replace G by a principal ideal domain A. If M is an n-dimensional Aoriented paracompact locally semialgebraic manifold over R (this defined in a rather obvious way [D2, III §3]), and to: X ~ , M is a triangulation, then the sum Z of all n-dimensional open suitably oriented simplices of X is an element of BIC~Cn(M,A) with boundary OZ = 0. The cap product with the "fundamental class" [Z] E BMHn(M,A) gives an isomorphism from Hq(M,A) onto BMtIq(M,A) for every q, as in classical Poincar6 duality theory [D, III §9]. If M is the space of real points V(R) of an n-dimensional algebraic variety V over R then Delfs defines again a fundamental class ~M E BMH,,(M,l / 2 ) in a similar way [D3]. ~M is characterized by the property that its restriction to any connected component Ni of the open subset Vreg(R) of regular points is the unique non-zero element of BMHn(Ni,Z/2). {In the case that M is complete this property had been verified before by other methods in [DK]].} Dells also constructs a fundamental class for h = l and M = Y = V(R(v"ZT)) with Y an algebraic variety (more generally, an

18

"isoalgebraic space") over R(xfZ-]-). Such fundamental classes have been introduced for real and complex analytic spaces by Borcl and Haefliger, and used by them to establish a topological intersection theory on these spaces [BH]. Let again M be a locally complete space and • be a family of supports on M. In order to define homology groups with supports in ¢ and, for M an oriented manifold, to establish Poincar6-duality with the cohomology groups with supports in ¢, one probably can use still combinatorial methods since we have a strong triangulation theorem at hand, but this would be more complicated than in the case ¢ = c l d . Instead of this Delfs returns to sheaf theoretic methods similar to those used in the classical theory [BM], [Bre]. Sheafifying the Borel-Moore chain complex Br~'.(M, A) he obtains a complex A. of sheaves of simplicial chains. For .T" a sheaf of A-modules he defines Hat(M, ~) as the q-th homology groups of F,I,(M, A.®A~") [D2, Chap. III]. For G a A-module and ~" the constant sheaf GM this group coincides with BMIIq(M, G) if ¢ = cld, and with the semialgebraic homology group Hq(M, G) if ¢ = c. Being able to vary ¢ and 9v, Dells has enough flexibility to establish Poincar6duality and semialgebraic intersection theory, the latter for M = V(R), A = Z/2, resp. M = V(R(v/:i-)), A = Z, with V an algebraic manifold over R or R(~'Z] -) respectively (even an isoalgebraic manifold in the latter case) [D2]. In particular Delfs obtains an important result about algebraic intersection numbers. Let C be an algebraically closed field of characteristic zero. Let V be a smooth algebraic variety over C, and let X1, X2 be irreducible subvarieties of V. Let Y be an irreducible component of X113 )(2 with codim Y = codim X1 + codim )(2. Then the algebraic intersection number i(X1 . X2, Y') is a well defined natural number. Now choose a real closed field R with C = R(x/-Z]'), which can of course usually be done in very many different ways. Then V = V(C) is also a geometric space over R, and Borel-Moore homology with A = Z gives us a semialgebraic intersection number in(X1 • X2, Y). Delfs proves that this number coincides with i(X1 • )(2, Y) for every choice of R. §9 B a s e field e x t e n s i o n a n d c o m p a r i s o n t h e o r e m s We fix a real closed base field R and a real closed field extension K of R. Let M be a semialgebraic subset of R n. Then we obtain from M a semialgebraic subset M(K) of K n by "base field extension" as follows. We write

M = ~] {x e R'~lfij(x) > 0, j = 1 , . . . , s ( i ) ; gik(x) > 0, k = 1 , . . . , t ( i ) } i=1

with some polynomials fij, gik in n variables over R. Then we define r

M ( K ) : = U {x e I("lfff(x ) > 0, j = 1 , . . . , s ( i ) ; gik(x) > 0, k = 1 , . . . , t ( i ) } . i=1

19

It is an immediate consequence of Tarski's principle that M(K) does not depend on the choice of the description of M above. If f: M -o N is a semialgebraic map between semialgebraic sets over R then, again by Tarski's principle, there exists a well defined semialgebraic map fK: M(K) --r N ( K ) whose graph F(/K) is obtained from r ( f ) by base field extension, r(fK) = r ( f ) ( K ) . Thus we have a functor M ~ M(K) from the category of affine semialgebraic spaces over R to the analogous category over K. This functor extends in a naturM way to a functor M ~-~M(K) from the category LSA(R) of geometric (= regular paracompact locally semialgebraic) spaces over R to LSA(K) [DK2, Chap. I]. This functor "base field extension from R to K" has an illuminating interpretation in the abstract setting. Let G denote the category of abstract spaces (a full subcategory of the category R0 considered in §6). Recall that the abstraction functor M ~-* M embeds LSA(R) into 6 . In the same way, LSA(K) is a subcategory of 6 . In particular G contains the one-point ringed spaces Sper R and Sper K. The inclusion R ~ K gives us a morphism SperK ~ Sper R. {By the way, SperR = SpecR and Sper It" = Spec K.} In ~ we have fibre products [S1], [$2], and M(~'~V) =/V/ XSper R SperK. For semialgebraic homology II,(-,G) with values in some abelian group G, the following comparison theorems are rather evident from the simplicial definition of H , ( - , G) (cf. [DK2, III §7] for a more precise version). First C o m p a r i s o n T h e o r e m . There is a canonical isomorphism

Hq(M, G) ...", Hq(M(K), G). S e c o n d C o m p a r i s o n T h e o r e m . If R = R there is a canonical isomorphism

tlq(M, G) 7 , Hq(Mtop, G). Here Mtop denotes the set M equipped with the topology which has the admissible open subsets of M as a basis, the so called "strong topology". Notice that this is the subspace topology of M in /17/. The groups Hq(Mtoo, G) are the singular homology groups. There is an important case, namely if M is "partially complete" (cf. §13 A below for this), in which Mtop coincides with the space M m~x of closed points in/~/. We have similar comparison theorems in sheaf cohomology due to Delfs [D2, Chap. II]. Let ¢ be a family of supports on M. It gives us a family of supports @(K) on M(K), namely the antifilter of closed subspaces generated by {A(K)IA E @}. In the

20 case R = R it also gives us a family of supports 'btop on Mtop, namely the antifilter of closed subsets generated by ¢. Let yr be a sheaf on M. It gives us a sheaf br(K) on M(K) by the rule ~'(K) ~ = ~r*(~), with ~r the natural projection from M(S) ~ = AS/XSpern Sper IV to 35I. In the case R = R we also obtain a sheaf ~top on A/top by the rule brtop = i*(5~) with i the inclusion Mtop ~/17/. F i r s t C o m p a r i s o n T h e o r e m [D2, II.6.1]. There are canonical isomorphisms

H~,(M,.T') ~, H$(K)(M(K), :F(K)). S e c o n d C o m p a r i s o n T h e o r e m [D2, II §5]. Let R = R. Assume either that 5r is locally constant or that M is partially complete. Then there are canonical isomorphisms H~,(M,.T) ~, H~,,o,(Mtop, 5rtop). Delfs points out that some restrictive hypothesis in the second theorem is indeed necessary [D2, II.5.6]. One should add that the first theorem also does not always give the result one would like to have. For cxample, if ¢ is the set of all complete subspaces of M, denoted previously by c, then gp(K) may be smaller than the set of all complete subspaces of M(K). Nevertheless Delfs proves that there exist canonical isomorphisms H~(M,.~') ,~~ Hqc(M(K), a'r'(K)) if M is locally complete [D2, II.6.10]. If M is locally complete then M(K) is locally complete, and, in the case R = R, the space Mtop is locally compact. Thus one may also look for comparison theorems in BoreI-Moore homology with supports and with sheaves as coefficients. Dells proves such theorems, again with a necessary restriction in the assumptions of the second comparison theorem [D2, III §10 and §ll]. §10 H o m o t o p y s e t s We fix a real closed base field R. A pair of geometric spaces (M, A) over R consists of a geometric space M and a subspace A of M. A morphism f from (M, A) to another pair (N, B) is a morphism f: M -~ g with f(A) C B. {Then the restriction IIA: A ~ B is again a morphism [DK2, 1.3.2].} It is now clear how to define a homotopy F between two morphisms f , g: (M, A) (N, B); namely, F is a morphism from (M × [0, 1], A × [0, 1]) to (N, B) with f(x,O) = f(x) and F(x, 1) = g(x) for every x e M. More generally all the basic notions of

2] the elementary classical homotopy theory make sense in the category LSA(R). But notice that we do not have something like spaces of morphisms at our disposal. Let [(M, A), (N, B)] denote the set of homotopy classes of morphisms from (M, A) to (N, B). One would like to know as much as possible about it. In the case that B --- {x} is a one-point set and ( i , A) = (Sq(R), {oo}), with Sq(R) the unit sphere in R q+l and o~ its north pole, we equip this set with a group structure, in the classical way which is of semialgebraic nature. This gives us the homotopy group ~rq(N,x). It is abelian for q > 2. For homotopy sets we have the following two comparison theorems. F i r s t C o m p a r i s o n T h e o r e m . If K is a real closed overfield of R then the natural map [f] ~ [fg] from [(M, g), (g, B)] to [(M(K), A(K)), (g(g), B(K))] is bijective. Second Comparison Theorem. If R = R then the natural map from [(M,A),(N,B)] to the set of homotopy classes of continuous maps [(Mtop, Atop), (Ntop, Btop)] is bijective. All this can be proved by using Tarski's principle, triangulations, and simplicial approximation techniques [DK2, Chap. III]. More generally such theorems hold for relative homotopy classes, where a subspace of M has to be fixed pointwise by the homotopies [loc. cit.]. If S = {x} and ( / , A) = (Sq(R), {co}) then the bijcctions in the theorems are, of course, group isomorphisms. The theorems now tell us what the homotopy groups of a pointed geometric space (N, x) over R look like "in principle". To he more precise, we choose a triangulation ~ : X U~ N such that ~ - l ( x ) is a vertex e of X. Let us assume for simplicity that N is "partially complete" (eft §13 A below), a property which forces the simplicial complex to coincide with its closure )(. Then X is the realization IL]tt of a locally finite abstract simplicial complex L (in the classical sense [Spa, Chap. 311. Let R0 denote the real closure of Q with respect to its unique ordering. This field has a unique embedding into any other real closed field. Of course, [L[R = [LIRa(R). Now the isomorphism ~ and the two comparison theorems give us group isomorphisms

7cq(N, x) ~ ~q(ILIR, e) ~ ~q(ILIR0, e) ~ ~rq([LIR, e)

~-

~q(ILlatop, e).

The last group, a classical homotopy group, can be written - somewhat tautologically - as the homotopy group ~rq(L, c) of the abstract simpliciM complex L.

22 ~11 W e a k l y s e m i a l g e b r a i c spaces, a n d q u o t i e n t s The comparison theorems stated in the last section, and analogous theorems for triples, quadruples, etc. of geometric spaces instead of pairs, allow us to transfer a considerable amount of classical homotopy theory to the category of geometric

spaces LSA(R), cf. [DZs, III §6] for examples. Nevertheless, homotopy theory in LSA(R) is hampered by the fact that many of the constructions used in the classical theory do not have counterparts here. In particular, quotients by seemingly innocent equivalence relations can cause problems. For example, CW-complexes exist in LSA(R) only to a limited extent. Even the (reduced) suspension SM of a pointed geometric space (M, x0) usually does not exist, since it may be impossible to contract the closed subspace M x {0} U M x {I} t.J {x0} × [0, 1] of M × [0, 1] to a point in some universal way. Thus we cannot start stable homotopy theory in LSA(R). Fortunately there is a way out of many of these difficulties. As has been shown in [K2], we can embed LSA(R) as full subcategory in the category WSA(R) of "weakly semialgebraic spaces" over R, where homotopy theory is more pleasant.

A weakly semialgebraic space M = (M, OM) over R is a set M equipped with a Grothendieck topology, consisting of admissible open subsets and admissible open coverings, and a structure sheaf OM, consisting of R-valued functions, such that M arises from affine semialgebraic spaces by gluing along closed subspaces carefully observing certain rules, cf. [Ks, p.3 f]. As for locally semialgebraic spaces, a morphism (M, OM) --* (N, ON) between such ringed spaces is determined by the underlying map f: M --* N, and will henceforth be identified with this map (cf. [Ks, IV §1 for the definition of morphisms). In WSA(R) there exist fibre products. Thus notions like proper morphisms and complete spaces make sense here. It turns out that every complete weakly semialgebraic space is a complete affine semialgebraic space, hence isomorphic to a polytope (= finite polyhedron), [K2, p. 44] and [DK2, p. 59 f]. Of particular importance for homotopy theory are the weakly semialgebraic spaces which can be obtained by gluing complete affine spaces along closed subspaces. They are called weak polytopes. Every weakly semialgebraic space is homotopy equivalent to a weak polytope [K~, V §4]. Let T be an equivalence relation on a weakly semialgebraic space M. Under which conditions can we expect that the quotient M/T exists in a reasonable sense? Certainly we should assume that T is a closed subspace of M x M. Let PT: T --* M denote the natural projection from T to the first factor in M x M. For M affine semialgebraic and PT proper ("proper equivalence relation") Brumfiel has proved the

23 important theorem that a "strong" quotient M / T exists [Bs]: the set theoretic quotient M / T can be equipped in a unique way with the structure of an affine semialgebraic space such that the projection map ~rT:M ~ M / T is "identifying" in a strong sense. This implies, in particular, that a'T is the categorical quoticnt of M by T in the category of affine semialgebraic spaces. In WSA(R) we can do even better. As an easy consequence of Brumfiel's theorem such a strong quotient 7rT:M ~ M / T exists if p: T ~ M is partially proper, i.e. the restriction of p to any closed semialgebraic subspace of T is proper [K2, IV §11]. At first glance, partially proper equivalence relations might not look much more general than proper ones. But in fact quotients by partially proper equivalence relations suffice for many important topics in homotopy theory. For instance, if A is a closed subspace of a weakly semialgebraic space M, and if f : A --* N is a partially proper map, then we have a space M UA,f N at our disposal, obtained by gluing M to N along A via f . In particular take M to be a direct sum of closed balls, and A the boundary OM, which is a direct sum of spheres. This is the essential step to build CW-complexes. Notice that now A is a weak polytope, and every morphism fi'om a weak polytope to a weakly semialgebraic space is partially proper. As a result we can construct CW-complexes and, more generally, relative CWcomplexes in WSA(R) in much the same way and for much the same purposes (e.g. killing of homotopy groups) as one is used to in classical topology. If (M, x0) is a pointed weak polytope then we obtain the suspension SM by choosing A = M × {0} 12 M × {1} O {x0} × [0, 1] and f as the morphism from A to the one point space *. Thus we can start stable homotopy in the full subcategory T~(R) of WSA(R) whose objects are the weak polytopes over R. If M is not a weak polytope then also A is not a weak polytope, and this means that the morphism A --* • is not partially proper. It turns out, that then SM does not exist as a strong quotient of M × [0, 1] in the sense indicated above. One can conclude from [Sch, §3] that SM does not exist even as a categorical quotient, at least if M is a geometric space. The existence of strong quotients by partially proper equivalence relations seems to be a best possible result. Indeed, C. Scheiderer has proved a "converse" to Brumfiel's theorem [Sch]: If T is a closed semialgebraic equivalence relation on a locally complete semialgebraic space M, and the strong quotient M/T exists, then, up to obvious modifications, T is proper. Scheiderer's method is very interesting. He makes essential use of abstract spaces and applies an extension of Tarski's principle to real closed fields with convex valuation rings [loc. cit.].

24 Weakly semialgebraic spaces often cannot be triangulated, but "patch techniques" have been developed to work with them nearly as easily as with triangulable spaces~ cf. [K2, Chap. V]. For every real closed overfield K of R we have a natural base field extension functor from WSA(R) to WSA(K) which extends the base field extension functor from LSA(R) to LSA(K) discussed above (§9). It turns out that the two comparison theorems for homotopy sets from above (§10) remain true for weakly semialgebralc spaces. Thus nothing is lost and much is gained in homotopy theory by passing from geometric spaces to weakly semialgebraic spaces. In classical homotopy theory one often works in the category" of topological spaces which are homotopy equivalent to CW-complexes, or variants of this category, which all look a little artificial, since a topological space may carry many structures of a CW-complex (cell decompositions), none of which is given in an intrinsic way. A nice fact about WSA(R) is that every weak polytope is homotopy equivalent to a CWcomplex [K2, V §7]. Thus here the category 79(R) is a satisfactory basic category for homotopy theory, the notion of a weak polytope being intrinsic and quite natural, cL §13 A below. What are the "abstractions" of wealdy semialgebraic spaces? Do they exist? Recently N. Schwartz gave a fascinating solution to this problem by inventing his "inverse real closed spaces" [$3]. In some sense the approach of Schwartz to inverse real closed spaces is simpler than my approach to weakly semialgebralc spaces in [K2], and thus throws new light on them. Inverse real closed spaces have passed early tests of usefulness successfully. For example they are instrumental in the homotopy classification of vector bundles and the definition of Stiefel-Whitney classes on real closed spaces (which helps to understand the old paper [B4] of Brumfiel, in particular) [Schwartz, talk at the Ragsquad seminar in Berkeley, October 1990]. I refrain here from a discussion of inverse real closed spaces, since this subject is in a less mature state than the other topics dealt with in this article. Instead I refer the reader to the paper [$3]. §12 G e n e r a l i z e d h o m o l o g y Let 79"(R) denote the category of weak polytopes with base points over R. In this category the direct sum of an arbitrary family of objects (Mx,)~ E A) is the wedge V Mx, obtained from the disjoint union U M.~ by identifying the base points of )~EA ~EA all the Mx. We also have the suspension functor S:T'*(R) ~ 79"(R). We pass from P*(R) to the associated homotopy category HT)*(R). It has the same objects as 79"(R), but the morphisms are the homotopy classes of morphisms in 79"(R). Now

25 we have all the tools we need to define generalized homology theories as one does in classical topology. There one uses the homotopy category H ~ * of pointed topological CW-complexes instead of HP*(R). D e f i n i t i o n (of. [Sw, p. 109 f]). A reduced semialgebraic homology theory over R is a family (knln E Z) of covariant functors k,, from H79*(R) to the category Ab of abelian groups, together with a family (an In E Z) of natural equivalences an: kn "~ kn+l o S, such that the following two axioms hold: E x a c t n e s s A x i o m . For every pair (M, A) of pointed weak polytopes over R, the sequence kn(A) , k,,(M) , k,,(M/A), induced by the inclusion A '--* M and the projection M ~ M/A, is exact. W e d g e A x i o m . For every family (M.xIA C A) of pointed weak polytopes, and every n E Z, the map

0 )~EA

kn(M.,.) .........., k,,(V M~.), AEA

induced by the inclusions Mx ~-~ V M~, is an isomorphism. Applying the homotopy theory developed in [K2] one proves that every reduced homology over R "extends" in an unique way to a non reduced homology theory (hnln E Z) on the category of all pairs of weakly semialgebraic spaces over R. Such unreduced homology theories are again defined in an axiomatic way analogous to classical topology, in other words, the Eilenberg-Steenrod axioms [ES, I §3] - with the exception of the dimension axiom - have to hold, cf. [K2, VI §4]. The relationship between ha and kn is given by ha(M, A) = kn(M/A) for any pair (M/A) of weak polytopes, the space M/A being equipped with its natural base point A/A. {Here I suppress a consideration of the boundary maps 0,,: h,,(M, A) ~ hn-l(A,O), which are present in the axioms of an unreduced homology theory.} It turns out that the functors hn satisfy a strong excision property [K2, VI.6.10], which is even better than excision in classical topology. A main result in [K2] about generalized homology is the following: Given a reduced homology theory (k~t°Pln E Z) on the homotopy category H ~ * of topological CWcomplexes, there exists, for every real closed field R, a reduced homology theory (k~ln E Z) on HP*(R) such that two comparison theorems analogous to those stated in §9 are true (cf. [K2, VI §3] for a more precise statement). One then also has comparison theorems for the associated unreduced homology theories [K2, VI.5.4].

26 Running parallel to all of this, there is a theory of reduced and unreduced cohomology with results analogous to those just stated. To appreciate the content of these results let us choose for (kt°p) the reduced singular homology theory with coefficients in some abelian group G. Then we obtain a homology theory (h R) for pairs of weakly semialgebraic spaces over R which satisfies all the Eilenberg-Stcenrod axioms. Given, say, a complete geometric space M over R and a triangulation ILIR ~~ M, with L a finite abstract simplicial complex, one verifies in the classical way that hqn(M) "~ Hq(L, G). Thus we have obtained anew the result of Delfs that (up to canonical isomorphism) the combinatorial homology group Hq(L, G) does not depcnd on the choice of the triangulation. In some sense, this new proof of Delfs' result is considerably easier than the two original proofs discussed in §4. Once one knows that weakly semialgebraic spaces exist with suitable formal properties, ill particular the two comparison theorems for homotopy sets, the proof works by straightforward homotopy methods, cf. [K2]. But notice that the new proof gives something less than the old ones: a connection with the sheaf cohomology groups H~(M, GM) is missing. From now on I call the groups Hq(M, A; G) which come from the topological singular homology theory ordinary homology groups. They coincide with the semialgebraic homology groups of Delfs discussed in §4 and later. There exist many prominent homology and cohomology theories in topology. For example, let (hnln E Z) be one of the classical cobordism theories [Sw, 12.24]. Then, by the results above, we have associated groups h~'(M, A) for every pair of weakly semialgebraic spaces (M, A) over some real closed field. {From here on I omit the subscripts R and "top".} Many real algebraic geometers will be interested in these groups only for (M, A) a pair of semialgebraic sets. But it seems to be difficult to understand the groups ha(M, A) working only with semialgebraic sets or, more generally, with geometric spaces. So wealdy semialgebraic spaces serve us well, even if we have no geometric interest in them. There remains the task of giving a geometric interpretation of the groups hr'(M, A) for, say, semialgebraic sets. They can probably be described using some semialgebrMc differential topology, of the sort that appears in the definition of the topological cobordism groups. The theory in [K2] does not give such an interpretation. To the best of my knowledge, nobody has tackled this task up to now. In the category 79(R) of weak polytopes over R there exists the realization ILII~of any simpIicial set L (= "semisimplicial set" in older terminology) [K2, Chap. VIII, with the formal properties one expects from classical topology [Mi], [May]. This makes it possible to solve a problem left open in Detfs' semialgebraic homology theory discussed in §4, namely the question of whether the semialgebraic singular chain complexes

27 give us the ordinary homology groups. For M a weakly semialgebraic space over R let SinM denote the singular set of M. This is the simplicial set consisting of the semialgebraic maps from the standard simplices over R to M. As in topology one has an obvious morphism JM: ISinM]R , M. One can prove directly that JM is a homotopy equivalence [K2, VII §7]. {In topology the analogous map is only a "weak" homotopy equivalence [Mi].} From this it follows immediately that the homology groups of the singular chain complex of M with coefficients in G are the ordinary homology groups tIq(M, G) [lot. cit.]. In particular, we now know that semialgebraic singular homology obeys the excision axiom. It would be desirable to have a more elementary proof of this, say, by nonlinear subdivision of singular chains. Certainly we would learn a lot from this about the geometry behind ordinary homology. §13 N o v e l f e a t u r e s of s e m i a l g e b r a i c t o p o l o g y : t h r e e examples. Much of what has been said up to now gives the impression that semialgebraic topology, at least in the geometric setting, is very similar to traditional topology. The definition of spaces, morphisms, etc. may look a little exotic to the classical topologist, but then the results in homology and homotopy are as he or she is used to. In fact semialgebraic topology has also features which may be unexpccted from the classical viewpoint, even if the base field R is the field R of real numbers. In the following I give three examples.

A. Proper and partially proper maps I start with some definitions. Let M be a locally semialgebraic space over R. A subset A of M can carry at most one structure of a subspace of M (cf. §5 above and [DK2, I §3] for the definition of subspace). If it does we call A a locally semialgebraic subset of M. Locally semialgebraic subsets are the only subsets of M which have a geometric meaning. If A carries the structure of a subspace which is even a semialgebraic space we call A a semialgebraic subset of M. If A and B are locally semiMgebraic subsets of M with B C A and A is semialgebraic then also B is semialgebraic. The image f(A) of a semialgebraic subset A of M under any morphism f: M --~ N is a semialgcbraic subset of N. All this justifies the idea that "semialgebraic" is something like "small". This notion of smallness is alien to classical topology. If f: M -+ N is a morphism and B is a locally semialgebraic subset of N, then the preimage f-l(B) is a locally semialgebraic subset of M. {It is not true in general that images of locally semialgebraic subsets of M are locally semialgebraic in N}. We call the morphism f semialgebraic if the preimages of semialgebraic subsets of

28 N are semialgebraic, and we call f affine semialgebraic, if the preimages of affine semiMgebraic subsets of N are affine semialgebrMc.

T h e o r e m 1. Every proper morphism f: M --* N is affine semialgebraic. This theorem has an interesting history. Dells and I gave an argument which proves that f is semialgebraic [DK2, p. 59 if]. This reduces the proof of the theorem to the case that N is affine semialgebraic and M is semiaIgebraic. In this case the theorem is due to N. Schwartz, who proved it by transition to abstract spaces [S], [$I], [$2]. It was one of the really remarkable early applications of his theory of real closed spaces. Then Robson gave a proof within the geometric setting in the special case that N is the one-point space, applying his embedding theorem, cf. [DK2, p. 60]. It was not until much later that R. Huber found a proof in general which stays in the geometric setting. This proof is contained in a joint papcr by Huber and Scheiderer [HuS], which contains related results about "locally proper maps", and which, in my opinion, also gives a good feeling for the interplay between the geometric and the abstract setting for such questions. In addition it contains a deepening of Schwartz's result on abstract spaces [$1, p. 83], [$2, V.5.7] which is behind his proof of Theorem 1 for M and N semialgebraic. If one views a morphism f: M ~ N as a "family of spaces" then Theorem 1 may be regarded as a negative result. Properness is commonly regarded as a condition which is often needed to ensure good behaviour in families of spaces. Theorem 1 tells us that we do not get anything new in essence if we study proper morphisms between geometric spaces instead of just affine semialgebrMc spaces. Fortunately there is another notion, alien to classical topology, but related to properness, which still gives us good families of geometric spaces which are not necessarily semiMgebraic. A morphism f: M --~ N is called partially proper if the restriction f[A: A ~ N to any closed semialgebrMc subset A of M is a proper morphism. A geometric space M is called partially complete if the morphism from M to the one-point space is partially proper. These notions are quite natural, as is illustrated by the following "relative path completion criterion" [DK2, I §6]: T h e o r e m 2. A morphism f: M --* N between geometric spaces is partially proper iff for every commuting square of morphisms (solid arrows) ]0, 1] ~

/

~M

iL J

[0,1]

~

N

29 with i the inclusion map, there exists a path & (dotted arrow) which extends a. {N.B. Since geometric spaces are "separated" there can exist at most one such path &, and lies over fl, f o & = ft.} In the special case that N is the one-point space the theorem tells us that a geometric space M is partially complete iff every "incomplete path" a: ]0,1] ~ M can be completed. Here the classical notion of sequential compactness comes to our mind, but sequences are a blunt instrument in semialgebraic topology. {There even exist real closed fields which do not contain any non-trivial zero sequences.} Incomplete paths may be regarded as the right substitute of sequences in the geometric theory. From this viewpoint partial completeness, and not completeness, is the right analogue of compactness. By the way, all the definitions in this subsection A make sense more generally for weakly semialgebraic spaces, and both theorems 1 and 2 remain true for them. The partially complete weakly semialgebraic spaces are precisely the weak polytopes discussed in §11. In the abstract setting the counterparts of path completion and path lifting criteria like Theorem 2 are valuative criteria similar to the ones in algebraic geometry. The books of Schwartz [$1], [$2], his paper [S4], and the paper [HuS] by Huber and Scheiderer contain several such criteria. Here the unit interval [0, 1] is replaced by the two point subspace {~, x} of the real spectrum Sper 0 = Spec 0 of a convex subring 0 of some real closed field, with ~ the generic and x the closed point of Sper 0, while ]0, 1] is replaced by {~}. B. D i f f e r e n t s e m i a l g e b r a i c s t r u c t u r e s on t h e s a m e classical s p a c e Let M be a locally closed semialgebraic subset of W'. Then we have on the set M the structure of a locally complete affine semialgebraic space, which we denote again by M. Let c(M) denote the direct system of all complete semiaIgebrMc subsets K of M. For every K E c(M) the interior ff is again semialgebrMc, thus both K and f( are affine semialgebraic spaces. We can form the direct limit Mlo¢=

lim [( Kec(M)

in the category LSA(R) [DK2, 1.2.6]. Mloc is also the direct limit of the spaces K [DK2, 1.7.8]. It is a partially complete geometric space over R which has the same underlying set as M. Even Mtop = (Mloc)tov. Assume that M is not complete. The identity map Mloc ~ M is a morphism but by no means an isomorphism. Indeed, a subset A of M is semialgebraic in Mloc iff A is

30 semialgebraic and bounded in R" and A has a positive distance from the boundary /17/- M of M, provided this boundary is not empty. In particular the space MIo¢ itself is not semialgebraic. A function f : M --+ R is a global section of the structure sheaf of Mloc iff f is continuous and, for every x E M, the restriction of f to a suitable semialgebraic neighbourhood of x in M is semialgebraic ("locally semialgebraic functions"). Thus C9(Mloc) is a bigger ring than O(M). The different nature of M and Mloc is perhaps best illustrated by looking at triangulations. For instance, if the set M is an open n-simplex in R n, then the space M has the tautological triangulation id: M --~ M, while we obtain a triangulation of Mtoc by subdividing the set M into infinitely many closed simplices. If M is an arbitrary semialgebraic subset of R" then we can still form the inductive limit of the spaces K with K E c(M) in the category WSA(R) of weakly semialgebraic spaces, and we obtain a weak polytope P(M). If M is locally closed in R n the ringed space P(M) equals Mloc. The space P(M) has the same underlying set as M, and the identity map JM: P(M) --+ M is a morphism in WSA(R). It has the following universal property: any morphism from a weak polytope to M factors through JM in a unique way [K2, IV §9]. It is evident from this that JM induces isomorphisms between the homotopy groups of P(M) and M. By a semialgebraic version of the Whitehead theorem [K2, V.6.10] it follows that jM is a homotopy equivalence. Intuitively, replacing M by P(M) means forgetting what happens in M "at infinity". Everything said in this subsection remains true if we replace R by a "sequential"real closed base field R, i.e. a real closed field which contains at least one non trivial zero sequence. Fortunately, most real closed fields occuring in practice are sequential. C. F i x e d p o i n t s Let R be an arbitrary real closed field and M a locally closed semialgebraic subset of R n. Then the direct limit Mloc of the spaces I~" with K E c(M) still exists in the category of all locally semialgebraic spaces over R. {Mloc is regular but perhaps not paracompact. ) It turns out that the abstraction /Y/loc of Mloc, as a locally ringed space, is an open subspace of l~/, namely the union of the open subspaces (/*()~ of M with g running through c(M). It follows that -~/loc (as a set) is also the union of the closed subsets

/? of 5I. A good understanding of /tT/loc as a subset of M is possible using ultrafilters of semialgebraic sets. Let 6 ( M ) denote the boolean lattice of all semialgebraic subsets

31 of M. As is well known, the points of ~4 can be identified with the ultrafilters in 6 ( M ) [Br, §4], [BCR, 7.2.4], [KS, III §5t, namely the ultrafilter F ( a ) corresponding to a point a E M consists of all S E 6 ( M ) with a E S. In this interpretation ~/loc is the set of all a E M with K E F ( a ) for at least one It" E c(M). Definition. The fringe Frin(M) of M is the set M -/tT/toc. Intuitively, Frin(M) consists of the points in 37/ "at infinity". We now turn to fixed points. Let f: M --* M be a semialgebraic map. It induces Qlinear maps Hq(f): IIq(M, Q) --* llq(M, Q) in ordinary (= semialgebraic) homology with coefficients in Q. The trace TrIIq(f) of flq(f) is well defined and an integer, since Hq(f) comes from an endomorphism of the finitely generated abelian group HdM, Z). We define the L fsd, tz h(f) as sual, A(f): = ~

( - 1 ) i TrHi(f).

i=0

Brumfiel has proved the following remarkable theorem ([B3I, cf. also [B21): T h e o r e m . Assume that A(f) # 0. Then the abstraction f : M --+ M of f has a fixed point. More precisely, either f: M --~ M has a fixed point or f has a closed fixed point in Frin(M). E x a m p l e . Let R = R, M = R2, f(x,y) = ( x + l , x y ) . Certainly f : M ~ M has no fixed point. But A(f) = 1, sincc M is contractible. Thus 9? must have at least one closed fixed point in Frin(R2). Such a fixed point is provided by the graph of the gamma function F(x). Let F denote the set of all A E ®(M) such that A contains the set {(x,F(x))lx > c} for some c > 0. This is an ultrafilter which gives us a closed point a E Frin (M) fixed under f . Brumfiel's fixed point theorem is important for his "real spectrum compactification" Mg of the Teichmfiller space/kIg of compact Riemann surfaces of genus g [B1], since it implies that every element of the Teichmfiller modular group has a fixed point in Mg.

§14 A n outlook What can be said about the situation in semialgebraic topology now, in the year 1991, and about prospects for the future, without too much speculation? Certainly abstract spaces contain a rich potential for further research. They should be "superior" to geometric spaces. But up to now nearly all the deeper results in homology and homotopy theory rely heavily on arguments in the geometric catcgory,

32 using triangulations, simplicial approximations, the homotopy extension property and the like. It would be highly desirable to have a homotopy theory for real closed spaces. There can be no doubt that we have found the "right" homotopy groups in the geometric setting. But all this does not tell us how to proceed in the abstract setting. It is only clear that an abstract homotopy theory should give us the geometric theory if we apply it to the abstractions of geometric spaces. For abstract cohomology the analogous problem seems to be easier. There exists an approach by N. Schwartz using inverse real closed spaces which looks promising. Schwartz talked about this at Luminy in October 1989, and then in the Ragsquad seminar at Berkeley in October 1990. In abstract homotopy one could proceed along the lines given in chapter II of Baues' book [Ba], where the hornotopy theory of a "cofibration category" has been developed. Or one could try to associate with a real closed space X (perhaps fulfilling some conditions like regularity or "tautness" [D2, I §3]) a simplicial set or, perhaps better, something like a simplicial space V X (in the classical sense, with Hausdorff topology), such that the homotopy type of V X encodes the honaotopy information about X. In his talk at Oberwolfach in June 1990 C. Scheiderer proposed a simplicial space V X (together with a "quasiaugmentation" V X --~ X ) which looks promising. The definition of V X employs chains of real valuations of commutative rings. Sche~derer was able to verify that, for any sheaf .T on X, the cohomology groups Hq(X,.T) can be computed from V X . Also, for X the abstraction of a geometric space M, V X represents the hornotopy type of M. These are hints that V X gives the right homotopy type. In the last years the interest in abstract spaces also has been enhanced by the discovery of the Spanish school (Andradas, Ruiz, ...), that questions in semianalytic geometry can be treated by considering real spectra. I just mention two basic observations: If A is a local ring of real analytic functions then the constructible subsets of Sper A are in natural one-to-one correspondence with the germs of semianalytic sets [Rz]. If A is the ring of global analytic functions on a compact real analytic manifold M then the constructible subsets of Sper A correspond bijectively with the globally semianalytic subsets of M [Rzl]. The homotopy or homology of such a constructible set (which is a real closed space) should have a close relation to the homotopy resp. homology of the corresponding semianalytic object. Concerning the two trends in the last ten years mentioned above (§4) we can safely say that the first one has run its course. Interaction between the geometric and the abstract theory is no longer a "trend" but a well established and widely used technique. However the first trend may continue in a wider context if we allow other meanings for the word "geometric".

33 Geometric spaces might be built, for instance, from semianalytic sets, or perhaps even from subanalytic sets. J. Denef and L. van den Dries have found a new approach to subanalytic sets [DvdD] which gives much hope that subanalytic sets are amenable to techniques similar to the ones we now have in the geometric semialgebraic topology. Subanalytic sets (more precisely, subsets of R" which are subanalytic in Pn(R)) and semialgebraic sets can both be subsumed under o-minimal structures, a notion stemming from model theory, cf. [PiS]. In his talk in the Ragsquad seminar at Berkeley, April 1991, van den Dries has outlined a "tame topology of o-minimal structures" with many of the features we are used to for semialgebraic sets, in particular N ° 1 - N ° 5 of the list in §3 for the geometric theory (cf. his forthcoming book [vdD], as soon as it appears). Thus o-minimal structures seem to provide a good framework for highly interesting lmw geometric spaces in the years to come. In 1990 R. Huber introduced "semirigid functions" [Hull as an offspring of his abstract approach to rigid analytic geometry [Hu]. They give us semirigid sets, which are vaguely analoguous to semianalytic sets in real analytic geometry. To understand these sets well, one definitely needs abstract spaces which are derivates of the real valuation spectra of commutative rings [Hul]. This, and the frequent occurrence of valuations in the theory of real closed spaces, are hints that valuation spectra will play a rote in a further development of semialgebraic topology, for which the word "semiatgebraic" is probably no longer appropriate. An introduction to valuation spectra has been given in [Hu, Chap. I] and [HuK]. From all of this it is pretty clear that tile second trend mentioned in §4, i.e., the employment of new spaces - both geometric and abstract, will persist. A c k n o w l e d g e m e n t s . I thank Roland Huber, Claus Scheiderer, Colm Mulcahy, and Victoria Powers for helpful criticism, both in mathematics (R.H., C.S.) and in language (C.M., V.P.). I further thank Michel Coste, Louis Mah~, and MarieFrancoise Roy for organizing the conference and thus providing - among many other things - a very pleasant opportunity to give a talk, of which this article is an expanded version.

34 References [Ba] t t . J . Baues, "Algebraic homotopy". Cambridge studies advanced math. 15, Cambridge Univ. Press 1988. [BCR] J. B o c h n a k , M. Coste, M.-F. Roy, "GSom~trie alg6brique r~elle". Ergebnisse Math. Grenzgeb. 3. Folge Band 12, Springer 1987. [BH] A. Borel~ A. Haefliger, La classe d'homologie fondamentale d'un espace analytique, Bull. Soc. Math. France 89,461 - 513 (1961). IBM] A. Borel, J . C . M o o r e , Homology for locally compact spaces. Mich. Math. J. 7, 137- 159 (1960). It. B r a k h a g e , "Topologische Eigenschaften algebraischer Gebilde fiber einem [Bra] beliebigen reell-abgeschlossenen GrundkSrpcr". Dissertation Heidelberg 1954. [Bre] G. B r e d o n , "Sheaf theory". McGraw Hill 1967. [Br] L. BrScker, Real spectra and distribution of signatures. In: "G~om6trie alg6brique rSelle et formes quadratiques" (Ed. J.-L. Colliot-Th~l~ne, M. Coste, L. Mah6, M.F. Roy), pp. 249 - 272. Lecture Notes Math. 959, Springer 1982. [B] G . W . B r u m f i e l , "Partially ordcred rings and semialgebraic geometry". London Math. Soc. Lecture Notes 37, Cambridge Univ. Press 1979. [B1] - - , The real spectrum compactification of Teichmfiller spaces. In: "Geometry of group representations" (Ed. W.M. Goldman, A.R. Magid), Contemporary Math. 74, A.M.S. 1988, pp 51 - 75. [B2] - - , A semi-algebraic Brouwer fixed point theorem for real affine spaces, ibid., pp 77 - 82. , A Hopf fixed point theorem for semialgebraic maps. Preprint Stanford [B3] University. [24] - - , Wittrings and K-theory. Rocky Mountain J. Math. 14, 733 - 765 (1984). [Bs] - - , Quotient spaces for semialgebraic equivalence relations. Math. Z. 195, 69 - 78 (1987). [CC] M. C a r r a l , M. Coste, Normal spaces and their dimensions. J. Pure Appl. Algebra 30, 227 - 235 (1983). [CR] M. Coste, M.-F. C o s t e - R o y , Le spectre rdel d'un anneau est spatial. C.R.A.S. Paris 290, 91 - 94 (1980). , Le topos ~tale r~el d'un anneau. In: 3 e colloque sur les categories [CR1] • , dedier a Charles Ehresmann, Cahiers de topologie et g~om. diff. 22-1, 19 - 24 (1981). , , La topologie du spectre r~el. In: "Ordered fields and real algebraic [CRy] geometry" (Ed. D.W. Dubois, T. Recio), Contemp. Math. 8, 27 - 59 (1982). [D] H. Delfs, Kohomologie affiner semialgebraischer R£ume. Dissertation Regensburg 1980. [D,] - - , The homotopy axiom in semialgebraic cohomology. J. reine u. angew. Math. 355, 108 - 128 (1985). - - , Homology of locally semialgebraic spaces. Lecture Notes in Math. 1484. Springer, to appear ( ~ Habilitationsschrift Regensburg 1984).

35

[D3] - - ,

Semialgebraic Borel-Moore homology, Rocky Mountain J. Math. 14, 987 990 (1984). [DK] H. Delfs, M. K n e b u s c h , Semialgebralc topology over a real closed field Ih Basic theory of semialgebraic spaces. Math. Z. 178, 175 - 213 (1981). [DK,] - - , , On the homology of algebraic varieties over real closed fields. J. reine u. angew. Math. 335, 122 - 163 (1982). [DK2] , --, "Locally semialgebraic spaces". Lecture Notes Math. 1173 (1985). [DKa] , , An introduction to locally semialgebraic spaces. Rocky Mountain J. Math. 14, 945 - 963 (1984). [De] C.N. Delzell, A finiteness theorem for open semialgebraic sets, with applications to Hilbert's 17th problem. In: "Ordered fields and real algebraic geometry" (Ed. D.W. Dubois, T. Recio), Contemporary Math. 8, 79 - 97 (1982). [DvdD] J. D e n e f , L. van den Dries, p-adic and real subanalytic sets. Ann. Math. 128, 79 - 138 (1988). [ES] S. E i l e n b e r g , N. S t e e n r o d , "Foundations of algebraic topology". Princeton Univ. Press 1952. [ci] B. Giesecke, Simpliziale Zerlegung abzghlbarer analytischer Rgume. Math. Z. 83, 177- 213 (1964). [EGA II A. G r o t h e n d i e c k , J. D i e u d o n n 6 , "l~lements de g6om~trie alg6brique h Le language des sch~ma.s. Publ. Math. N ° 4, Inst. Hautes Etudes Sci. 1960. [EGA I*] , , "Elements de gdom~trie alg~brique I". Grundlehren math. Wiss. 166, Springer 1971. [Hal R. H a r d t , Semialgebraic local triviality in semi-algebraic mappings. Amer. J. Math. 102, 291 - 302 (1980). [Ho] M. H o c h s t e r , Prime ideal structure in commutative rings. Trans. Amer. Math. Soc. 142, 43 - 60 (1969). [HM] J. H o u d e b i n e , L. Mah6, S~paration des composantes connexes r~elles dans le cas des vari~t6s projectives. In: "Gdometrie alg4brique r6elle et formes quadratiques (Ed. J.-L. Colliot-Thdl~ne, M. Coste, L. Mahd, M.F. Roy), pp 358 - 370. Lecture Notes Math. 959, Springer 1982. [Hu] R. H u b e r , "Bewertungsspektrum und rigide Geometrie". Habilitationsschrift, Regensburg 1990. [Hul] - - , Semirigide Funktionen. Preprint Regensburg 1990. [HuK] R. H u b e r , M. K n e b u s c h , On valuation spectra. To appear in the proceedings of the RAGSQUAD-seminar Berkeley 1990/91, Amer. Math. Soc. [HuS] R. H u b e r , C. S c h e i d e r e r , A relative notion of local completeness in semialgebraic geometry. Arch. Math. 53, 571 - 584 (1989). [El M. K n e b u s c h , An invitation to real spectra. In: "Quadratic and hermitian forms" (Ed. C.R. Riehm, I. Hambleton), Canadian Math. Soc. Conference Proceedings 4, 51 - 105 (1984). [KI] - - , Some open problems. In: "Conference on quadratic forms - 1976" (Ed. G. Orzech). Queen's Papers Pure Appl. Math. 46, 361 - 370 (1977). [K21 - - , "Weakly semialgebralc spaces". Lecture Notes Math. 1367, Springer 1989.

36 [K3] - - , "Semialgebraic fibrations and covering maps". In preparation. [KS] M. K n e b u s c h , C. S c h e i d e r e r , "Einffihrung in die reelle Algebra", Vieweg,

[L] [M] [May]

[Mi] [P] [PiS] [Rol [RI

[ 11 [Sch]

iS] [Si]

IS3] [S4] [Spa] [Sw]

Braunschweig-Wiesbaden 1989. S. Lojasiewicz, Triangulation of semi-analytic sets. Ann. Scuola Norm. Sup. Pisa (3) 18, 449- 474 (1964). L. Mah~, Signatures et composantes connexes. Math. Ann. 260, 191 - 210 (1982). J.P. M a y , "Simplicial objects in algebraic topology". Van Nostrand Math. Studies 11, Van Nostrand 1967. J. M i l n o r , The geometric realization of a semi-simplicial complex. Annals of Math. 65, 357 - 362 (1957). M. P r e c h t e l , "Endliche semialgebraische R~.ume". Diplomarbeit Regensburg 1988. A. Pillay, C. S t e i n h o r n , Definable sets in ordered structures, I. Trans. Amer. Math. Soc. 295, 565 - 592 (1986). R. Robson, Embedding semialgebraic spaces. Math. Z. 183, 365 - 370 (1983). M . - F . Roy, Faisceau structural sur le spectre r~el et fonctions de Nash. In: "G~om~trie alg~brique r~elle et formes quadratiques", pp. 406 - 432. Lecture Notes Math. 959, Springer 1982. J . M . Ruiz, Basic properties of real analytic and semianatytic germs, Publ. Inst. Recherche Math. Rennes 4, 29 - 51 (1986). -- , On the real spectrum of a ring of global analytic functions, ibid., 84 - 95. C. S c h e l d e r e r , Quotients of semi-algebraic spaces. Math. Z. 201, 249 - 271 (1989). N. Schwartz, "Real closed spaces". Habilitationsschrift Ludwig-MaximiliansUniversit£t Mfinchen 1984. -- , "The basic theory of real closed spaces". Memoirs Amer. Math. Soc. 77, N ° 397 (1989). -- , "The basic theory of real closed spaces". Regensburger Mathematische Schriften 15, Fakult~t ffir Mathematik der Universit~t Regensburg 1987. , Inverse real closed spaces. Illinois J. Math., to appear soon. , Open morphisms of real closed spaces. Rocky Mountain J. Math. 19, 913 - 939 (1989). E . H . Spanier, "Algebraic topology". McGraw-Hill t966. R . M . Switzer~ "Algebraic topology - Homotopy and homology". Grundlehren math. Wiss. 212, Springer 1975.

A d d r e s s : Fakult£t f/Jr Mathematik, Universit£t Regensburg, Universit£tsstr. 31, D-8400 Regensburg

Algebraic Geometric Methods in Real Algebraic Geometry R. P a r i m a l a

This article is based on the talk given at the conference. It is a brief overview of certain recent developments in the study of the topology of a real algebraic variety through a combination of techniques from algebraic geometry and the theory of quadratic forms. Our underlying theme is to illustrate how the higher dimensional variations of some basic theorems of Witt ([W]) for real curves have encompassed a wide range of mathematical activities in the last decade. Thus, this article is in no sense a comprehensive survey of the results linking algebraic geometry and real algebraic geometry. For instance, an area left out is the construction of real moduli spaces. Let X be a real variety and X(JT~) its set of/R-rational points, equipped with the strong (Euclidean) topology. We consider principally two classes of maps which relate geometric invariants of X to topological invariaats of X(~). I. Suppose X is smooth and X ( ~ ) compact. We discuss the cycle map from the Chow group of X to the homology groups of X ( ~ ) with 2~/2 coefficients, its kernel and image. II. We discuss two kinds of fibre maps: 1) The signature map from the Witt group of X to the group of 2~-valued continuous functions on X ( ~ ) . 2) The "mod-2" signature map from the unramified 6tale cohomology of X to the cohomology of X(L~). The study of these maps for real curves is contained in the above mentioned paper of Witt.

38 We fix the following notation for the rest of this article. By a variety X over a field k, we mean a reduced irreducible scheme over k and by a subvariety a closed subscheme which is reduced and irreducible.

1

T h e cycle m a p

Let X be a nonsingular variety o v e r / R such that X ( ~ ) is compact. Let Zk(X) denote the group of )~-cycles on X; namely the free abelian group on k-dimensional sub-varieties of X. Let Pk(X) denote the subgroup generated by cycles which are rationally equivalent to zero. Let z*kh(x) denote the subgroup of Zk(X) generated by "thin cycles", i.e. z~h(x) is generated by k-dimensional subvarieties Y of X with Y(/R) not Zariski dense in Y. The cycle map clk: Zk(X) ~ H~,(X(J~),2g/2) is defined as follows: Let Y be a k-dimensional subvariety of X. If Y is thin, then we define clk(Y) = O. Otherwise, let cry E Hk(Y(J~£),~/2) be the fundamental class of Y ( ~ ) . Then c4(Y) = i(ay), where i : Hk(Y(JT~), ~ / 2 ) --, Hk(X(~), 2g/2) is induced by the inclusion Y(/R) ---, X ( ~ ) (cf. [Bor-H] 5.12). The image of the cycle map is called the group of algebraic homology cla88e~, denoted H."tg(X(~), 2~/2). If p: H*( X (1R), 2~/2 ) ~ H.( X (1R), 2g/2 ) denotes the Poincar@ duality isomorphism, then p-l(H~.tg(X(~),~,/2)) = H*la(X(~),2g,/2 ) is called the

group of algebraic cohomology classe~. We first look at the ken,tel of the cycle map. This map respects rational equivalence ([Sor-H] 5.13) so that clk(Pk(X)) = 0. Further, by definition, c4(Z~h(x)) = 0. The question whether the kernel of the cycle map is precisely the group Pk(X) + z~h(x) for smooth projective varieties X over remained open until recently. We first indicate how the results of Witt imply that for a smooth projective real curve X, kernel of clo is equal to Po(Z) + z~h(x). Let X(/R) = UCi, {Ci} denoting the set of connected components of X ( ~ ) . Each Ci is topologically a circle. Suppose we axe given an even number of points {P1, P 2 , . . . , P2,~} on some Ci. Witt proves that there is a rational function f E K~(X), regular on X(/R) which changes sign on Ci exactly at the points Pi. Knebusch ([K]x 4.5 a, p. 61) refines this into the following theorem: There exists a rational function f E / R ( X ) ,

39

regular on X(K/) such that divf=

Pi+D,

~ l<_i<2n

where D is a divisor on X with (supp D) N X ( ~ ) = ¢ (Indeed one may choose supp D to be any given complex point of X). This implies that kernel clo = z~h(x) + Po(X). Colliot-Wh616ne and Ischebeck [C-I] prove that ker clo = z~h(x)+ Po(X) for varieties X of arbitrary dimension and their method of proof is to reduce to the case of curves. Brhcker [Br]l proves the same result for the case of divisors, namely, k = dim X - 1. Ischebeck and Schiilting [I-Sc] prove more generally that ker clk= ztkh(x) + Pk(X) for all varieties X proper over gi~. Their proof has two main steps : 1) If f : Z ---* X is a morphism of smooth projective varieties of dimensions k and n respectively, with f / R : Z ( ~ ) ~ X(~r~) bordant, then f.([Z]) is rationally equivalent to a thin cycle. 2) If z C Zk(X) with clk(z) = 0, then there is a morphism f : Z --. X of smooth projective varieties with f ~ : Z(/R) --* X ( ~ ) bordant and f.([Z]) = z + z', Z t e Z~,a(Z). This is achieved by relating H~Ia(X(K~),2F,/2) to the algebraic part of the k-bordism group. Schiilting [Sc] studies the behaviour of the cokernel of the cycle map under birational transformations of smooth projective varieties. Let Tk(X) = Hk(X(.~),2g/2)/H~tg(X(J1:~),Z/2). If X is a smooth projective variety of dimension n and rr : Y --, X is a blow-up with smooth center Z of dimension n - d, then he shows that

Tk(Y)", T,,(X)•

E3

Tj(Z).

k-d+l<j<_k-1

In particular, TI(X) = HI(X(1R),2~/2)/H[tg(X(~),~,/2)is a birational invariaat for smooth projective varieties. If dim X = n, Tn(X) ~ (2g/2) s-l, where s is the number of connected components of X ( ~ ) , which is also a birational invariant for smooth projective varieties over ~ (cf [Bo-C-R] Cor 10.4.6. p. 227 and [D-K] §13). Thus, for a smooth projective surface X, T.(X) is a birational invariant of X. Sithol ([si] Th. 5) proves that for real rational surfaces (i.e., surfaces for which X ~ is birational to iP~), TI(X) = 0 .

40

2

Image of the cycle map; C o n n e c t i o n s w i t h K-Theory

Let X be an atCfine algebraic variety over ~ with X ( ~ ) compact. Let KF(X) denote the Grothendieck group of topological F-vector bundles on X ( ~ ) , F being ~ , ¢ or ~-/. By a theorem of Swan ([Sw]l p. 268), KF(X) ~Ko(F®C(X)), the Grothendieck group on finitely generated projective F® C(X)-modules, C(X) denoting the ring of continuous functions on X ( ~ ) . We define a certain algebraic part I4,~tg(X) of K ~ ( X ) which, under the Stiefel-Whitney map

w: KK~(X ) ~ H*(X(~),2g/2), maps into H~tg ( x ( ~ ) , 2g/2). Let ~ ( X ) denote the ring of regular functions on X; i.e. rational functions f/9 E ~ ( X ) such that g is non-vanishing on X ( ~ ) . A continuous vector bundle ( on X ( ~ ) is said to be strongly algebraic if its shea£ of sections is associated to a projective module over ~ ( X ) . The group KF_,~g(C(X)) is simply the image of the natural homomorphism Ko(F ® T~(X)) --+ KF(C(X)). The Stiefel Whitney map w maps K1R_,~,g(X) into H2tg(X(J~),2~/2 ) ([Be-T]2 2.4) and the two groups K~_,~ta(X ) and H*t.q(X(~),2~/2 ) are objects of extensive study for compact real varieties X. By a theorem of Swan ([Sw]2 Th. 2.2) for compact varieties X, Ko(T~(X)) --~ K g~(X) is injective. Further, a vector bundle ~ on X ( ~ ) is strongly algebraic if and only if its class in K~:~(X) belongs to Kn:~_alg(X). Thus, we may identify the two groups Ko(T~(X)) and K~_~,tg(X ). One of the basic problems is to characterise strongly algebraic vector bundles on X ( ~ ) . Typical examples of varieties X for which I(1R_~,tg(X) = K.g~(X) are spheres (IF]) and projective spaces ([G-R]). Swan shows [Sw]3 that for an n-dimensional sphere X over 1R, Ko(X) " Ko(T~(X)) "2_ K ~ ( X ) . Benedetti mad Tognoli [Be-T]1 prove that if X is a smooth afflne variety over/R, homeomorphic to S '~, K~_ata(X ) = K ~ ( X ) . In [Bo-K]3, there is a characterization of strongly algebraic vector bundles on X ( ~ ) in terms of Stiefel-Whitney invariants, for low dimensional varieties. More precisely, Bochnak and Kucharz prove that if dim X _< 3, a continuous vector bundle of constant rank on X(/R) is strongly algebraic, if and only if its StiefelWhitney classes are algebraic.

41

Bochnak, Bfichner and Kucharz ([Bo-B_LK] Th.I.1) exhibit a large family of real varieties X for which the group Kf-alg(X) is rather small. They show that if X '--* ~ P ~ is a real variety of dimension at least 1 such that X ( ~ ) is connected and X ¢ is a non-singular complete intersection, KF_,Ig(X) is either finite or a direct sum of a finite group and 2~. If dim X is odd and H*'*~(X, 2~) is torsion free, they show that K¢_~lg(X ) = O. However, the groups KF(X) may generally have arbitrarily large rank (cf. [So-B-K]).

3

C e r t a i n t y p e s of algebraic m o d e l s for compact C°%manifolds

After the solution of the Nash conjecture by Tognoli [T], the following natural questions, related to approximations of algebraic models by non-singular algebraic sets, were raised (cf. [So]). Let M be a compact (connected) Coomanifold. An algebraic model of M is a non-singular affine scheme X with X ( ~ ) diffeomorphic to M. Q1) Given M as above, does there exist an algebraic model X with

HT~,g(X(1R),~/2 ) = H*(X(~),Z/2)? Q2) Characterise all compact Coo-manifolds M satisfying the following condition: For every algebraic model X of M,

H*lg(X, 2g/2 ) = H*(X(~),2g/2). It was shown in [A-K] and [Be-T]2 that there exists an algebraic model X for M with H~,g(X(R=~),2~/2) = Hl(X(J~),2~/2). In general,however, Q1 has a negative answer. Benedetti and Dedo ([Be-D]) construct a l l dimensional compact C°°-manifold M with H2~g(X(~), z~/2) ¢ H2(X(~),2~/2) for any algebraic model X of M. Regarding Q 2 , / R i p2~ are examples of such manifolds. For any algebraic model X of / R ~ 2~, HI(X(I~), 2~/2) ~ 2~/2, with the non-trivial element given by the first Stiefel-Whitney class of the determinant of the tangent bundle of/RiP 2". Thus the non-orientability of ~ i P 2" implies that H~tg(X(Kt),2Z/2 ) = H'(X(K~),2~/2). However, until recently, it remained open whether ~ 3 has this property. In ([So-B-K]), examples of 3-dimensional varieties X,

42 diffeomorphic to / R ~ 3 were constructed, which have the property that 1 r K~,,g(X) = 0. In fact, for such an X,r H~o(X(~),~/2 ) = 0 (cf. [Bo-BK], Prop. 7.8). Thus ~ 3 does not belong to the class defined in Q2. The question, given a subgroup G C HI(M, 2~/2),M a compact Ca-manifold, whether there exists an algebraic model X with H~lg(X(~),2~/2 ) = G, has been studied by Bochnak and Kucharz in [BoK]4. A necessary condition to be satisfied by G, for non-orientable M, is that it should contain the first Stiefel-Whitney class of M. It is shown in [Bo-K]4 that if M is a compact connected orientable C~-maxfifold of dimension _ 3, for any G c Hi(M, ~ / 2 ) , there exists an algebraic model X such that H~tg(X, 2~/2) = G. The same conclusion holds for non-orientable M , provided the first Stiefel-Whitney class of M belongs to G. For a detailed bibliography on applications of the a~alysis of the groups H~lg to problems in various branches of mathematics, we refer to [BoK]4. We mention that Bochnak and Kuchaxz use their computations of KF-~lg(X), (cf. §2) to construct algebraic models X ,--, ~2k+1 for a compact C ~ hypersurface in ]R2k+1 with the property that every regular mapping from X into S 2k is null-homotopic [Bo-K]2. They contrast this result with the odd dimensional case, where for any real algebraic set of dimension 2k + 1 in ~ , compact, connected and orientable, either every smooth mapping into S 2k+~ is homotopic to a regular map or precisely those maps of even topological degree have this property [Bo-K]I.

4

W i t t g r o u p s o f real v a r i e t i e s a n d s i g n a tures

Let X be any variety over a real closed field R and let SpecrX denote the real spectrum of X (cf. [Bo-C-R], §7.1). We recall that the points of SpecrX are pairs (x,u), where x E X and u is an ordering of R(x). Let W(X) denote the W i t t group of X. The total signature homomorphism sgn : W(X) ~ C(SpecrX, ~ ) assigns to any quadratic bundle q on X, the function sgn(q) : Spec,.X --* z~, defined by sgn(q)(x,u) = signature of the restriction of q to the fibre at x, with respect to the ordering u. Let s denote the number of semi-algebraic components of X(R). Then C(Spec~X, 2~) = ~ and in the case R = ~ , s is the number of connected

43 components of X ( ~ ) . The question whether the signatures separate semialgebraic components of X(R) was raised by Knebusch in [K]3. The separation of connected components of curves over/R by signatures is due to Witt. Knebusch himself proves [K]I that the signatures separate semi-algebraic components of curves over real closed fields, using Geyer's algebraisation of Witt's results [Ge]l. In the beautiful paper [M]I, Mah~ gives an attCirmative answer to this question in the most general setting of any atiine scheme SpecA with 2 invertible in A. He proves in fact that given an open and closed subset U C SpecrA, there exists a quadratic bundle q over A and an integer N > 0 such that sgn(q) ] U = 2N and sgn(q) [ SpecrA \ U = 0. The separation of semi-algebraic components of a smooth projective variety over a real closed field was proved in Houdebine-Mah6 [H-M]. Thus the cokernel of the total signature map is 2-primary torsion. A natural question is whether the exponent 2g of the cokernel of the signature map is bounded by a function of the dimension of X. Mah6 [M]I proved that N can be bounded by a function f(s, m), where s is the number of inequalities necessary to define a semi-algebraic basic open subset of X(R), and m is the least integer such that every totally positive element of R[X] divides an element of the form 1 + ~]1<,<,~ x~,x, C R[X]. Srhcker ([Br]2 and [Br]3) proved that s is bounded by a function of the dimension of X. Mah6 [M]2 proves that m < n - 1 + 2"+1, where n = dimX. This settles the above question in the affirmative. It would be interesting to have more precise bounds for this exponent in terms of the dimension of X. For instance, one could ask whether this exponent is bounded by 2"+1, for smooth varieties of dimension n over _~. This is true for curves, which is due to Witt and for smooth projective surfaces and abelian varieties, which is due to Colliot-Th~l~ne and Sansuc ([C-S], Th. 4.1 and Th. 5.4). They use Geyer's duality theorem for abelian varieties X over ~ [Ge]2 which gives the bound s < 2g where g is the dimension of X - a n d s denotes the number of connected components of We now mention a refinement of Mah~'s result due to Brumfiel [Bru], which compares the Witt group W(X) and the topological W i t t group W(X(R)). Let ,S°(X) denote the ring of semi-algebraic continuous functions on X(R). Since every semi-algebraic bundle on X(R) has a unique positive definite structure upto isometry (cf. [Bo-C-R] 15.1.11, p.331),

44

there is an isomorphism Ko(S°(X)) ~ W(Spec~X). The natural inclusion R[X] ~ ,S°(X) induces a homomorphism i : W ( X ) --* W ( S ° ( X ) ) = W(Spec~X) = Ko(S°(X)) and we have a commutative diagram

w(x)

,

sgn \

Ko(S°(X))

~/ C(Spec~X, 2~)

Brumfiel proves that the cokernel of i is 2-primary torsion. If R = / R , we have Ko(S°(X)) ~_ Ko(C(X))= I(.fl:t(X), ([Bo-C-R], 12.7.8, p. 287). Thus the theorem above compares the Witt group of X with the K-theory of X ( ~ ) and asserts that the cokernel of this map is 2-primary torsion. We record a few computations for the Witt groups of real varieties. The computation of the W i t t groups of real curves, or more generally of curves over real closed fields is due to Knebusch [K]2. The Witt group is a birational invariant for smooth projective surfaces. In [Su], Sujatha computes the Witt group of a real projective surface X in terms of certain birational invariants of X. For smooth projective real rational surfaces X (that is X~ is birational to lP~7), the number of connected components s of X(fl~) determines W ( X ) completely: if X(I~) is not empty, W ( X ) ~2g 8 (~(2g/2) (8-1) and if X(I~) is empty, W ( X ) ~ 2~/4, being generated by < 1 > (cf. [Su] § 4), noting that Theorem 4.2 needs a slight modification). We also mention here some computations of the symplectic Witt group of a real variety in terms of algebraic cohomology classes. Let X be a compact affine real variety of dimension at most 3. Let W S p ( X ) denote the symplectic Witt group of X. An element [(1 E W S p ( X ) is represented by a rank 2 symplectic bundle ~ on X, since every symplectic bundle of rank _> 4 on X has a non-vanishing section and hence splits off a hyperbolic plane 2 ;" ([B], §4.12). Let a(~) E H~,,(X (/R), 2g) denote the Euler class of (. Indeed,

the image of c~(~) in H2(X ( ~ ) , 2g/2) is simply the second Stiefel-Whitney class w~(~) of ~. Let /3 : H ~1 t g (rX ( ~ ) , N / 2 ) ~ H~tg(X 2 ( ~ ) , 2g ) denote the Bockstein homomorphism. Then there is a homomorphism

induced by a. Barge and Ojanguren [Ba-O] show that if dim X _< 2, 5. is an isomorphism, observing that in dimension 2, H~2tg= H ~. The mod 2 version

45 of this result, namely that W S p ( X ) / 2 "" (2~/2)", s denoting the number of connected components of X ( ~ ) , is due to Pardon ([P], Th.B) and a refinement of Pardon's result through different methods is in [O-P-S]. Bochnak and Kucharz ([Bo-K]a Prop. 2.3) show that for dim X _< 3, a induces an isomorphism 6 : W S p ( ~ ( X ) ) ~- H~,g(X(1R),2~)/flH~,g(X(~),2F,/2), where "Fg(X) denotes the ring of regular functions on X.

5

Etale

cohomology

and

real cohomology

Let X be a smooth quasi-projective curve over ffL Let Br(X) denote the Brauer group of X. Given an element ¢ C Br(X) , represented by an Azumaya algebra A on X, for any rational point x C X ( ~ ) , the restriction of A to the fibre at x defines a central simple algebra over ~ which is the class of i f / o r zero in the Brauer group of ~ . This defines a map Br(X) -~ C(X(1R),2~/2). Witt's construction of regular functions on X with prescribed signs on each connected component of X ( ~ ) shows that this map is surjective. We now explMn a higher dimensional analogue of this theorem. Let X be any quasi-projective scheme. We define (cf. [C-P]) a fibre map, called the rood-2 signature map,

h, : H ~ ( X , #2) --* C ( S p e % X , Z[, /2) as follows: Let (x, u) C Spec,.X, 1., : k(x) "--+L being an imbedding in a reM

closure. Let

: Ha(X,m) -, H"(k(x),m) z+ H"(r,m)

2a/2 be

the restriction to the fibre. We define h,~({)(x,u) = ~b(x,,)({). Let X be a smooth quasi-projective variety over a real closed field R. Let 7-{" denote the Zariski sheaf associated to the presheaf U H H~(U,#2). Since every section of 7~n extends uniquely, at a point x C X, to an 6tale cohomology class, the naturality of h,~ yields a homomorphism

hn : H ° ( X , TI '~) ~ C(SpeGX, 2~'/2 ). It is shown in [C-P] that, if X is a smooth quasi-projective variety of dimension d over a real closed field, then for n >_ d + 1, the maps h. are

46

isomorphisms. In particular for n _> d + 1 the groups H°(X, T/") are isomorphic to (z~/2) ~, s denoting the number of semi-algebraic components of X(R), and hence they are finite. We point out a consequence of this result for quadratic forms. Knebusch [K]3 raised the following question: "Is the Witt group of a smooth affine algebraic variety over ~ finitely generated?" The answer to this question is positive for dimX ~ 2, the case of curves being treated by Knebusch [K]2, and the dimension 2 case by Ayoub [Ay]. It was shown in [Pa] that for a smooth atYine threefold X over/R, W(X) is finitely generated if and only if CH2(X)/2 is finite, CH 2 denoting the group of codimension two cycles, modulo rational equivalence. The finiteness of CH2(X)/2 is a wide open question in dimension 3 and the above theorem, which establishes the equivalence of this problem in geometry to one on quadratic forms, is proved via real algebraic geometry, namely through the finiteness of H°(X, TI4). We describe an interpretation of the map hn, due to Scheiderer, which leads to calculations of the higher cohomology groups of the sheaf 7-/n in terms of the semi-algebraic cohomology of X(R). Let ~2 " SpecrX ---* X be the natural map, ~(x,v) = x. The maps hn : g~(x,#2) ---* C (SpecrX, 2~/2) are functorial and yield a sheaf morphism hn : 7"/n ¢.(2~/2). The theorem of [C-P] can be restated as an isomorphism h,~ : 7"/" _~ ~.(2~/2) of sheaves for n > d + 1, where X is a smooth quasiprojective variety of dimension d over a real closed field R. In particular, for i > O,n >__d + 1, H i ( X , ~ n) ~ H~(X,T.2~,/2). The beautiful remark of Scheiderer (cf [S]1) is, that for any variety X over a real closed field R, even possibly singular, RqT.(2~/2) = 0, so that Hi(X, ~.2~/2) ~_ H i (Spec~X, Z~/2) = H~,(X i (R), 2~/2) (cf [D-K]). Specializing R to the field of real numbers, we obtain an isomorphism h,~: H'(X,7-/'~) ~ H'(X(~),2~/2), for n k d + 1, i _> 0. In particular, if X is a smooth surface over ~ with X(/R) compact, and s is the number of connected components of Z ( ~ ) , the degeneracy of the Bloch-Ogus spectral sequence ([C-P]) gives, for i _> 4,

Hi(X) ~_ H°(X, 7-14)o g 1 ( x , ~ 3 ) o H2(X, 7-t2) ,.,

(cf [C-P] 3.2.1 and [S]1).

e

47

The following observation of Colliot-Th61~ne [C] allows us to dispense with the smoothness assumption on X in the above theorems. Cox in [Co] proves that, for any variety X over/R, there is a natural isomorphism

H:t(X, 2g/2) ~m

G

O
for m >__2 d + 1. This yields an isomorphism 7-/r~ "~ T.2~/2, for m >__2 d + 1, in view of the above mentioned remarks of Scheiderer. Since for an affine scheme X, and for m >__d + 1 , H ~ ( X ¢ , 2 ~ / 2 ) = 0, the Kummer sequence yields a long exact sequence in etale cohomology:

H'~(X~T,z~/2) ..-, Hm(X, 2~/2) uS.-;' Hm+l(X,z~/2)---, H'~+I(X(T,2~/2), which, for m > d + 1, yields an isomorphism 7"/m -% 7-/m+] of sheaves and which commutes with h, so that one has hm : 7-/"* & ~.2~/2, for m > d + 1. In [S]2, Scheiderer tries to interpret the results of Cox in the general setting of the real spectrum of any scheme. This leads to a natural isomorprism lJ.m H'~(X, 2~) '2_@ Hi(Spe%Z, 2~/2), -'-+n i>0

for any affine scheme X in which 2 is invertible. This places in an abstract setting, the results of Arason-Elman-Jacob for fields ([A-E-J], 2.4). One of the interesting consequences of this analysis is that the connected components of the real spectrum of any commutative ring are separated by the @tale cohomology classes. A c k n o w l e d g e m e n t s . I thank J.-L. Colliot-Th6l~ne for his help in the preparation of this talk. I also thank M.A. Knus and J. Barge for critically reading the manuscript.

References [A-K] S. Akbulut, H. King, A relative Nash theorem, Trans. Amer. Math. Soc. 267 (2), 465-481 (1981). [Ar-E-J] J.Kr.Arason, R.Elman, B.Jacob, The graded Witt ring and Galois cohomology, CMS.Conf. Proc. Vol.4, Providence: AMS, 1984, 17-50.

48

lay]

G. Ayoub, Le groupe de Witt d'une surface r6elle. Comment. Math. Helv. 62, 74-105 (1987).

[B]

H. Bass, Unitary algebraic K-theory, Hermitian K-theory and Geometric Applications, Battelle Institute Conference 1972. Algebraic K-theory III, SLN 34:3, 1973, pp. 57-265.

[Ba-O] J. Barge, M. Ojanguren. Fibr6s a/g6briques sur une surface r6elle. Comment. Math. Helv. 62, 616-629 (1987). [Be-D l R, Benedetti, M. Dedo, Counterexamples to representing homology classes by real algebraic subvarities up to homeomorphism. Compositio Math. 53, 143-151 (1984). [Be-T]1 R. Benedetti, A. Tog:noli, Sur les fibr~s vectoriels alg~briques rfiels. C.R. Acad. Sc. Paris, t. 287, S~rie A-831-833 (1978). [Be-T]2 R. Benedetti, A. Tognoli, Remarks and counterexamples in the theory of real algebraic vector bundles and cycles, Gfiom~trie alg~brique r~elle et formes quadratiques, Rennes 1981, SLN 959, pp. 198-211.

[Bo]

J. Bochnak, Topology of real analytic sets - Some open problems. Rennes 1981, SLN 959, pp. 212-217.

[Bo-B-K] J. Bochnak, M. Buchner and W. Kucharz, Vector bundles on real algebraic varieties, K.-theory, 3, 271-298 (1990). [Bo-C-R] J. Bochnak, M. Coste, M-F.Roy, G6om6trie alg6brique r6elle, Ergeb. der Math. 3. Folge, Bd. 12, Springer 1987. [Bo-K]I J. Bochnak, W. Kucharz, Representation of homotopy classes by algebraic mappings, J. Reine. Angew. Math. 377, 159-169 (1987). [Bo-K]2 J. Bochnak, W. Kucharz, On real algebraic morphisms into even dimensional spheres, Ann. of Math. 128, 415-433 (1988). [Bo-K]3 J. Bochnak, W. Kurcharz, K-theory of real algebraic surfaces and threefolds, Math. Proc. Camb. Phil. Soc. 106, 471-480 (1989).

49

[Bo-K]4 J. Bochnak, W. Kucharz, Algebraic models of smooth manifolds, Invent. Math. 588-611 (1989).

[Bor-H]

A. Borel, A. Haefliger, La classe d' homologie fondamentale d'un espa~e analytique. Bull. Soc. Math. France, 89, 461-513 (1961).

[Br]l L. BrScker, Reelle Divisoren, Arch. Math. 35, 140-143 (1980). [Br]2 L. BrScker, Minimale Erzeugung von Positivbereich. Geom. Dedication 16. 335-350 (1984).

[Br]

L. BrScker, Spaces of orderings and semialgebraic sets, Quadratic and hermitian forms, CMS Conf. Proc. Vol. 4, Providence: AMS, 1984, pp. 231-248.

[Bru] G.W. Brumfiel, Witt rings and K-theory. Math. 14, 733-765 (1984).

[c]

Rocky Mountain J. of

J.-L. Colliot-Th61~ne, Abstract of a talk at OberwoKach, 1990.

[c-i]

J.-L. Coltiot-Th@l~ne, Ischebeck, L' @quivalence rationnelle sur les cycles de dimension z@ro des vari@t@s alg@briques r@etles. C.R. Acad. Sci. Paris, Serie I, 292, 723-725 (1981).

[c-P]

J.-L. Colliot-Th@l~ne, R. Parimala, Real components of algebraic varieties and @tale cohomology, Invent. Math. 101, 81-99 (1990).

[c-s]

J.-L. Colliot-Th@l&ne, J.-J. Sansuc, Fibr@s quadratiques et composantes connexes r@elles, Math. Ann. 244, 105-134 (1979).

[Co]

D.A. Cox, The @tale homotopy type of varieties over ~ , Proc. Amer. Math. Soc. 76, 17-22 (1979).

[D-K] H. Delfs, M. Knebusch, Semialgebraic topology over a real closed field II: Basic theory of semialgebraic spaces. Math. Z. 178, 175-213

(1981).

IF]

R. Fossum, Vector bundles over spheres are algebraic. Invent. Math. 8, 222-225 (1969).

50

[G-al

A.V. Geramita, L.G. Roberts, Algebraic vector bundles on projective space. Invent. Math. 10, 298-304 (1970).

[Ge]l

W.D. Geyer, Ein algebraischer Beweis des Satzes yon Weichold fiber reelle algebraische FunktionenkSrper. In : Algebraische Zahlentheorie, Oberwolfach, 1964.

[Ge]

W.D. Geyer, Dualit/~t bei abelschen Varietiiten fiber reelle abgeschlossen K5rpern, J. reine, angew. Math. 293-294, 62-66 (1977).

[H-MI

J. Houdebine, L. Mah6, S6paration des composantes connexs r6elles dans le cas des variet6s projectives, G6om6trie alg6brique r6elle et formes quadratiques, Rennes 1981, SLN 959, p. 358-370.

[I-Sc]

F. Ischebeck, H-W. Schfilting, Rational and homological equivalence for real cycles, Invent. Math. 94, 307-316 (1988).

[K]I

M. Knebusch, On algebraic curves over real closed fields I. Math. Z.

150, 49-70 (1976). M. Knebusch, On algebraic curves over real closed fields I1. Math. Z.

151, 189-205 (1976).

[KI3

M. Knebuseh, Some open problems, Queen's Papers in Pure and Appl. Math. 46, 361-370 (1977).

[M]I

L. Mah6, Signatures et componentes connexes. Math. 191-210 (1982).

[MI2

L. Mah6, Th6or&me de Pfister pour les vari6t6s et anneaux de Witt r6duits, Invent. Math. 85, 53-72 (1986).

Ann. 260,

[O-P-S] M. Ojanguren, R. Parimala and R. Sridharan, Symplectic bundles over affine surfaces, Comment. Math. Helv. 61,491-500 (1986).

[PI

W. Pardon, A relation between Witt groups and zero cycles in a regular ring, Algebraic K-theory, Geometry and Analysis, SLN 1046, 1984, pp. 261-328.

51

R. Parimala, Witt groups of affine threefolds, Duke Math. J. 57, 947-954 (1988).

[Sll

C. Scheiderer, A remark on the paper, Real components of algebraic varieties and 6tale cohomology by J.-L. Colliot-Th616ne, R. Parimala, Preprint, 1990.

[S]2 C. Scheiderer, Real and 6tale cohomology, Preprint, 1991. [Sc] H.W. Schfilting, Algebraische und topologische reelle Zykeln unter birationalen Transformationen, Math. Ann. 272, 441-448 (1985).

[si] R.

Silhol, Cohomologie de Galois et cohomotogie des vari6t6s alg6briques r6elles; Applications aux surfaces rationnelles, Bull. Soc. Math. France 115, 107-125 (1987).

[Su] R. Sujatha, Witt group of real projective surfaces, Math. Ann. 288, 89-101 (1990). [Sw]l R. Swan, Vector bundles and projective modules. Math. Soc. 105, 264-277 (1962).

Trans.

Amer.

[Sw]2 R. Swan, Topological examples of projective modules, Trans. Amer. Math. Soc. 230, 201-234 (1977).

[Sw]3

R. Swan, K-theory of quadric hypersurfaces, Arm. of Math. 122, 113-154 (1985).

IT]

A. Tognoli, Su una congettura di Nash. Ann. Scuola Norm. Sup. Pisa 27, 167-185 (1973).

[w]

E. Witt, Zerlegung reeller algebraischer Funktionen in Quadrate, SchiefkSrper fiber reelle Funktionenk6rper. J. Reine Angew. Math. 171, 4-11 (1934).

School of Mathematics Tara Institute of Fundamental Research Bombay- 400005 India. E-mall : [email protected]

ON THE CLASSIFICATION OF DECOMPOSING PLANE ALGEBRAIC CURVES G.M. Polotovskii (Nizhni~ Novgorod, USSR} This r e p o r t represents a survey of results on the arrangement of curves which are decomposed into product of curves "in general position"

in

real

projective

plane

P2 ~,

The

problem

consideration is in close connection with the classical

under

question

of the Hilbert 18-th problem on the topology and

arrangement

of

non-singular algebraic varieties. Note

some

we

that

receive, in fact, the classification of the

in

cases

non-singular

affine

plane algebraic curves (see sections I I I - VI below}. Let's denote the non-singular plane real curve of degree m by symbol C m, and the set of points of this curve in P0~a by the same symbol. Let's denote the union of transversal intersecting curves C m and Cn ( ~ n ) are

by C muCn *>

equivalent

if

Two of such curves C m u C n

topological

triples

( R ~ , C ~ u C ~ , C '} are homeomorphic,

or

m=n

and

C'm u c'n

(R'J~a,OmUCn,Cm} and

C=C"

and

and

6 =C

(i.e. curves differ only by the sequence of writing the factors}. A

De/~n~t~on. The c u r v e Cmu C n is called the

curue if C a is (M-9}-curve, Cn

is

(M-~}-decompos£n~

(M-~)-curve,

the

number

of

A

common real points of C m and CniS equal to mn-r and p+@+r=~. The similar definitions are used for the case C m U C n U C k ; particular,

these

curves

have

only

non-degenerate

in

two-fold

singular points. The problem of classification of decomposing curves is (as in the case of non-singular curves} most interesting and complicated ~> The notations of the factors are distinguished m-n by "the hat",

in

the

case

53 in

"maximalities

cases".

We

shall

M-,(M-I)-

discuss

(M-£)-cases mainly.

and

A

I. C m u C nin cases m+n<6 and CmuCn u C k in cases

m+n+k<7

In this cases

problem. For

the

classification

is

not

a

example, restrictions arising from Harnack

and

Bezout

theorems

allow only 7 types of topological models for M-decomposing curves C uC

- see Fig. l where the last model is non-realizable.

De/~n~t£on. If topological model satisfies

the

restrictions

arising from Harnack and Bezout theorems it is called odm~ss~bLe.

\

/

/ Fig. 1 LII. The decomposing curves of 6-th degree The

classification

of

Cm u C n

non-trivial case. This problem

has

where

m÷~=5

is

the

first

been

solved

in

my

papers

odd

branch

[i]-[3]. If all common points of C, and C of C

then all realizable types of

belong to

M-decomposing

curves

C uc

are shown in Fig. 2. Two of these types were unknown earlier. For example, some realized types of M- and

(M-2)-decomposing

curves C u C 4 are shown in Fig. 3 (the first and the second curves are Hilbert ones; the symbol } denotes "one oval in the other")

54

~rnack, 1876

Gudkov,196g Fig. 8

Hilbert, 18gl a.] M-decomposing curves C u c

B t

1 b} C 4 is (M-R)-curve with scheme I-

Fig.3

Flg. 4

•

55

When all common points of C

and C

these curves all realized types of

belong to odd branches of A M-decomposing curves C u c 3

are shown in Fig. 4. The total statistics of results for the case ~+~=6 is in Table I. I shall remark that 156

models

are

given

admissible

but

non-realizable - examples are shown in Fig. 5. I shall remark also that all models of Fig. 5 and some others 1980 restrictions

(Petrovski~

satisfy

inequality,

all

known

Kharlamov

and

by Viro

inequalities and so on) which were applicable in our situation. Table 1 The number oi realizable models cm M-decompo.slnq n total C uC 15 14g C-~ u c 4 c

total

'1

~

321

42

170

46

~340

103

(M-2)

~---"

(M-4}

Fig. 5 To obtain the classification of the 6-th curves

I

used

the

following

degree

techniques:

I)

decomposing "Geometric

56

combinatorics" for Quadratic

the

enumeration

transformation

~o=xx,y,=xox,

RP~-~

of RP ~

admissible defined

y =xox ; 3) Hilbert-Rohn method,

models; by

2)

formulae

i.e. study

of

bifurcation of curves during the motion in the space R2 :v of real sextics. III,The applications of the decomposing curves of S-th degree Recall

that

the

classification

of

Ci u C n

is

the

classification of non-singular affine curves of degree n: take C 1 as a line at infinity in affine plane, see Fig. 6.

0 >

Fig. 6 i) The first application is the smoothing

of

non-degenerate

five-fold point (singularity NI~ by Arnold notation in [4]). In ig80 Viro suggested the gluing of charts of polynomials as a method of perturbation details

in

[5]

-

of

[7]).

singularities

(you

can

particular,

Viro

obtained

In

find

the some

collection of smoothings of singularity N 1 8 In ig83 Shustin proved in [8] - [i0] that any singularity N may be smoothed by the gluing of any non-singular affine quintic. Therefore,

the

classification

of

C Iu C

gives

the

complete

was

advanced

classification of smoothings of singularities N l e Hilbert 16-th problem on essentially by such

non-singular

smoothings.

In

curves

particular,

Viro

[6]

and

Shustin [ii] constructed M-curves, and I [IE] - [14] constructed (M-I)-, (M-2)- and (M-3)-curves of degree 8 by this

method.

You

57

can see the example of such construction in Fig. 7. The

curve

with singularity Nie was constructed by the hyperbolism from Gudkov

quintic

from

paper

[15], and

then

by

singularity Jto by Viro method. In these

pictures

denote the

Greek

number

of

empty

ovals

by

smoothing and

C the of

below

letters

in

I the

corresponding regions. %%

I I

o

•

/

~er~O~s m ÷

g /

\

~

~-~

smoo~h~

~ z'-~,

n~ethod ~~ ~ - -

I

/xT[

_x / /

~,

~/ - ' x

,~ig.7" _ r . . "

,'

values by[i3-~

6 0

ng

s o m .e

_ ~-t,

realized

va1~e<-.

c,l~A-"

/-'-"-

\ °

wil:h the scheme _

,, ~ - ~ + &~- +
.

58

By analyzing the results of such

constructions

ig81 (see [18]) that Viro conjecture on

M-schemes

I of

found

in

degree

8

with the deep nest (see [7]) and Rohlin conjecture on schemes type I (.see [17]) contradict of each other. Later that both

conjectures

were

false.

Counterexamples

conjecture were constructed in [18], [19]. One construction

was

shown

in

it

Fig. 7.

of

was

found

to

Rohlin

variant

of

Counterexamples

the

to

Viro

conjecture were constructed by Korchagin in [20]. 2) The second application is elementary constructions of

the

4-th degree non-singular surfaces in ~D~. I recall that the problem on the begun in Rohn paper[~l].

4-th

degree

surfaces

was

Hilbert considered this problem in [~2].

The following construction belongs to Viro [23] used for obtaining a rigid isotopic classification

and of

it

was

the

4-th

degree surfaces by Kharlamov in [~4]. Let C u C 4 be in P2 z and i=O be the equation of plane P,2a RP ~, The surface of degree ~ in P22 having the equation

L~+6C=O,

where 16] is small, is the ellipsoid or hyperboloid of one depending on sign of 6. The 4-th degree surface in P23

into which the

cylinder

sheet

with

the

result

of

C4=0 divides

the

equation ( L2+~C )~+6C4=0, where lal is small, is doubling one " h a l f " ,

in

the

surface Lz.~C=O. Fig. 8 gives the example of such constructions. 3) The next application is the

classification

of

the

degree non-singular curves on the surfaces of degree 2. I that this problem was investigated by Hilbert

8-th recall

[2g] exactly

i00

years ago. In[23]

Viro suggested to use decomposing sextics in order to

construct the 8-th degree curves on hyperboloid by Hilbert method of the projection.

Figures

g

and

i0

i11ustrate

Hilbert-Viro

59

\

<

,

Fig.8 by me in [i]; we obtain the sextic smoothings of 7 ~ots.

in

Fig. g

on

It is possible by Brusotti

Now we make the projection of C

with

knots

A

the

right

theorem and

B

by

[28].

on

the

hyperboloid H intersecting plane P~P~ along the conic which passes through points A and B. The center of projection is the

point

P

of rectilinear generators passing through A and B. As a result we obtain M-curve of degree 8 on H, By this method, Gudkov [27] obtained

the

classification

the 8-th degree non-singular curves on hyperboloid, and

of

toseth@r

60

ill

Fig. g

>'i,

i

"

~

4.*

~.

~ 1 5

-

--

5:,7

--

J

Fig. lO with Shustin [28] - on ellipsoid. 4~ By analysis of classification of

decomposlng

sextics,

I

found a formulation of the below theorem on complex orientations.

61

Let P be point of intersection of branches A and B which provided with orientations. The smoothing of P called

consented

with

orientation,

and

in

in

are

Fig. lla)

Fig. llb)

-

is non-

consented.

a)

Fig. ll

Let curves C m and Cn be provided with ol-ientations, Cm+ n be non-singular curve obtained by the

consented

and

let

smoothings

of knots of C m U C n, Let ~m*n be obtained also if we preliminarily replace the orientation of one of curves C m or Cn by the opposite one.

(I recall that according to Rohl!n [17], the curve is of type I

(resp, II)

if

it

divides

(resp.

not

divides)

its

own

complexification; the complex orientation of the curve of type is

the

induced

border

orientation

of

one

of

half

I

of

its

Theore~ A. Let C m and Cn be of type I and let the number

of

complexification).

common real points of C m and C n be ~n - 2. /el C'm+n be

obtained

A

from C m U C n by smoothings of all knots. Then C'men is iff C~+n=Cm+ n and ~m+n is of type II

er,

vice

of

type

I

versa,

and Cmt n is of type i!. Moreover, the arising orientation is

the

complex orientation of C' The proof of this theorem was reported to me by

Fiedler

and

by Makeev independently. Earlier Fiedler [~g] and Marin [30] have obtained similar results in the case

when

the

number

of

real

common points of C m and C n equals ~n, and later Viro and Zvonilov

62 (see [ 7 ] ) have found the formulation which contains all cases. IV. The problem on a type C i f CI u c

is given

Let the curve IV. Now I s h a l l consider the f o l l o w i n g problem: l e t there is C u c

be given. What type ( I or I I ) can the curve

C be of ?

It is clear that this question has a sense only when the ~nde/~n~te type scheme ( i . e .

the real scheme

of

C

C

has

may

be

realized by the curve of type I and by the curve of type II ). It is known that such quintic consists of odd branch and four

empty

C uc

are

shown in Fig. 12 where the number of distinct situations is

equal

ovals. In this case all realizable

arrangements

of

to 30.

_/ Fig. 12 The following theorem [31] gives the answer to

the

question

above.

Theorem B. The curve C l)can not be of A2,B3,Ea;

2)can

not

be

of

type

type I

II in

in

cases cases

A ,A ,A4,Bo,B,B,C,D ~, EI,E ; 3)may be of an arbitrarily type I or II, both situations are realized) in all other 17 Fis. 12.

cases

( of

63 CoroLLary. Pairs ( C u C , C )

make

up

~

149+17=166

classes

with respect to the rigid isotopy in P-22 Techniques used for the proof of theorem B are the following: quadratic transformations and theorem A mentioned above Remark. Later in [3~], Khariamov and Viro obtained Cby

another

F

way} the result from which the part authors remarked in [32] that my

of

theorem

classification

B

follows.

of

But

decomposing

sextics was one of the initial points for their paper I also remark ~~ ,-, a "~

some of

curves were

my

used

for proving the exactness of some inequalities

as

examples

See, for example,

Zvonilov paper [33]. There are some other applications So, the decomposing Hilbert

16-th

problem,

sextics and

classification

it

is

was

useful

in

naturally

to

investigate

intersection

of

C i and

decomposing curves of degree 7 Y. The curves C uC G a) In case when all points belong to one of the

ovals

of

of C e,

M-decomposing

investigated by Korchagin and Shustin [34] statistics of the results

The

were

The Table 2 gives the

In the first column

of intersecting branches are shown

curves

Ce

the

arrangements

topological

models

for

which the question on realizability is open are listed in Fig. 13

"-" L~ "JC~ I

/4

3 1 5"

2 -

5" 5"

6 2

-

.,-3 3

I i

8 i ~5

3

2'5

9 01 5 0 5 .........

Fig. 13

~

0.9

64

Table 2 The arrangement of interoecting branches

The number of admissible models total

realized

prohibited

the question is open

61

17

30

4

"serpent"

27

g

i6

3

"snail"

33

23

3

"camel"

18

0

18

0

total

12g

33

86

10

"comb"

/Ah

b) The next case was investigated by Korchagin,

Shustin

me. Let C e be a M-curve but there are imaginary common points C

and Ce or common points belong to more then one oval. For

admissible models we realized 127 and

prohibited

33

and of 167

ones. The

question on realizability for ? models of Fig. 14 is open. c) Korchagin realized 4 of (M-f)-curves of

type

"camel"

of

Fig. 16. M-schemes of such types were prohibited in [34],[36]. For obtaining the results of this section the techniques

were

used. First, the

method

of

following

new

"reduction

of

65 singularities" which was found by Korchagin [36] and [37] independently. Second, the complex orientation

Shevaillier theory

[i7] and [38]). Finally, homological Viro method. (You

(see

can

find

details in [34],[35].)

]SE I~,3,#} Fig. 14

0 0 4

14

#,

#2

#,

g B 5 3

0 3 0

0 0 0 0

E

Fig. 15 Affine sextics were used in [18]

for

constructions

of

the

8-th degree (H-i)-curves by smoothings of six-fold points. The next sections

of

my

report

have

not

been

published

because they are not finished. VI, The problem on a type C e if C u c e is given Let C uce be given. What type ( I or II ) can the curve

Ce

be of? This question is similar to the problem of section and has a sense only when C e has the indefinite type

IV

above

scheme.

By

Rohlin [!7] then C8 has one of the following schemes:

Now I know that 361

models

must

be

investigated.

For

80

models I prove that if the model is realizable then C e belongs to type II. 86 models are realized by curves of type I. Ther-e exist

66 arrangements in which Co can be of an a r b i t r a r y type ( I or I I ). VII. The curves C2 u c In t h i s section I consider

only

M-decomposing

t h a t a l l I0 common real points of C2 and C belong

curves to

such

the

same

branch Q of C.

a) If fl is an oval then lO

arrangements

of

admissible. In Fig, 18 the shaded regions lying contain different

another

("free")

arrangements

of

ovals these

of

C.

free

Co out

Taking ovals

and of into

we

admissible models. I succeeded in prohibiting of I0

Q ~

are can't

account

obtain of

them

80 by

complex o r i e n t a t i o n s .

Fig. 16 (everywhere a+~=5) b) I f ~ is the odd branch of C we assume t h a t l i n e L doesn't i n t e r s e c t the oval C2 and we denote by s the number of components of ~\(~ n L) which contain common points of C2 and ~. I t is clear that s may be equal to I , or 3,

or

5.

Examples of

admissible

67 models you can see in Fig. 17.

s=l

s=5

s=3 Fig. 17 at

The results which I succeeded in obtaining

present

are reflected in Table 3.

time

Table 3

The number of admissible models S

.........

C u~ oval

-

odd branch

Total

10

C u c prohibited realized the question ~$ ooen 6o

io

$o

-

1

33

aa83

144g

19

705

3

15

1375

107S

43

257

5

2

14

8

4

2

6O

3672

2598

88

1014

VIII, M-decomposing curves C uc 4 Tn this section I consider only two cases: a) All common points of C 3 and C4 belong to the and to one of the ovals of C. 4

In this case

16

oval

of

C

arrangements

of

intersecting ovals are admissible. Taking into account free ovals of C4 we obtain G4 possibilities in which I prohibited Ig by complex orientations and realized 4 models which you

models can

see

in Fig. 18. The question on realizability of 41 others is open. b) All common points of C

and C 4 belong to odd branch of

C

and to one of the ovals of C 4, and this oval is convex. There are

3 admissible models for intersecting branches,see Fig lg. Taking into account free ovals we have again 64 arrangements in which

I

68

f

/ Fig. 18

Fig. 19

prohibited 14 by complex orientations. A

IX. M-decomposing

curves

C uC uC

where

all

10

common

points of C and C u C ~ belong to odd branch of C Such

curves

appeared

in

Brusotti

paper

"bifrontal generating") and were used for

[ 3 9 ] (so-called

constructions

of

the

10-th degree M-curves in [40]. The curve C, u C , separates P2" in two disks A and B (Fig. 20). Let s (resp. t) be the

number

and C in the disc A (resp. B). or

Fig. 20

Since A and B

are

then

sufficient

to

s>t.

It

is

examples

of

(s, t)

in

it

consider

evident that s+~ is even and 2<-s+t<8. You arrangements of C ,C and (~ for Fig. 21.

different

is only can

see

values

equivalent

of

89

f?

Fig. 21. i

Fig. 21.2

(s,t)=(3,1)

Fig. 21, 3

Fig, 21.4

(s, L):(3,3)

Fic]. 21.5

(s, t)=(1,1)

(Total ig pictures?

(s, t)=(5,1)

(Total 13 pictures)

(S, t):(5,3)

70 The s t a t i s t i c s of r e s u l t s I know is given in Table 4 brackets. In the

case

Cs, t ) : ( l , l )

the

constructions

without are

the

work of Korchagin in the main. Table 4

(s,t?

The number of admissible models z%

A

fi.uct u C t (I,i) (3,1)

8

C uC

uC

prohibited

~

realized

the

question is open

{8>

'71 {561}

8 {157}

18

45

19 {19}

14~ {854}

- {225}

-

142

3

(5,1)

{3)

i~

{27}

-

{-}

12 {27}

-

{->

i,

(3,3)

(5,3) Total

93 {525)

13 {!3) i

{i>

- {142>

{2g}

4

-

322(1998>

44 {44)

-

93

34

28O

<8>

8 {~30} A

X. CM-2)-decomposing curves C uc u c JI,

We shall discuss the curves such as in section IX, but now C is

(M-2)-curve,

i.e.consists

of

odd

branch

and

four

ovals. The part of statistics of results is shown in brackets.

Table

There is a sense, as in section IV and YI, in

empty 4

in

question A

on type C I or I I ) of C in the given arrangement of

C uc~ uc,.

! remark t h a t my theorem B above was very e f f e c t i v e f o r obtaining of these r e s u l t s .

In p a r t i c u l a r ,

only in 55 cases the question on

the type of C is open. REFERENCES I. Polotovski~ G.M. Catalo~ o/ de~ree.-

DokL.Akad. Nauk

M-decomposed

SSSR,

236:3

curues C19779,

o/

6-th 548-551.

CEn~Ltsh transl, tn 5ov. Math. D o M 18:5C19779,p. 1241-!245.9 2. P o l o t o v s k i l

G.M.

CompLete

curues o~ 6-th de~ree tn real

classification projecttve

o/

plane.-

M-decompostn~ deposited

tn

VINITI ~I349-TaDep, I-t03. 3.

Po!otovski-i G.M.

CM-I.)- and CM-2.)-decompostn~ curves o/

6-th

71

de~ree.-Methods o/ the 62uatttattve

Theory

o/

Dt//.

Equattons,

6os. Univ. ,6orky (I978), I30-I48. 4. Arnold V. I. ,Varchenko A.N. ,Gusetn-Zade S.M. 5tn&utea'tttes

dt//erenttabte r~pptngs, Vot. I, "Nauka",

Moscow, I982

o~

(Engttsh

t rans t . , Bt r~ASuser, I985. ) ,5.

O . Y . G~utng o/

Viro

ptane

rear

algebraic

constructtons o/ curves degrees 5 and 7 . -

curves

a~d

Lect. N. Math. ,Vot. I950

C1984), 187-200. B. Viro

O . Y . GLutn~ algebraic

s~ngutczrt t tes

constructions

and

hypersur/aces, o~ curves.-

smoothtn~s Trudy

o~

Lentngr.

Mezhdur~r. Totmt. Con/er., Lentngrad, I989, I49-I97. 7. Viro O.Y. Progress tn topotoE9 o~

rea~

a~geOratc

vertettes

over the last s i x years. - Usp. Nat. Nauk. ,4I:3 (19869, 45-57. 8. Shustin

E.I. The

complicated

singular

Httbert-Rohn method and points

o/

curves

o/

smoothtngs

o/

8-th

de~ree.-

5-lord

singular

Usp. Mat. Naul~.,38:6 619839, 157-158. g. Shustin

E.I.

Real

sraooth~n~s o/

stmpte

point.-deposited tn VINITI bJ6872-83Dep, 1-21. 10. Shustin

E.I.

The

Ht tbert-RoAn

method

singular points o/ rear atgebratc curves.-

and

saoothtn~

o/

Dokt. Akad. Nauk 555R,

281 : I C1985.), 33-36. i l . Shustin E.I. Independence o/ sro~oothtn~s o/

stngutar

points

and new N-curves o~ degree 8 . - Usp. Mat. Nau~, ,40:5 CI985), 212. la. Goryacheva T.V. ,Polotovskil G.M.

Constrttcttons o/

CM-1)-

curves o/ 8-th de~ree.- deposited tn VINITI bJ4441 85Dep, 1-30. 13. Polotovskii G.M. CM-29-curves o/ 8-th de~ree:

constructions,

open probtems.- deposited tn VINITI Pd185-85Dep, 1-194. 14. Polotovskii G.M. ,Tscherbakova A.V.

(M-3)-curves of 8-th degree.- depost ted l -~D.

On. the cortstructton of tn VINITI

b14440-OSdep,

72 15.

Gudkov D.A. Construction

o~

new

series

o~

M-curves.-

DokL.Akad. Nauk SSSR, 200:6 ( I g 7 i ) , 125g-1878. 16. P o l o t o v s k i l G.M. On the

c[asst/~cat£on

o/

(M-2)-curves

o/

order 8 . - Selecta Math. Soutettca g:4 (lggob,403-4og. 17. Rohlin

al6ebratc

V . A . Complex

ctxves.-

topoto6tcal

charactertsttcs

Usp. Mat. Nauk.,33:5

(1978),

o/

77-89

real

(EnEltsh

transl, tn Russta~ Math. Surveys, 33 (1978) .) 18. Shust i n E.I. Counterexaaples to RohLtn

m,

,

conjecture.-

Funkt.

Anal. AppL., 19:2 ( I 9 8 5 ) , gd-g5. 19. Polotovski~ G.M.,Shustin E.I. Construction o/ counterexamples

to Rohltn c o n j e c t u r e . - Usp. Mat. Nauk 39:4 CIg849, II3. nO, Korchagin A.B. Hew

M-curves

o/

de6rees

8

9.-

and

Akad. Nauk SSSR, 306:5 (fg8g), 1038-1041 (En6ltsh transl, tn

Do~I. Soy.

Math. Dok[., 39:3(fgsg),p. 559-5-T&) 81. Rohn K. Dte Fl~che 40rdnun6 h t n s t c h t l t c h threr und t h r e r Gestaltun~.- P r e t s s c h r t / t der F~rstl. 6eselscha/t,

Knotenpun~te

Jabtor~o~ktschen

LetI~ta, Ia85.

88. Hilbert D. Uber die Gestalt ether

FtEche

vterter

Ordnun~.-

65tt. Nachrtchten ( I 9 0 g ) , 308-313, 83. Viro O.Y.

Construction

o/

aulttcompo~ent

real

alaebratc

sur/aces.- Dokt. Akad. NauR SSSR, 248:2 (19793, 279-282. 84. Kharlamov V, M. On the c l a s s t / t c a t t o n of non-stn6ular" surfaces

o/

de~ree

4

tn

P,I'3

wtth

respect

to

rt~td

tsotoptes.-

Fun~t. Ana~.AppL., 18:1 (1984), 49-56. 85, Hilbert D. Ueber

die

ree~len

EdEe al6ebratscher

Kurvem-

Math. Ann., 38 (18910, II5-138. 86. B r u s o t t i L. g u l l s "ptccola vartaztone"

dt

una

curva

plans

atffebrtca r e a l t . - Rend. Rom.Acc. ttnc. (59 30 (lg21), 375-379. 87.

Gudkov

D.A.

On the

topotoEy

o/

at6ebratc

hyperbo~o~d. - Usp. Mat . Nauk. ,34:6 (19799, 26-32.

curves

on

73 28. Gudkov D.A.,Shustin E . t .

The c l a s s t / t c a t t o n

non-singular

o/

curves o/ 8-th degree on e l l i p s o i d . - Methods o/

the

Qualitative

Theor~ of Dtf/.Equattons, Gos. Untv.,Gor~y (1980), I04-107. 2g. Fiedler T. Etne Beschr~nku~6 /ur die Lage yon r e e l t e n

a tgebrat scher~ Kurven. -

Bettr~ge

uad

Atgebra

zur

ebenen

6eorretrte,lI

c190i ,7- 9. i

30. Matin A.

reaarques

Quelques

sur

~es courber

atg~brt~ues

planes ree[les.-Pubt. Math. Untv. Parts VII, g C1980), 51-58. 31. Polotovskli G.H. The connection between r i g i d isotopic o/ r#nstngular curve o/ 5-th degree tn RP~ and

its

crass

arrangeaent

with respect to t t n e . - Funkt. Anat. Appt., 80:4 C1986), 87-88. 3~. Kharlamov V.H.,Viro

O . Y . Extensions

o~

the

Gudkov-Rohttn

congruence.- Lect. N. Math., I345 CI9880, 357-405. 33. Zvonilov V.I. Strengthened Petrovsktt and Arnold tnequaLtttes

/or curves o~ odd degree.- Funkt. Anal. AppL.,i3:4 (igVgO, 32-39. 34. Korchagin A.B.,Shustin E.I. A//tne curves D

o/

degree

5

and

,,

sraoothtngs o~ a

~on-degenerate

sixth

order

singular

Isv. A~ad. NauR sgs£, Ser. Matea., 58:6 C1988), II8I-I199.

potat.(Eagltsh

transl, ta Math. USgR I s v e s t t y a , 33:3 CigSg),p. 501-580.3 3$. Shustin

E.I.

On the

isotopic

classification

o/

M-curves o/ degree 5.-Method~ o/ the QuaLitative Theor~

a/fine

o/

Dill.

and

the

Eq~ttoas, Gos. Uatv., Gor~y Cig88), 97-i05. 36. Korchagin A.B. On the reductton o/

stngutart ties

i

ctasst/tcatton

o/

nonstr~gu~ar

afftne

sexttcs.-

deposited

tn

VINITI ffl1OT-B (1986), 1-16. 37. w,,

Chevaiilier

B.

A propos

des

courbes

de

degree

5.-

C. R. Acad. Sct. Parts, Ser. A 302:1 (1986), 33-38. 38. Fiedler

T.

Pen.ctts

o/

trues

and

the

tolx~toEy

atgebratc curves.- Isv. Akad. Nauk SSS/~, Ser. Matem. ,46:4 853-863. CEngl~sh transl,

o/

rear

£i982.),

tn Hath. USSR Isvest~ya, 21 C1983}.D

74

3g, B r u s o t t i

L. Nuovt a e t o d t

asseKr~ato,

dotate

[~end. t s t. to~7~b. , Ser. I I a

deL

coatrutttvt aasstrr~o

dt c u r v e piano rataero

dt

d'ordtne c t r c t t t t t. -

47C 19I 4)-49C 1916).

4,0. Chisienko Y.S. M-cttr~es o [ de~ree ter~. - Zap, r~auchr~, se~tn, nLO~fI, ~. C1982.),

i 4 6 - i 6 1 . CErz~Ltsh

CI 984.) , no. l )

transL,

t~

J. S o t l t e t

Math.,

26

Real Algebra and Its A p p l i c a t i o n s to G e o m e t r y In the Last Ten Years: S o m e Major D e v e l o p m e n t s and Results

Claus Scheiderer Fakult£t fiir Mathematik, Universit£t Regensburg, 8400 Regensburg, Germany

This is the slightly expanded version of a talk I have given at the conference. Its object was to report on some progress obtained in real algebra and geometry during the past decade. I want to stress right at the beginning that no attempts have been made to give a survey of encyclopedic character. This cannot and will not be a comprehensive account; rather choices had to be made about which topics to include. They reflect the personal taste of the organizers and of myself, and are not meant to imply any kind of ranking. The presentation is divided into three parts. The first is concerned with problems on sums of squares in fields and rings, both qualitative and quantitative ones. The second assembles results on sums of higher (even) powers, together with related subjects like real valuations and real holomorphy ring. The last part presents some applications of real algebra techniques to geometric questions. Here I had to be even more selective, since geometric problems have always been a strong incentive for the development of Mgebraic methods, and so the applications abound. On tile other hand, it is a pleasure and sometimes a surprise to see how this can be turned around, so that geometry is used to solve purely algebraic problems. We shall meet this phenomenon in several instances. Among the topics which had to be excluded are pure quadratic form theory (although quadratic forms are very much present, at least behind the scenes), Galois theory and Galois eohomology of fields in connection with reality phenomena, and the extensive work on semialgebraic topology and its abstract generalizations in the real spectrum. It is a pleasure for me to express my gratitude to D. Gondard, M. Knebusch, T.Y. Lam, L. Mah6, A. Prestel and D. Shapiro for their generous help with the preparation of this talk.

76 1. S u m s o f s q u a r e s

As my point of departure I want to take the famous 17th Hilbert problem. It is well known that through its solution by Emil Artin it catalysed the development of specific real algebraic techniques. Though Artin solved it completely in its original form, both the problem and its solution lead naturally to a variety of related questions, many of which are still unsolved. Hilbert originally had asked if a rational function f E Q(X), X := ( X l , . . . , X , ) , which takes non-negative values where it is defined, is a sum of squares of rational functions. Artin's proof showed much more generally, for any ordered field (k, P), that every rational function in k(X) which takes non-negative values over the real closure kp can be written f = ~ i aig2 with ai E P and gi C k(X). Thus, a natural way to generalize the question to arbitrary fields k (with possibly many orderings) is to ask: If k is a field and T is a preordering of ~, when is it true that every rational function f E k(X), which over k takes values in T, can be written f -- ~itig~ with ti E T and gi E k(X)? This has been called the weak Hilbert property for (k, T). In the cases where T is an ordering, respectively where T -- Ek 2 and [k* : T*] < oc, it had been shown in the 70s that this property is equivalent to k being dense in its real closure with respect to T, respectively in each of its real closures (Gondard, Ribenboim, McKenna, Prestel). However the general solution was given only recently by Zeng Guangxin in [Zeng 88], see also [Zeng 91]. He showed that generally the weak Hilbert property is equivalent to a "local density" property of (k, T). He also proved that this property can be axiomatized in the language of preordered fields. Fix a real closed field R, and let Pn,d be the set of all forms f in n variables of degree d over R which are positive semidefinite (psd), i.e. which over R take nonnegative values only. One may regard Pn,d as a subset of R N, where N = (n+dd-1) is the number of degree d monomials in n variables. As such, it is a closed convex semialgebraic cone. Identifying a form f with its tuple of coefficients in R N, one may ask if there is a solution to Hitbert's question which holds uniformly for all psd forms f and whose entries depend rationally on the parameters, i.e. on (the coefficients of) f . To make the question sensible, one has to allow linear combinations of sums of squares with non-negative coefficients (which should depend rationally on f , too). Also one may want to know if such a uniform solution exists which depends at least continuously on f . These questions were posed by G. Kreisel in 1962 (in a more general form), after Daykin had shown that a rational solution exists which is almost uniform in that it needs only finitely many "case distinctions". The answers to Kreisel's questions were given by Ch. Delzell in his 1980 Stanford thesis: On the one hand, a uniform rational solution is impossible whenever d > 2. This is proved by showing that the closed set Pn,d is not "basic", i.e. cannot be presented by a conjunction of inequalities (see §3). On the other, there is always a uniform solution which depends semialgebraically and continuously on the parameters. For these results, and for more precise formulations, see [Delzell 82a+b] and [Delzell 84]. The existence proof in [Delzell 84] is a nice application of results from real algebra and geometry, like real Nullstellensatz (in a homogeneous version) and semialgebraic triangulation.

77

Hilbert himself had already shown that a psd form is not in general a sum of squares of forms. Thus one is led to study En,d, the closed convex subcone of P,,,d consisting of the sums of squares of forms, and the relations between the two cones. This has been done in several articles by M.D. Choi, T.Y. Lam and B. Rezniek, starting already in the 70s. It is in general a non-trivial task to exhibit psd forms which are not sums of squares (aos), and to verify this. Historically, the first (published) example was given only in 1967, by Motzkin. Of particular interest is it to find the extremal psd forms, since they generate P,,,d by convex combinations, and since such a form is not sos unless it is a perfect square. Reznick observed that m a n y of the known examples of forms in Pn,d -- En,d could be obtained by substituting monomials into the arithmetic-geometric inequality (which compares arithmetic and geometric mean). In [Reznick 89] he studies these "agiforms" in great detail, thereby obtaining conditions for agiforms to be sos or to be extremal. These conditions can be used to produce m a n y new examples of psd forms which are not sos. T h e y are of geometric nature and involve the counting of integral points inside certain polytopes associated with such forms. T h e techniques partially extend and generalize methods used earlier by Motzkin, Choi and Lam to study isolated examples. A very explicit ease study is made for d = 6 in [Choi Lam Reznick 87], where the intersection of the two cones with the space of even symmetric sextics is determined completely. As a by-product, a family of largely new symmetric polynomial inequalities is obtained. See also [Choi Lam Reznick 91], where new psd ternary sextics are discovered which are extremal and which generalize classical inequalities of Lehmus and Schur. The constructions are inspired by considerations in the euclidean geometry of triangles. I also want to mention Reznick's study of the "Hurwitz form" H n = x'~ + .. • + x ~ - n x l " • x n which is closely related to the arithmetic-geometric inequality [Reznick 87]. While Hurwitz had already shown that Hn is sos (for n even), his construction required a great number of squares; Reznick reduces this length considerably by his "parsimonious construction". I now pass to quantitative problems. A celebrated theorem of Pfister says that the level of a field (also called S t u f e ) , if finite, is a power of 2, and that every power of 2 occurs. Recall that the level s ( A ) of a (commutative) ring A is the minimal n such that there are a l , . . . , a n E A with - 1 = a 2 + - . . + a2n (and is <x~ if - 1 is not a sum of squares). It is natural to ask what happens over rings which are not fields; in particular, what type of integer can be the level of a ring? This question was raised by R. Baeza [Knebusch 77, Problem 13]. It was answered - - and, in fact, much more t h a n that - by Z.D. Dai, T.Y. Lam and C.K. Peng at the beginning of the 80s [Dai Lam Peng 80]. Later, a more elaborated and comprehensive theory was set up by the first two of them [Dai Lam 84]. Plainly, the answer is that any positive integer is the level of a suitable ring A. In fact, A can be chosen to be a domain whose field of fractions has as its level any preassigned power of 2 not exceeding n. But at least as interesting as the answer itself are the methods, which extend way beyond this specific application. T h e basic idea is t h a t there is a natural relationship between the arithmetic of sums of squares in rings and the topology of spheres. At quite an elementary level, this is present already in the short and very elegant proof of the fact that A = N I X 1 , . . . , Xn]/(1 + X ~2 + - - - + X~) has level n: Assuming s ( A ) < n, one takes a representation by less than n squares and plugs in purely imaginary arguments. But then a direct application of the Borsuk-Ulam theorem to the odd degree part gives the desired contradiction!

78 The Borsuk-Ulam theorem states that for any map f : R n ~ R n-1 which satisfies f ( - x ) = - f ( x ) for all x there is x C S n-~ with f ( x ) = 0. Shortly after the above application to the level of rings appeared, algebraic proofs were found for the polynomial version of this theorem [Knebusch 82] [Arason Pfister 82]. Arason and Pfister actually obtained a strong generalization of one of the results of [Dai Lam Peng 80], which in particular gives the formula

s(]g[Xl,...,Xn]/(1Av Z 2 -[-"'+ X2))

= min{n,s(k)}

for any field k. But I want to return to the work of Dai and Lam. Given a topological space X with an involution e (i.e. an action of the group Z2 on X), they form the R-algebra A x of all equivariant (continuous) maps X --~ C (the involution on C being complex conjugation). Generalizing the above argument, they show the Level T h e o r e m [Dai Lain 84, Theorem 3.1]: (Ax) --

> 1:

there is an equivariant m a p X -* S

8(X).

The involutions on the spheres are the antipodal maps x ~-~ - x . The invariant s ( X ) of a space with involution, and its dual (for obvious reasons called the level and the colevel of X), had been studied by topologists before, under the names coindex and index; but the relation to algebra had not been observed. But Dai and Lam do not stick to spheres and to consequences of Borsuk-Ulam. More generally, they consider Stiefel manifolds together with various natural involutions on them. Equivariant maps into these spaces are related to quadratic forms over rings, more specifically, to the study of subforms of the type r(1)_ks(-1) of forms n(1). Applying deep theorems from topology, this yields striking results. For example: T h e o r e m . I r A is any (commutative) ring, and i f ( - 1 ) is a suhform of n(1}, then also p(n)(-1) is a subform of n(1}. But in general, (1 + p(n))(-1} is not a subform of n(1). Here p(n) is the Hurwitz-Radon function, defined by p(24a+bu) : 8a-b2 b for integers a, b, u with u odd and 0 < b <: 3. The first assertion is more or less well known, but the second is quite deep and rests on Adams' results about vector fields on spheres. Theorem. Given q < n, there are (honest!) subform ofn(l>, but not so (q + 1)(1>_k(-1>.

rings A over which q(1)±(-1 / is a

In particular, a Pfister form can be isotropic without being hyperbolic. Many other interesting results are obtained by Dai and Lain, which unfortunately cannot be mentioned here. The level of the odd-dimensional real projective spaces RP 2m-1 (on which, by identification of R 2m with C m, multiplication with i induces a natural fixed point free involution) has later been computed in [Stolz 89], see also [Pfister Stolz 87]. It is related to systems of complex quadratic forms whose imaginary part is anisotropic. Another beautiful application of topology to sums of squares is given in [Dai Lain Milgram 81]. Again it was a question of Baeza [Knebusch 77, Problem 12] who asked whether in a ring A the units which are sums of 2 n squares do always form a group (i.e.

79 are closed under products). Of course, this is known to be true for fields (Pfister), mad more generally for semilocal rings (Knebusch). By classical identities it is also true for 2" = 1, 2, 4, 8 in any ring. In [Dai Lain Milgram 81] it is shown that there are no other integers m for which U S m ( A ) , the units which are sums of m squares, are a group in every ring A. So Pfister's multiplicativity result does not in general extend beyond the classical cases. Not surprisingly, the counterexample is the "generic" R-algebra for this question, namely A = N I X 1 , . . . , Xm, Y1,. •., Ym, ( ~ Z ~ ) -~ , (53Yi~)- 1]. If (EX~)(~Yi 2) is a sum of m squares, the authors produce a map S 2~-1 ---* S m with odd Hopf invariant, which, according to Adams, can exist only for m = 1, 2, 4, 8. In fact, significantly stronger results are obtained. Using results of Adams on the J - h o m o m o r p h i s m in K theory, the authors show that a bi-skew ( = bi-equivariant) mapping S ~-1 x S '~-1 --~ S "-1 can exist only if r <_ p(n). T h e y conclude that U S r ( A ) . U S n ( A ) C U S ~ ( A ) is true in every ring A (if and) only if r < p(n). The special case r = n gives back the first result. With this, one is close to the composition problem for sums of squares, a classical and fascinating subject which dates back to Hurwitz. It asks for formulas =

+ . +

2

z.

(,)

where the xi, yj are variables and the z~ = z~(x,y) are supposed to be bilinear in x and y. Originally, this was asked over C or IR, but the problem may be considered over any commutative ring k. Hurwitz and Radon completely settled the "classical" case s = n, proving that ( , ) exists (over a field of characteristic not 2) if and only if r < p(n). In general the smallest n for which (*) exists is written r *k s. Closely related is the question of the existence of a nonsingular bilinear map N r × R ~ ~ R n. Here the minimal n is written r ~ s. In general, these numbers are very hard to compute. For upper bounds one needs more or less explicit constructions, while lower bounds have mostly been obtained by finding topological obstructions (this goes back to the work of Stiefel and Hopf in the 1940's). I want to mention briefly some remarkable progress which has been made in the last few years. An excellent survey of these matters, as of 1984, is [Shapiro 84]; see also [Lam 84]. Concerning the existence problem, S. Yuzvinsky claimed to have constructed three new infinite series of composition formulas ( . ) , starting respectively with (r, s, n) = (10, 10, 16), (12, 20, 32) and (14, 40, 64) [Yuzvinsky 81], [Yuzvinsky 84]. However, T. Smith and T.Y. L a m discovered serious flaws in his reasoning. Fortunately, they managed to fix the proofs in the first two of the three cases, the third remains open. T h e construction, which also yields further new pairings, uses ideas from group representations {Lam Smith 90} 1 . On the non-existence side, K.Y. Lam presented new and beautiful geometric ideas which have since then been used to improve many of the previously known lower bounds ([Lam 85], see also [Lam 84]). Associated with a composition formula (*) (over ll~) is its Hopf map H : S r+s-1 --* S n. Lam shows that for any point Q E S n, the 1 References given in curly braces had, to the best of my knowledge, not (yet) appeared in print when this article was typed (July 1991). In these references, the numbers indicate the year when the corresponding preprint was (probably) written. References in square brackets have been published before July 1991, and here the year of publication is given.

80 fibre H - I ( Q ) is a "linear" subsphere, i.e. H - I ( Q ) = W Cl S t + s - ' where W is a linear subspace of 11{r+s, and that the tangent mapping to H induces a nonsingular bilinear map W x W ± --+ TQ(Sn). One should think of these bilinear maps as "hidden" behind (*). Moreover, he proves that there are points where H has rank q _> r # s. So, from the existence of a formula (*) one can conclude that there is a nonsingular bilinear map of type ( r + s - - q , q, n) for some q in the range r # s _< q < n. This gives potential new obstructions to the existence of (*). Many applications show the power of this discovery, see also [Yiu 86], [Yiu 87]. For example, Lam uses it to prove 10 *IR 11 > 20, thus showing for the first time that r *R s ~ r # s is possible (since 10 # 11 = 17). By going more deeply into the geometry of composition formulas, and by using advanced topological tools like secondary cohomology operations, Lain and Yiu were later able to show 16 *IR 16 > 29 [Lain Yiu 89]. Until [Lam 85] appeared, 23 had been the best lower bound! Since a composition formtfla (16, 16, 32) is known classically, one seems to be quite close to the solution of the "16-square problem". Over the integers, the corresponding problem has been solved by Yiu, who showed that indeed 32 squares are necessary [Yiu 90]. Tile arguments are purely combinatorial and highly involved. Reportedly, Yiu has recently managed to determine r • s over Z for all pairs r, s < 16 ! I return to purely algebraic methods. The Pythagoras number P(A) of a ring A is the smallest number q (if one exists) such that every sum of squares in A is actually a sum of q squares, and oo otherwise. It is a very delicate invariant which is usually hard to get a hold on, and is most interesting if s(A) = oc (i.e. if A has some formally real residue field), since otherwise it lies in the range s(A) < P(A) <_s(A) + 2 (and even P(A) < s(A) + 1 if 2 is a unit). Which integers can be the Pythagoras number of a field is still an open problem. The precise value of P(K) is unknown even for the fields K = IR(X1,..., X~) of rational functions in more t h a n 2 variables, although one has the bounds n + 2 < P(K) <_ 2 '~ from work of Pfister and Cassels-Ellison-Pfister. This already illustrates how little accessible this number is in general. This being said, [Choi Dai Lam Reznick 82] make a thorough study of this invariant and obtain a variety of previously unknown results. Particular attention is given to affine algebras over a field, and to local rings. One of the basic results is that the polynomial ring R[X, Y] has infinite P y t h a g o r a s number, for any real closed field R; the same is true for Z[X]. Before, it was not even known if there was any atone R-algebra with P(A) = oo. Among the consequences drawn from this theorem is that there exists a principal ideal domain A (containing 1/2) and a unit of A which is a sum of squares in the quotient field, but not in A itself. In contrast, P(A) is finite for one-dimensional affine R-algebras; but P(A) can be arbitrarily large, even for the coordinate rings of plane rational (singular) curves. It seems to be unknown if the situation is different for regular curves. In the case of local rings, the threshold value lies one higher. For regular local rings A of dimension _> 3 with real residue field it is proved that P(A) = co, by using the associated graded ring and the result for polynomial rings. This is used to show that, for any affine R-variety V of dimension _ 3 with a smooth R-rational point, P(R[V]) = oo. On the other hand, local rings of dimension < 2 tend to have P(A) < oo. This is proved, for example, for certain such rings of geometric type, where the theorems of Cassels and Pfister even allow to give upper bounds for P(A). However, in the dimension two global ease, it seems to be unknown for general surfaces V / R whether P(R[V]) is finite or not. Beyond its main

81 results, the paper contains a great amount of detailed complementary information, and numerous examples are used to discuss and illustrate the general theorems. Detailed studies of Pythagoras numbers have been made in the case of real APcurves, that is, one-dimensional noetherian local real domains with real closed residue field which have Artin's approximation property (i.e. are henselian and excellent). For example, analytic and Nash curve germs give such rings. The Pythagoras number of a real AP-curve is always finite. A key invariant of such a ring is its value semigroup. [Ruiz 85a] characterizes those semigroups for which all, respectively no, corresponding curves are pythagorean. [Campillo Ruiz 90] show that a real AP-curve is pythagorean if and only if it is Aft; this is also equivalent to the Pythagoras condition for sums of 2n-th powers, for any n > 1. [Ortega 91] studies the algebroid (= complete) case. Fixing an additive semigroup F of positive integers, the corresponding curves are parametrized by an algebraic set. This set is partitioned into finitely many semialgebraic subsets, according to the Pythagoras numbers of the curves. Both upper and lower bounds for P(A) are given. In particular, it is shown that P(A) can be arbitrarily large. I would like to add a remark on recent deep results about Pythagoras numbers, although they do not belong to real algebra. They are concerned with function fields F over a number field k. Colliot-Th616ne observed that Kato's Hasse principle for (Galois cohomology of) such function fields in one variable has immediate consequences for their Pythagoras numbers. Indeed, it follows that P(F) <. 7; hence also s(F) < 4 if F is non-real, and P ( F ( X ) ) < 8 [Colliot-Wh616ne 86]. Recently, Jannsen has proved Kato's conjecture for function fields in two variables. With the help of results of Merkur'evSuslin and Rost, this gives similar applications: If F / k is a 2-dimensional function field, then P(F) < 8 and P ( F ( X ) ) < 16 [Colliot-Th616ne Jannsen 91]. The authors remark that Kato's general conjecture, together with well-known conjectures on Mitnor K-theory and Galois cohomology, would imply P(F) <_2d+l for any function field F / k of dimension d > 2. I now turn to L. Mah6's quantitative results for rings of geometric nature. In [Mah6 86] he considered affine varieties over a real closed field R and succeeded in bounding the cokernel of the global signature map (compare Parimala's talk). For this he needed quantative results on the level of affine R-algebras which axe of independent interest. In [Mah~ 90] the bounds are improved, and new results are added. If V is an R-variety, the ring Ti[V] of real regular functions on V consists of those rational functions which are regular in all real points. A function f E 7¢[V] is totally positive if it takes strictly positive values on V(R), the real points of V. In general rings, the notion of totally positive elements is defined via the real spectrum. One of the main results in [Mah6 90] is T h e o r e m . Let V an al//ne R-variety of dimension d. If A is either T4[V] or a semilocadization of R[V], then every totadly positive unit of A is a sum o£2 d squares in A. So, under this aspect, these rings behave like the corresponding function fields. In the semilocal case the remarkable thing is that no regularity condition is needed. A corollary is that in Pfister's theorem, saying that a psd polynomial function on V is a sum of 2 g squares of rational functions, one can in fact find such an expression where the denominators have no more zeros in V(R) than the given function.

82 T h e other main result gives quantitative versions of the Positivstellensatz, of the following type. If f E R[V] is totally positive then there is an equation f'M

= 1+[]

inR[V],

with m = 2 d + 7 and n = 2 d+a + d - 4, if d > 4. (Here [ ] stands for a sum of s squares.) If f is a unit, then even the significantly b e t t e r bounds m -- 2 d and n -- 2 d + d - 5 hold (d > 4). For d < 3 one has (re, n) -- (2 d+l - 1, 2 d + l - 1), and (2 d, 2 a - 1) in the unit case. 2 In particular, this gives immediate bounds for the level of R[V] in case V ( R ) = 0. W h e t h e r or not such bounds exist in terms of d had been an open problem in [Dai Lam 84]. Similar, but weaker estimates appeared before in [Mah~ 86]. The proofs proceed inductively over the dimension; Mah6 starts with Pfister's theorem to get representations "with denominators", and then by clever arguments gets rid of the denominators, thereby slightly increasing the number of squares. Throughout, weak versions of mtfltiplicativity properties are used for Pfister forms over such rings. One can derive quantitative Stellens/itze of more general type from the above ones. See e.g. [Simon 89] where this was done using the weaker estimates of [Mah~ 86]. 2. S u m s o f h i g h e r p o w e r s , r e a l v a l u a t i o n s a n d r e a l h o l o m o r p h y r i n g Recall t h a t E. Becker, in the late 70s, found the notion of an ordering of higher level of a field, and showed that it bears a close relationship to sums of higher (even) powers. This appears most suggestively in the fact that the sums of 2n-th powers in K are precisely the elements contained in every ordering of level dividing n (assume char K -- 0). This is completely analogous to Artin's characterization of the sums of squares in a field. From the beginning, the idea was that this similarity should only be part of a more comprehensive extension, to higher levels, of classical Artin-Schreier theory with all of its features. There has been considerable work towards this program, and I will try to give a very brief sketch of some of the main lines. A very clear exposition of this program, together with a survey of what had been done by 1984, can be found in [Becker 84]. An ordering of higher level is defined to be an additively closed subgroup P of K* for which K * / P is cyclic (necessarily of finite even order); the (exact) level of P is ~[KI-. : P]. Quite soon a shift in emphasis was made from orderings of higher level to signatures of K , i.e. characters X : K* --~ # --- Q / Z whose kernel is closed under addition. This point of view has many advantages, as is already demonstrated by the study of extension theory in [Becker H a r m a n Rosenberg 82]. It leads naturally to the definition of the higher level reduced Witt rings [Becker Rosenberg 85@ which completely parallels the description of the reduced W i t t ring of quadratic forms as a ring of Z-valued functions on the space of orderings. In the level n case, one takes the space of all signatures which vanish on E K *2n, and the ring of (continuous) C-valued functions on it generated by the evaluation maps X H x(a), a E K*. As in the level one case, it is technically important to extend the definition to preorderings more general t h a n E K 2n. It is amazing to see how the analogy to the quadratic forms case remains 2 According to Mah6, the figures in the printed version are incorrect in the non-unit cases; here I have given the corrected ones, communicated by Mah6.

83 valid for almost all aspects of the theory; e.g. prime ideals of the reduced Witt ring and their relation to orderings {Becker Rosenberg 85b}, the characterization of fans, and the representation theorem, which characterizes the reduced Witt ring as a subring of the ring of continuous functions. ~-hrther work on higher level Witt rings can be found in [Schwartz 87] and {Pecker Schwartz 89}. However, it is still not clear how much these reduced Witt rings have to do with actual higher degree forms, and if there is a good definition of non-reduced Witt rings of higher level. Marshall's spaces of orderings, the abstract frame for reduced quadratic form theory, have been generalized as to also incorporate the higher level reduced theory. Much of Marshall's original theory has been transferred to these so-called spaces of signatures by C. Mulcahy, V. Powers and M. Marshall. See [Mulcahy 88a+b], [Marshall Mulcahy 90], [Powers 87], [Powers 89]. Signatures also proved helpful in clarifying the relations between real closures at higher level orderings. Such real closures exist, but are in general not unique up to isomorphism, contrary to the case of usual (level one) orderings. By introducing chains of (higher level) orderings, J. Harman was able to give a satisfying treatment for 2power levels [Harman 82]. After further investigation of this concept by R. Brown [Brown 87a+b], N. Schwartz formulated the notion of chain signatures [Schwartz 84] which seems to be quite an effective tool. See also [Pecker 84, § 4] for a summary of some of its features. Fields which are real closed with respect to some higher level ordering have been called generalized real closed fields. A lot of work has been devoted to the study of their properties, and in particular, their model theory. It was first investigated by B. Jacob, who showed that if R is a field which is real closed at an ordering of level n, the theory Th(R) of R is decidable and model-complete, if the language of fields is enriched by suitable predicates which only depend on n [Jacob 81]. The crucial and remarkable point is his discovery of a certain elementarily definable henselian valuation ring which has an archimedean real closed residue field. His results were used in [Pecker Jacob 85] to derive a Nullstellensatz and a solution of a generalization of Hilbert's 17th problem for function fields over generalized real closed fields. The model theory of chain closed fields and of chainable fields (these notions refer to Harman's concept) was much studied by F. Delon, M. Dickmann and D. Gondard; see e.g. [Delon 86], [Dickmann 88], [Gondard 87], [Gondard 90] and literature cited there. Another branch of recent research tries to understand the role of higher level orderings for real algebraic geometry. See [Barton 88], [necker Gondard 89], [Berr 89], {Bert 91} for steps in this direction. I would like to point out two recent papers in which the general theory is applied to give concrete and attractive results. Generalizations of Hilbert's 17th problem to sums of 2n-th powers are studied in {Becker Berr Delon Gondard 90}, both in a weak and a strong version. The question is whether a regular function on an irreducible kvariety is a sum of 2n-th powers of rational functions if the values it takes (in closed resp. in k-rational smooth points) are sums of 2n-th powers. The integers n for which either version holds form a multiplicative semigroup generated by primes, and conversely every such semigroup occurs (for the weak version, at least). The case where k is a generalized real closed field is studied in detail; also classes of base fields k are given (namely function fields over totally archimedean fields) for which one or both versions

84 hold for all exponents. I also want to hint at Berr's recent work {Berr 90} about sums of mixed powers in fields. A nice application is the previously unknown existence of "mixed" Hilbert identities. The structure of higher level orderings and signatures of a field F is crucially related to the real valuations of F. In this way, also the real holomorphy ring H(F) -- the intersection of all residually real valuation rings of F - - plays an important role in the understanding of sums of higher powers. H(F) is a Priifer domain, and has many other remarkable properties. In [Becker 82b] it is amply demonstrated how a good understanding of H(F) and of its ideal structure will have direct consequences for sums of higher powers in F. Not surprisingly, the most interesting results on the holomorphy ring and on real valuations have been obtained in the function field case. If the base field is R, then H(F) consists of all functions which are regular on the real points of ,orae projective model (which one can take smooth by Hironaka). A similar description holds for the relative real holomorphy rings H(F/R) over arbitrary real closed fields R. In this way, birational geometry and resolution of singularities become important tools. See [Sch/ilting 82a+b] for this and for other foundational results on the holomorphy rings. An important point is to reduce the vast class of all real valuations to a more manageable subclass. Since valuations of a function field chosen at random are ugly in many respects, one would like to dispose of a family of valuations with good arithmetic and geometric properties (value group, residue field, center). On the other hand, since various important criteria refer to the entire class of all real valuations, such a family should be large enough to be sufficient for those criteria. For instance, of central importance is Becker's characterization of ~ F 2n as those sums of squares f for which the value v(f) is divisible by 2n for every real valuation v of F. Results of this kind have been obtained by H.-W. Schiilting. Working with function fields over the real numbers, he showed [Sch/ilting 86a] that the class of real prime divisors shares all these good properties: It suffices for the Becker criterion just described, for representing the real holomorphy ring as an intersection of valuation rings, and also for the local global principle for weakly isotropic quadratic forms (due to Brgcker and Prestel). As an application, a geometric and very handy description of E F 2~ is given, for F = IR(X) the function field of a surface, which involves the multiplicities of infinitely near points of X on (the nonreal part of) the divisor of f . Sch/ilting also shows that signatures of a function field F/1R can be identified with characters of local divisor groups in smooth projective models. From a more general point of view, valuations of function fields F/k are studied in [Br6cker Schiilting 86]. Most results apply also if the theory of formally real fields is replaced by a suitable other one, like that of formally p-adic fields, and the base field k is existentially closed in the respective theory. One of the main results is an existence theorem for chains of valuations of F/k with prescribed centers (in a fixed model), value groups and, to some extent, residue fields. Other parts of the paper generalize earlier work of Schiilting on holomorphy rings. The essential and non-trivial tool, used throughout in both papers, is Hironaka's resolution of singularities. It is interesting to compare this to the article [Kuhlmann Prestel 84], written at about the same time. Its main theorem is in its spirit quite close to Schiilting's results just mentioned, and in particular, to the existence theorem in [Brgcker Sch/ilting 86].

85 It also says that in a function field extension F/k of characteristic O the "well-behaved" valuations are dense, in a very strong sense, among all valuations of F/k. Moreover, one is still given considerable freedom of how to specify welI-behavedness, as demonstrated by various modifications. This has several applications in common with the Br6ckerSch(ilting theorem. However, Kuhlmann and Prestel proceed model-theoretically: Instead of using Hironaka, their main tool is the Ax-Kochen-Ershov theorem on the model theory of hensetian valued fields. Another theorem on chains of valuations in function fields has been proved by C. Andradas in [Andradas 89] (see also [andradas 85]). Working in the real spectrum, he also prescribes centers, dimensions, and rank and rational rank of the value groups. It is remarkable that he does not need desingularization theorems nor model completeness results. Instead, he uses elementary methods like curve selection lemma and Noether normalization. The results from [Kuhlmann Prestel 84] are applied in [Prestel 84] to show the following: In order to cheek if a polynomial f E R [ X I , . . . , X d ] has value divisible by 2n .p.der all real valuations (the essential part of the Becket criterion), it is sufficient to check oniy those valuations which come from plugging Laurent polynomials (in one variable) into f. But then Prestel gives an instructive example to show that this becomes false if R is non-archimedean and n > 1; namely the polynomial f(X) = X 2 " + ~ X 2+ 1, where w is a very large element. Thus for n > 1, to be a sum of 2n-th powers is not an elementary property in the coefficients of f , contrary to the case n = 1! Another way to put this is that in representations fro(x) = x

+ mx

+ 1 =

\ hr. /

'

i=1

for positive integers m, one cannot keep the Nm and the degrees of the hm bounded at the same time for ra -+ ~ , although this is possible for the Nm alone. The substitution criterion was later refined in [Prestel 87], where the degrees of the Lanrent polynomials were bounded by computable functions. At this point I want to digress for a moment, to mention recent work on R(X), the field of rational functions in one variable. Becker had already shown P4 (R(X)) < 36 [Beeker 82b] where P4 is the Pythagoras (or Waring) number for sums of fourth powers. This was improved to 24 in [Schmid 88]. But now in {Choi Lain Prestel Rezniek 91}, P4(R(X)) _< 6 is shown! Also for higher exponents the upper bounds from [Becker 82b] are slightly improved. This uses a new characterization of ~R(X) 2n. In addition to the obvious necessary conditions for a polynomial f (initial coefficient positive, multiplicities of real roots and degree divisible by 2n) - - which by Beeker's criterion suffice if R is archimedean --, a new condition is needed. If one partitions the upper half plane of R(v/'L'i) into equivalence classes, by defining two points to be equivalent if their hyperbolic cross-ratio and its inverse are bounded in absolute value by some integer, the condition says that the number of nonreal roots of f lying in each equivalence class must be a multiple of n. Obviously, this condition is implied by the others if R is archimedean. If one keeps these cross-ratios in each class bounded (and fixes the degree of the polynomial), then also the complexity of representations as sums of 2n-th powers can be bounded. A related result had been proved before over IR in [Prestel Bradley 89].

86 I conclude this part with a series of beautiful results about ideals in real holomorphy rings. Here once more geometrical methods are used (coming from algebraic geometry and topology) to obtain purely algebraic theorems. There was a long-standing question by Gilmer, who had asked if any finitely generated ideal in a Priifer domain has actually two generators. Sch/ilting disproved this by taking the real holomorphy ring H(F) of F = R(X, Y) and showing that the fractional ideal (1, X, Y) is not generated by two elements [Sch/ilting 79]. Let #(H) be the minimal number of generators sufficient for all finitely generated ideals of H. Since H(F) has Krull dimension d, for F/]R a ddimensional real function field, a general result of Heitmann gives #(H(F)) < d + 1. After SchNting had given his counterexample in 1979, people began to wonder whether equality might hold in general. In the end, this turned out to be true, but the general result was established only quite recently: After R. Swan had shown equality for F = R ( X 1 , . . . , Xd) [Swan 84] and Schiilting had proved # (H(F)) = 3 for all function fields of real surfaces {Schiilting 86b}, W. Kucharz settled the general case in [Kucharz 89a], showing # (H(F)) = d + 1 for F / N real of dimension d. As Becker had observed much 2n earlier, this implies that there are f l , . . - , f d + l C F such that f~" + . . . + f~+l is not a sum of d 2n-th powers for any n _> 1. If F / R is purely transcendental one can take 1 , X 1 , . . . , X d (this came out already from Swan's proof), and thereby obtain a generalization of a classical theorem of Cassels! Kucharz's proof uses the description of H(F) as those rational functions which are defined on the real points of some smooth projective model. Since his techniques build upon the ideas of Schiilting and Swan, I will skip over the details of their work. Kucharz proceeds roughly like this: A fractional ideal of H(F) corresponds to a line bundle on some smooth projective model X of FIR. Blowing up X in a real point x0 one gets ~r : X I --, X. For the class a of the exceptional fibre E = 7r-l(x0) one shows that a d 7~ 0 in H d (X'(R), Z/2). If the fractional ideal corresponding to E were generated by d elements, a calculation with Stiefel-Whitney classes shows that there would be a multi-blow-up a : X " --* X' for which a d maps to zero in H d ( x " ( R ) , Z / 2 ) . But one can show that the map in cohomology induced by such a a is always injective. Actually, Kucharz's paper provides much more information than just it(H) = d + 1. For example, Sch/ilting had studied the binary class group BE of F, i.e. the quotient of the class group of H(F) by the classes of those ideals which admit two generators. This group plays also a role in Becker-Rosenberg's theory of reduced Witt rings of higher level {Becker Rosenberg 85b}. Sch/ilting had found that BE ~ (Z/2) s if d = 2, and IBFI = oo for d > 3, where s is the number of connected components of a smooth projective model {Schiilting 86b}. These results are widely generalized by Kucharz to other quotients of the class group. The result it(H) = d + 1 was later generalized to relative real holomorphy rings H = H ( F / R ) over arbitrary real closed fields R {Kucharz 89c}. [Buchner Kucharz 89] also studied relative real holomorphy rings H(F/V), where V is an affine irreducible smooth R-variety and F its function field. Here one has it(H(F/V)) < dim V if and only if V(R) has no compact connected components. Also this is a generalization of the above main result of Kucharz.

87 3. S e l e c t e d a p p l i c a t i o n s o f real a l g e b r a to g e o m e t r y I want to start with a few words on the work of J.M. Ruiz on global semianalytic geometry. He not only made available real spectrum methods and applied them with great success; rather, on his way to this goal, he established remarkable results of algebraic nature which can certainly claim independent interest. The key result, which makes the real spectrum a highly useful tool in semianalytic geometry, is an Artin-Lang property for global analytic functions on a compact analytic set X. It means that the globally semianalytic subsets of X correspond bijectively, via an operator "tilda" which preserves set-theoretic operations, to the constructible sets in the real spectrum of O(X). The Artin-Lang property in turn rests on the study of the real spectrum formal fibres of local analytic rings. The basic result, whose use pervades the whole theory, is the following extension theorem of Ruiz. Suppose A is an excellent local domain and a is an ordering of its field of fractions. If ol makes the maximal ideal of A convex, then it extends to an ordering of the total ring of quotients of A, the completion of A. Other forms of this result are a formal curve selection lemma and a real dimension theorem, both for excellent rings. See [Ruiz 85b], [Ruiz 86a+b]. As an immediate application, his theorem allowed Ruiz to solve the 17th Hilbert problem for meromorphic functions in the amrmative, on a compact irreducible analytic set [toc.cit.]. Before, this had been known only in very special cases. Another application is the real Nullstellensatz for global analytic functions. It must be pointed out that the same results for compact analytic sets (ArtinLa~g property, Hilbert 17, real Nullstellensatz) have been proved independently by P. Jaworski. In his approach, part of the algebraic reasoning of Ruiz is replaced by the use of Hironaka's theorems [Jaworski 86]. But more effort was needed to establish basic properties of well-behavedness for the real spectra of (local or global) rings of analytic functions. Ruiz showed that the class of excellent rings is a suitable environment: It contains the rings arising in analytic geometry, and on the other hand has "excellent" properties also for the real spectrum. Apart from the above mentioned theorem (which is central for many arguments), this includes dimension theory, the going-down property for regular homomorphisms, constructibility of closures of constructible sets and (under suitable conditions) of connected components. See [Ruiz 86b], [Ruiz 89a+b], [Alonso Andradas 87], [Andradas Brhcker Ruiz 88]. By Artin-Lang, this gives corresponding geometric theorems: If X is a compact analytic set and Z is a globally semianalytic subset of X, then closure and connected components of Z are again globally semianalytic; and there is a "Finiteness Theorem" saying that the operator tilda preserves closures [Ruiz 86b], [Ruiz 88]. (The compactness hypothesis on X is weakened in [Ruiz 90a].) Another application will be mentioned later in this talk. A recent very nice result is the characterization of sums of 2n-th powers of meromorphic functions, on a compact analytic set X, by their values on curve germs [Ruiz 90b] (in the case of analytic surfaces this was also proved in [Kucharz 89b]): A necessary and sufficient condition for f to lie in Efl4(X) 2n is that for every analytic curve germ a : ]-e, e[ ~ X one has f(a(t)) = at2"k+ higher powers, with a > 0, as long as a ~ Sing X and f o a is neither undefined nor identically zero. This is a remarkable result, and is of course very much reminiscent of Prestel's substitution

88 criterion mentioned before. Tim similarity is "explained" and embedded into a more general context in {Ruiz 91}. In the remainder of this talk I want to record some progress in questions on the complexity of systems of inequalities. A common feature of the following is that it rests on t h e - apparently completely abstract - - theory of preorderings and fans in fields, and on its even more abstract generalization, the spaces of orderings. I will not go here into the purely algebraic aspects of these matters; it seems that the majority of important results in this theory had already been obtained by the beginning of the 80s, and that since then the applications have been more striking. A noteworthy exception are the important papers [Schwartz 83] and [Marshall 84] on saturation and local stability. The problem is how to describe semialgebraic sets as economically (or "cheaply") as possible, i.e. how to minimize the number of functions in a description, regardless of their complexity. The first major step was done by L. Brhcker [Brhcker 84a]. He considered affine algebraic varieties V over a real closed field R, and what was later called basic open (semialgebraic) sets in V(R); that is, sets which can be written S = {x E V ( R ) : fl(x) > 0 , . . . , f r ( x ) > 0} with polynomials fi C R[V]. The question is, how many inequalities are really needed to present S in this form? Brhcker gave the surprising answer that there is a common upper bound for all basic open sets in V(R), which depends only on d = dim V. In fact, writing s(d) for the least upper bound valid for all d-dimensional varieties, he showed that d(d-2)(d-4)...2 deven, d < s(d) <_ d!!:= d(d 2)(d 4) .1 d odd. In particular, s(d) = d for d = 1,2, 3. It is worthwile to recall the main steps of his proof. Given a basic open set S, one starts by looking only at generic presentations of S, t h a t is, at basic open sets T for which the symmetric difference of S and T has dimension < d. This is a problem on the orderings of the function field R(V) (suppose V irreducible, for simplicity), which had been solved by Brhcker long before, actually in much greater generality [Brhcker 74]. Namely, d = dim V inequalities suffice for a generic presentation. And - - assuming that R(V) is formally real - - this number is the best possible, even if one considers only generic presentations. This becomes the starting point for an induction argument, since one can now pass to a subvariety of smaller dimension. However, in the end one has to glue together these generic presentations on subvarieties, to obtain a true one. For this, Brhcker proved special pasting lemmas, which are consequences of the Hhrmander-Lojasiewicz inequality for polynomial functions. The estimate d !! comes from an extra argument which allowed to approximate S even up to codimension two by a generic presentation with d inequalities. A somewhat simplified version of the proof was later presented in [Mah~ 86], where Brhcker's results were used in an essential way to bound the exponent of the cokernel of the global signature map. By improving the pasting techniques, Brhcker later gave better bounds for s(d), roughly equal to (vz2) d+l {Brhcker 85}. He also proposed similar questions for basic closed sets (replacing "> 0" by " ~ 0"), and also for general open (resp. closed) semialgebraic sets. Here one tries to write an open or closed set as a union of few basic (open

89 or closed) sets. (The Finiteness Theorem asserts that this is always possible.) In this way one defines corresponding invariants ~, t and t. In [Br6cker 84b], finiteness of *(d) and ~(d) was shown (actually, bounds were given), and {Br6cker 85}, {Br6cker 87} gave the estimate ~(d) <_ ½d(d + 1), with equality for d = 1, 2. As in the basic open case, the upper bounds for ~, t, ~ are obtained by pasting from the corresponding generic bounds. However, in contrast to the s-invariant, the t-invariant is even generically little understood, and even the generic bounds are probably much too high. The H6rmander-Lojasiewicz inequality has proved useful also for different kinds of problems. For example, it is needed for other characterizations of the property of being basic: An open semialgebraic set is basic open iff it does not meet the Zariski closure of its boundary and is generically basic in each irreducible subvariety. For closed sets, the same holds mutatis mutandis, with the condition on the Zariski closure dropped {Br6cker 87}, {Br6cker 88b}. The advantage of such criteria is that properties like being basic or being representable by a conjunction of r inequalities are generically well understood, in terms of fan theory of the function field; and under favorable conditions, this fan theory can be reformulated in geometric terms. Other applications of the Lojasiewicz inequality are to separation questions, where also spaces of orderings are very instrumental. I want to digress into this for some moments before returning to the description of semialgebraic sets. The separation problem starts with two closed disjoint semialgebraic sets A, B on aal a ~ n e _R-variety V, and asks if there is a polynomial f with fIA > 0 and f i b < O. One quickly realizes that this is not in general possible. But. it becomes so if one allows square roots of everywhere positive polynomials (theorem of Mostowski and BochnakEfroymson). Br6cker showed that separation by polynomials is possible if A, B are basic closed and one of them has dimension ~ 2; the dimension hypothesis cannot be dropped [Br6cker 88a]. This uses both pasting and spaces of orderings; in fact, the results of Schwartz and Marshall on saturation cited above are an essential tool. Pasting techniques are also applied in [Fortuna Galbiati 89], where conditions are studied under which separation up to a closed subvariety implies actual separation by some polynomial. Quantitative aspects of the separation problem were studied by Stengle. He asked for the minimal number of square roots that have to be adjoined to make separation possible, and defined the Mostowski number ra(d) to be the smallest m such that on all ddimensional varieties, separation always is possible after adjoining m (suitable) square roots as above. From the standard proof of Mostowski's theorem one gets the upper bound re(d) <_~(d)~(d) [Mah6 86]. Stengle was interested in lower bounds, and related them to a notion of complexity for semialgebraic sets. In his notion, one can use the relations = and # for free, but one has to pay for every use of <, < etc. In this way a lower bound for re(d) was obtained which is roughly logarithmic in d [Stengle 84], [Stengle 89]. It was improved in [Br3cker Stengle 90], where rn(d) >_ d - 1 was shown. Here fans were used again, together with trace forms. Another remarkable separation result was proved in [Ruiz 84], namely for closed semianalytic germs, in ease one of them has dimension < 1: The (punctured) germs can be separated by a polynomial. Again, the Lojasiewicz inequality is used. Ruiz gave nice applications to semialgebraie geometry, thereby partially answering some questions of T. Recio in [Brumfiel 82].

90 Returning to the description of basic sets, the next important step was done in [Andradas BrScker Ruiz 88]. Here it is shown that Br6cker's ideas apply also to compact analytic sets X, namely to the description of basic open sets defined by global analytic functions. The main result is that the number of inequalities needed is bounded in terms of d = dimX. Actually the same bounds are given as in the semialgebraic case. Similar to the semialgebraic case, the problem can be translated into the real spectrum of the ring of global analytic functions O(X), thanks to Artin-Lang property and finiteness theorem. This fact lies at the base of the proof. Generally, one can consider basic open subsets in the real spectrum of any commutative ring A, and ask for the minimum number of inequalities (of the form a > 0, with a E A) needed to present them. The supremum of these numbers is an invariant of A which, generalizing terminology from the field case, is called the (geometric) stability index of A. As a consequence of the Positivstellensatz, an "abstract" Lojasiewicz inequality is available on the real spectrum of any ring. So the pasting lemmas hold in general, and the main problem becomes to find the generic minimal bound - - which again turns out to be the dimension. Using a refined version of Weierstrafi preparation theorem, the stability index of the fraction field of a complete local domain is estimated. The transition to henselian local (excellent) rings is accomplished by the use of Artin's conjecture, as proved by C. Rotthaus. Then I succeeded in showing that indeed s(d) = d holds in general [Scheiderer 89], i.e. the stability index of a real algebraic R-variety is just its dimension. Also Brgcker found a proof which is slightly different {Brgcker 88b}. The proof makes essential use of the fact that each semilocal ring gives rise to a space of orderings, and of a structure theorem about fans in these spaces of orderings. These results axe due to [Kleinstein Rosenberg 78, 80] and [Knebusch 81]. Thus the proof is less geometric than BrScker's former approach. The main difference is that the pasting techniques are completely left aside, and instead the generic presentations are improved inductively. Actually, the proof generalized without trouble to noetherian rings whose (real) singularities behave reasonably well (e.g. excellent rings). Surprisingly, in such a ring the (minimal) length of a generic presentation is already the (minimal) length of an actual one. (If the ring is not regular, one has to modify the meaning of "generic", according to the singularities.) In particular, also for basic open semianalytie sets as considered in [Andradas Br6cker Ruiz 88], the dimension of X is the exact upper bound for the number of inequalities. On the other hand, it was soon realized that BrScker's proof had the advantage not to need any conditions on the singularities [BrScker 90], {Marshall 89}. There is no more a reduction to a generic situation, but a local global principle for the presentation of basic open sets still holds with respect to the class of all residue fields. In particular, one has the "stability formula": The stability index of a noetherian ring is the supremum of the stability indices of its residue fields. It should be mentioned that also the g-invariant was determined in [Scheiderer 89] for all R-varieties. Here BrScker's previous upper bound ½d(d + 1) turned out to be best possible, for any real d-dimensional R-variety. Again, fan theory was used for this. But that's not yet the end of the story! Soon after the above proofs were circulated, Mah4 came up with a new proof of s(d) = d [Mah6 89a], {Mah4 89b}. It is more elementary in that it doesn't use spaces of orderings, nor the Lojasiewicz inequality. If a basic open set S is given, Mah4 first inverts all polynomials which are strictly positive on S. This doesn't affect the problem, and has the advantage of producing sufficiently

9] many units, so that one can do some of the things with quadratic forms which one is used to from fields. For example, a weak form of transversality holds. Mahd then proceeds by a careful analysis of Pfister forms over such rings. In this way, the proof uses only explicit calculations with forms! The only exception to this is the Tsen-Lang theorem, which is used to get the stability index of the function fields. The very latest progress is a recent preprint of M. Marshall {Marshall 91}. He proves the stability formula in perhaps the greatest generality one can think of, namely for arbitrary commutative rings A: If a basic open subset S of the real spectrum can be presented by r > 1 inequalities in each residue field of A, then S itself has a presentation by r inequalities. (The real spectrum of A can even be replaced by any intersection of basic subsets.) Marshall essentially follows Mahd's proof; but where Mah~ argues by induction over the dimension to get a weak representation, Marshall uses a more abstract tool, namely" a local global principle (due originally to [Brbcker 82]) for modules over preorderings in rings. So after all, the former "problem of the stability index" has been settled in a truely elementary and satisfying way, and in the greatest generality, with the help of the machinery of abstract real algebra.

References

In the text, references to articles are by name(s) of author(s) and year. For the difference between curly braces and square brackets see footnote 1 in the text. [GARFQ] Gdomdtrie Algdbrique Rdelle et Formes Quadratiques, Proc. Conf. Rennes 1981, ed. J.-L. Colliot-Thdl~ne, M. Coste, L. Mahd, M.-F. Roy, Lect. Notes Math. 959, Springer, 1982. [OFRAG] Proc. Special Session on Ordered Fields and Real Algebraic Geometry, ed. D.W. Dubois, T. Recio, Contemp. Math. 8, 1982. [RAAG] Real Analytic and Algebraic Geometry. Proc. Conf. Trento 1988, ed. M. Galbiati, A. Tognoll, Lect. Notes Math. 1420, Springer, 1990. M.E. Alonso, C. Andradas: [87] Real spectra of complete local rings. Manuscripta math. 58, 155-177 (1987). C. Andradas: [85] Real places in function fields. Comm. Algebra 13, 1151-1169 (1985). [89] Specialization chains of real valuation rings. J. Algebra 124, 437-446 (1989). C. Andradas, L. Br6cker, J.M. Ruiz: [88] Minimal generation of basic open semianalytic sets. Invent. math. 92,409-430 (1988). J.Kr. Arason, A. Pfister: [82] Quadratische Formen fiber affinen Algebren und ein Mgebraischer Beweis des Satzes yon Borsuk-Ulam. J. reine angew. Math. 331,181-184 (1982). S.M. Barton: [88] The real spectrum of higher level of a commutative ring. Ph.D. thesis, Cornell Univ., Ithaca, 1988. E. Becker: [82a] Valuations and real places in the theory of formally real fields. In [GARFQ], pp. 1-40.

92 [82b] [84]

The real holomorphy ring and sums of 2n-th powers. In [GARFQ], pp. 139-181. Extended Artin-Schreier theory of fields. Rocky Mountain J. Math. 14, 881-897 (1984). E. Becker, R. Bert, F. Delon, D. Gondard: {90} Hilbert's 17-th problem for 8urns of 2n-th powers. Preprint 1990. E. Becker, D. Gondard: [89] On rings admitting orderings and 2-primary chains of orderings of higher level. Manuscripta math. 65, 63-82 (1989). E. Becket, J. Harman, A. Rosenberg: [82] Signatures of fields and extension theory. J. reine angew. Math. 330, 53-75 (1982). E. Becket, B. Jacob: [85] Rational points on algebraic varieties over a generalized real closed field: A model theoretic approach. J. reine angew. Math. 357, 77-95 (1985). E. Becker, A. Rosenberg: [85a] Reduced forms and reduced Witt rings of higher level. J. Algebra 92,477-503 (1985). {85b} On the structure of reduced Witt rings of higher level. Preprint 1985 (unpublished). E. Becket, N. Schwartz: {89} Reduced Witt rings of infinite level. Preprint 1989. R. Berr: Reelle algebraische Geometrie hSherer Stufe. Dissertation, Ludwig-Maximilians-Uni[89] versit~it Mfinchen, 1989. {90} Sums of mixed powers in fields and orderings of prescribed level. Preprint 1990. {91} The p-real spectrum of a commutative ring. Preprint 1991. L. BrScker: [74] Zur Theorie der quadratischen Formen fiber formal reellen KSrpern. Math. Ann. 210, 233-256 (1974). [82] Positivbereiche in kommutativen Ringen. Abh. Math. Sem. Univ. Hamburg 52, 170178 (1982). [84a] Minimale Erzeugung yon Positivbereichen. Geom. Dedicata 16, 335-350 (1984). [84b] Spaces of orderings and semialgebraic sets. In: Quadratic and Hermitian Forms, Conf. Hamilton 1983, ed. I. Hambleton, C.R. Riehm, Canad. Math. Soc. Conf. Proc. 4, 1984, pp. 231-248. {85} Description of semialgebraic sets by few polynomials. Lecture CIMPA, Nice 1985. {87} Characterization of basic semialgebraic sets. Preprint 1987. [88a] On the separation of basic 8emialgebraic sets by polynomials. Manuscripta math. 60, 497-508 (1988). {88b} On basic semi-algebraic sets. Preprint 1988 (to appear Expos. Math.). [90] On the stability index of noetherian rings. In [RAAG], pp. 72-80. L. Br5cker, H.-W. Schiilting: [86] Valuations of function fields from the geometrical point of view. J. reine angew. Math. 365, 12-32 (1986). L. BrScker, G. Stengle: [90] On the Mostowski number. Math. Z. 203, 629-633 (1990). R. Brown: [87a] Real closures of fields at orderings of higher level. Pacific J. Math. 127, 261-279 (1987). [87b] The behavior of chains of orderings under field extensions and places. Pacific J. Math. 127, 281-297 (1987). G. Brumfiel: [82] Some open problems. In [OFRAG], pp. 19-25.

93 M.A. Buchner, W. Kucharz: [89] On relative real holomorphy rings. Manuscripta math. 63, 303-316 (1989). A. Campillo, J.M. Ruiz: [90] Some remarks on pythagorean real curve germs. J. Algebra 128, 271-275 (1990). M.D. Choi, Z.D. Dai, T.Y. Lam, B. Reznick: [82] The Pythagoras number of some affine algebras and local algebras. J. reine angew. Math. 336, 45-82 (1982). M.D. Choi, T.Y. Lam, A. Prestel, B. Reznick: {91} Sums of 2m-th powers of rational functions in one variable over real closed fields. Preprint 1991. M.D. Choi, T.Y. Lam, B. Reznick: [87] Even symmetric sextics. Math. Z. 195,559-580 (1987). [91] Positive sextics and Schur's inequalities. J. Algebra 141, 36-77 (1991). J.-L. Cotliot-Th~l~ne: [86] Appendix to: K. Kato, A Hasse principle for two dimensional global fields. J. reine angew. Math. 366, 181-183 (1986). J.-L. Colliot-Th~l~ne, U. Jannsen: [91] Sommes de carr~s dans les corps de fonctions. C. R. Acad. Sci. Paris 312, sdr. I, 759-762 (1991). Z.D. Dai, T.Y. Lam: [84] Levels in algebra and topology. Comment. Math. Helvetici 59, 376-424 (1984). Z.D. Dai, T.Y. Lam, R.J. Milgram: [81] Applications of topology to problems on sums of squares. Ens. Math. (2) 27, 277-283 (1981). Z.D. Dai, T.Y. Lain, C.K. Peng: [80] Levels in algebra and topology. Bull. Am. Math. Soc. (N.S.) 3, 845-848 (1980). F. Delon: [86] Corps et anneaux de Rolle. Proc. Am. Math. Soc. 97, 315-319 (1986). C.N. Delzell: [823] A finiteness theorem for open semi-algebraic sets, with applications to Hilbert's 17 th problem. In [OFRAG], pp. 79-97. [82b] Case distinctions are necessary for representing polynomials as sums of squares. In: Proc. Herbrand Syrup., Logic Coll. 1981, ed. J. Stern, North Holland, 1982, pp. 87103. [84] A continuous, constructive solution to Hilbert's 17th problem. Invent. math. 76, 365-384 (1984). M. Dickmann: [88] The model theory of chain closed fields. J. Symb. Logic 53, 73-82 (1988). E. Fortuna, M. Galbiati: [89] S~paration de semi-alg~briques. Geom. Dedicata 32, 211-227 (1989). D. Gondard: [87] Th~orie du premier ordre des corps chMnables et des corps chaine-clos. C. R. Acad. Sci. Paris 304, s~r. I, 463-465 (1987). [90] Chainable fields and real algebraic geometry. In [RAAG], pp. 128-148. J. Harman: [82] Chains of higher level orderings. In [OFRAG], pp. 141-174. B. Jacob: [81] The model theory of generalized real closed fields. J. reine angew. Math. 323,213-220 (1981).

94 P. Jaworski: [86] Extension of orderings on fields of quotients of rings of real analytic functions. Math. Nachr. 125, 329-339 (1986). J.L. Kleinstein, A. Rosenberg: [78] Signatures and semisignatures of abstract Witt rings and Witt rings of semilocal rings. Canad. J. Math. 30, 872-895 (1978). [80] Succinct and representational Witt rings. Pacific J. Math. 86, 99-137 (1980). M. Knebusch: [77] Some open problems. In: Conf. Quadratic Forms, Kingston 1976, ed. G. Orzech, Queen's Papers Pure Appl. Math. 46, 1977, pp. 361-370. [81] On the local theory of signatures and reduced quadratic forms. Abh. Math. Sem. Univ. Hamburg 51,149-195 (1981). [82] An algebraic proof of the Borsuk-Ulam theorem for polynomial mappings. Proc. Am. Math. Soc. 84, 29-32 (1982). W. Kucharz: [89a] Invertible ideals in real holomorphy rings. J. reine angew. Math. 395,171-185 (1989). [89b] Sums of 2n-th powers of real meromorphic functions. Monatsh. Math. 107, 131-136 (1989). {89c} Generating ideals in real holomorphy rings. Preprint 1989. F.-V. Kuhlmann, A. Presteh [84] On places of algebraic function fields. J. reine angew. Math. 353, 181-195 (1984). K.Y. Lam: [84] Topological methods for studying the composition of quadratic forms. In: Quadratic and Hermitian Forms, Conf. Hamilton 1983, ed. I. Hambleton, C.R. Riehm, Canad. Math. Soc. Conf. Proc. 4, 1984, pp. 173-192. [85] Some new results on composition of quadratic forms. Invent. math. 79, 467-474 (1985). K.Y. Lain, P.Y.H. Yiu: [89] Geometry of normed bilinear maps and the 16-square problem. Math. Ann. 284, 437-447 (1989). T.Y. Lain, T.L. Smith: {90} On Yuzvinsky's monomiM pairings. Preprint, Berkeley 1990. L. Mah6: [86] Th@or~me de Pfister pour les varidt@s et anneaux de Witt r~duits. Invent. math. 85, 53-72 (1986). [89a] Une d~monstration 61@mentaire du th~or&me de BrScker-Scheiderer. C. R. Acad. Sci. Paris 309, s@r. I, 613-616 (1989). {89b} On the geometrical stability index of a ring. Lecture, Quadratic Forms Conf., Oberwolfach 1989. [90] Level and Pythagoras number of some geometric rings. Math. Z. 204, 615-629 (1990). M. Marshall: [84] Spaces of orderings: Systems of quadratic forms, local structure, and saturation. Comm. Algebra 12, 723-743 (1984). {89} Minimal generation of constructible sets in the real spectrum of a Noetherian ring. Preprint 1989. {91} Minimal generation of basic sets in the real spectrum of a commutative ring. Preprint 1991. M. Marshall, C. Mulcahy: [90] The Witt ring of a space of signatures. J. Pure Appl. Algebra 67, 179-188 (1990).

95 C. Mulcahy: [88a] An abstract approach to higher level forms and rigidity. Comm. Algebra 16, 577-612 (1988). [88b] A representation theorem for higher level reduced Witt rings. J. Algebra 119, 105-122 (1988). J.J. Ortega: [91] On the Pythagoras number of a real irreducible algebroid curve. Math. Ann. 289, 111-123 (1991). A. Pfister, S. Stolz: [87] On the level of projective spaces. Comment. Math. Helvetici 62, 286-291 (1987). V. Powers: [87] Characterizing reduced Witt rings of higher level. Pacific J. Math. 128, 333-347 (1987). [89] Finite spaces of signatures. Canad. J. Math. 41,808-829 (1989). A. Prestel: [84] Model theory of fields: An application to positive semidefinite polynomials. Mdm. Soc. Math. France (N.S.) 16, 53-65 (1984). [87] Model theory applied to some questions about polynomials. In: Contributions to General Algebra 5, Proc. Salzburg Conf. 1986, Wien 1987, pp. 31-43. A. Prestel, M. Bradley: [89] Representations of a real polynomial f(X) as a sum of 2m-th powers of rational functions. In: Ordered Algebraic Structures, ed. J. Martinez, Kluwer 1989, pp. 197207. B. Reznick: [87] A quantitative version of Hurwitz' theorem on the arithmetic-geometric inequality. J. reine angew. Math. 377, 108-112 (1987). [89] Forms derived from the arithmetic-geometric inequality. Math. Ann. 283, 431-464 (1989). J.M. Ruiz: [84] A note on a separation problem. Arch. Math. 43, 422-426 (1984). [85a] Pythagorean real curve germs. J. Algebra 94, 126-144 (1985). [85b] On Hilbert's 17th problem and real Nullstellensatz for global analytic functions. Math. Z. 190, 447-454 (1985). [86a] C6nes locaux et compl6tions. C. R. Acad. Sci. Paris 302, s6r. I, 67-69 (1986). [86b] On the real spectrum of a ring of global analytic functions. Publ. Inst. Rech. Math. Rennes 4, 84-95 (1986). [88] On the connected components of a global semianalytic set. J. reine angew. Math. 392, 137-144 (1988). [89a] A dimension theorem for real spectra. J. Algebra 124, 271-277 (1989). [89b] A going-down theorem for real spectra. J. Algebra 124, 278-283 (1989). [90a] On the topology of global semianalytic sets. In [RAAG], pp. 237-246. [90b] A characterization of sums of 2nth powers of global meromorphic functions. Proc. Am. Math. Soc. 109, 915-923 (1990). {91} The substitution test for sums of even powers. Preprint 1991. C. Scheiderer: [89] Stability index of real varieties. Invent. math. 97, 467-483 (1989). J. Schmid: [88] Eine Bemerkung zu den hSheren Pythagoraszahten reeller KSrper. Manuscripta math. 61, 195-202 (1988).

96 H.-W. Schfilting: [79] Uber die Erzeugendenanzahl invertierbarer Ideale in Prfiferringen. Comm. Algebra 7, 1331-1349 (1979). [82a1 On real places of a field and their holomorphy ring. Comm. Algebra 10, 1239-1284 (1982). [82b1 Real holomorphy rings in real algebraic geometry. In [GARFQ], pp. 433-442. [86a] Prime divisors on real varieties and valuation theory. J. Algebra 98, 499-514 (1986). {86b} The binary class group of the real holomorphy ring. Preprint 1986. N. Schwartz: [83] Local stability and saturation in spaces of orderings. Canad. J. Math. 35, 454-477 (1983). [84] Chain signatures and real closures. J. reine angew. Math. 347, 1-20 (1984). [87] Chain signatures and higher level Witt rings. J. Algebra 110, 74-107 (1987). D.B. Shapiro: [84] Products of sums of squares. Expos. Math. 2,235-261 (1984). 0. Simon: [89] Aspects quantitatifs de Nullstellensgtze et de Positivstellens£tze. Nombres de Pythagore. Comm. Algebra 17, 637-667 (1989). G. Stengle: [84] A lower bound for the complexity of separating functions. Rocky Mountain J. Math. 14, 925-927 (1984). [89] A measure for semialgebraic sets related to boolean complexity. Math. Ann. 283, 203-209 (1989). S. Stolz: [89] The level of real projective spaces. Comment. Math. Helvetici 64, 661-674 (1989). R.G. Swan: [84] n-generator ideals in Prfifer domains. Pacific J. Math. 111,433-446 (1984). P.Y.H. Yiu: [86] Quadratic forms between spheres and the non-existence of sums of squares formulae. Math. Proc. Camb. Phil. Soc. 100, 493-504 (1986). [87] Sums of squares formulae with integer coefficients. Canad. Math. Bull. 30, 318-324 (1987). [90] On the product of two sums of 16 squares as a sum of squares of integral bilinear forms. Quart. J. Math. Oxford (2) 41,463-500 (1990). S. Yuzvinsky: [81] Orthogonal pairings of euclidean spaces. Michigan Math. J. 28, 131-145 (1981). [84] A series of monomial pairings. Linear Multilin. Algebra 15, 109-119 (1984). G. Zeng: A characterization of preordered fields with the weak Hilbert property. Proc. Am. [881 Math. Soc. 104, 335-342 (1988). [91] On preordered fields related to Hilbert's 17th problem. Math. Z. 206, 145-151 (1991).

TOPOLOGY OF REAL PLANE ALGEBP~IC CURVES E.i. Shustin (Math. Dept., Samara State University, ul. Acad. Pavlova 1, 443011 S~nara, USSR) In this article we'll consider some ideas, methods, results found over the last ten years in topology of real plane algebraic curves. We'll also speak of some unsolved problems. Topology of real curves is expounded in many known surveys ~ , 4, 9, 18, 26, 28, 29~. We'll consider matter not included in them. The text consists of 4 parts: in the first part we speak of methods of getting restrictions, in the second part we speak of real curves constructions, in the third part we speak of connection between algebraic curves and singularities smoothings, in the fourth part we formulate some conjectures. Used below (without refering) classical results in real curves topology are described in ~I, 4, 9, 18, 26, 28, 29J quite completely. Introduction By plane real algebraic curve we mean a homogeneous real polynomial

F C~o~cI~

~m)

(being considered modulo a constant

factor) and the set

Connected non-singulaz' two-sided components of ~ are called ovals. The classification problem of real curves of a given degree can be considered from three points of view: (i) The real cl+~ssification - up to continuous isotopies in ~ p z (i.e. the ovals arrangements classification) (ii) The equivariant one - up to equivariant isotopies in ~p~ connectin~ the given cu~.ves complexifications

(iii) The rigid one - up

to

isotopies consisting of algebra-

ic curves of a given degree. We'll deal with the first problem. I. Restrictions All the kno~m restrictions to an ovals arrangement arise from investigation of various structures on objects connected with a real curve. Such objects are

(i) the pairs (~p2 2 ~ F ) ~

(~P~j C F ) 3

g8

(ii) the pair ('5 ~

S ~= t P ~

4 ; IRF~={eeRP21EF(e)>--D3,

and also the double covering of

~2

branched a t

E=_+, ~F

or at

(~F/~H~) U ~

, if a curve degree is even. The technical means of investigation are algebraic-topological (Smith theory, complex orientations, quadratic form of intersection index in 2-dimensional homology of double covering etc.) or algebraic-geometrical (Bezout's theorem, a position of a curve with respect to special straigh~ lines or conics etc.). Here we'll give the detailed description of one method of getting restrictions (see [15, 24, 28, sec 4.15]), which is based on some topological and geometrical properties of curves. Namely, let F be a real non-singular curve of the even de&Tee p/~ , let p (resp. kW ) denote the number of ovals lying inside even (odd) nmnber of other ovals. Put j---~#- 1 - p -- ~ ; where $ is a curve genre. Let ~ ~ ~ --~ ~ p ~ be the double covering with branching at ~ F The complex conjugation C~H~'~ ~"p~--p #~' p a is covered by tv¢o antihoiomorphic involutions ~ H ~ + 2 ~ ' on ~ . Assume that ~ £ X ~_) is non-orientable. It is well-knovaa [2, 11] that ~ a ( ~ is the unimodalar lattice (without torsion) with respect to intersection index, and it contains the maximal sublattice H ÷ ~ H _ ~ H where

~,

H= { d, e H= ( ~.) l r..o~4j±;.~:-,~ _.

•

"

"~ H= 1', "~ H+ = (~2_ 3 ~ ) / 2

~'

.

~'

--

.4"

+ p -in,

z4H_ = (~"-sm)/Z * 2 -p+~;

6"*/H)= -/, 6--(H)=O: ~-+(~)=0~<6.*~)/8, ~-(~/.) =

3{~-~)/8 2

+ p-~,

Here ( ~ 6" ) ~-o) i s a signature. To get restrictions we'll first construct the sublattice ~ - c H~. , which is connected with ovals arrangement, and second verif Z relations

99

+- ( T )

.~- s ° ( 7 - )

_< ~ +- ( H E )

( ~)

and

oLsc~. ' T = ~:~. 2~scz H E

if

z~, "T= ~

H a. .

(2)

The first approacli to construction of cycles realizing classes in H a ( ~ ) is as follovJs. Let /~ be an immersed in ~ p 2 surface with boundar$ ~ A4 ~ d~ F such th~vt

(~) ( m \ 9 , ~ ) ~O CF= ~;

c-o~j'M= A4,,

Then the surface ~-i~/~d) is orientable and represents (being equipped with a fixed orientation) some integral 2-cycle in ~ . Consider examples of such membranes ~ giving classes in ~ E : I. Arnold's construction [2, 29]. As /$I it is possible to take any connected component o f ~ E ~ orientable if ~ = O ~ o S ~ ) . 2. Rokhlin's construction [17J. Let ~ be any connected component of ~ F_ E , let ~4 c (~F \ [ . ~ and ~z ~ ~ t be arcs connecting two points on ~ ~ . There exists an immersed in ~ P ~ \ ~ disc ~Z) with boundary ~ t / ~ 2 . Then put 3. Viro's construction (see [15, 28~). Consider the segment ~] = [ L ~; L z ] of a real straight lines pencil, where L 1 2 / ~ are tangents to ~ and don't cross ~ F \ ~ ~ . Assume that C ~ Z ~ ( ~ L /3 (~F \ ~ r ) ~< L/ for ~ Le [7 . The set

U

(CL\~L)

L e 17 transposing by C o ~

S 1 = (£•-!f3 E F ;

consists of two connected components . Denote ~ I U one of them. Put

R/7 = U

R L ,~ RF_

~..

L e l-! Suppose that points of contact of F with lines L E / 7 are connected pairwise in ~ I ~ by smooth arcs ~¢~ ...) ~ z

ca~,~ (~'~ C) De L ) -~ "f,, L E / 7 , , Then S = S I U.~y U, , . (.J~'Z

,: = % , , . , meets ~'L

Z .

at two points , and at one point if / = L ~ orL~; • Connect the points ~ ~ ~ L by an arc in • L , which depends on / continuously. The union of these arcs is a disc with boundary 5 . Put /%4 = ~ U ~ ° ~ The Viro const-

iz l ~ [7\ { L o L z ~

100

ruction gave maximal sublattices in I~E for the first time, and allowed to obtain restrictions from (2) E28~, what stimulated the search of other constructions° The fourth construction was suggested by Kharlamov and author (see ~24]). It differs from above constructions° Namely, let ~lj .. "2 ~ be the set of different straight lines and conics crossing ~ I--- transversally and only in real points. Then ~-~(~L)

\ F~x c 4 ~ j ~

t

has two connected components

transposing by involution

ent ~

?

so that the chain ~ ,

COs~ E IL ~

. We'll oriis inva-

riant with respect to co~jE . Then ~ ) = ~ = ~ is 1-cycle on FLX~J4~a. , : f ~ ~- [ ~ , ] = 0 EH~CFzx~uJ'E) then there is 2-chain S , consisting of closures of the set F~× c~I~ £ \ ~ ~ connected components such that ~S--~ repre. Then 2-cycle ~ ~ I ) ~ - ~ ~Z~2~ ) \j ~ S = ~_--~ sents some class in HE , ~or ~ = O it is Arnold's construction. The cycles appearing in [2] gave the Arnold inequalities, and the second construction allowed to give an elementary proof of strong Arnold-Petrovsky inequalities (cf. [29S, where the proof is based on Smith theory):

p-~ S ( ~ - ~ - ~ , 3 / 8 a < 3

Cm'~-~)/8

p ~ C~ ~- ~ 4 ~

~- I + ~ ;

.+

C~ z-6~+

-r- p - ?

(~a) (3b)

1~_ -+ 9. C - ~ ) ' / ~ . ) / ~ "1" P.,*

(3c)

8)/o °

Od)

-e- ~0 ~?

where p * ) p-~ p ° (resp. 1/)~ ~)- ~1o )_are numbers o£ even (odd) ovals bounding the components of ~ \ ~ with positive, negative and zero Euler characteristic from the outside. Namely, let ~ P ~ , , ~ ) ~ be all the components of ~ , homeomorphic to a circle or a ring. It is clear that ~ = p ~ _ o Consider the sublattice T - ~ H ~ , generated by classes ~:-- ~ ( ~ p ~ ) ] ) ~_- ~ , , . ) ~ , According to [2~

;~£~Z= -2~ if q ~ O?

if

~

is a circle is a ring

(4)

101 For every component ~ of ~ F _ , let ~4~,,,~ ~ / denote all the adjoining conponents of ~ ~+ so that ~ ( ~ ) ~,.. ~< 2~ ( ~/j. If ~ ( T ~ ) >I ~ and ~ ~ then we connect ~ with ~ j ,.,~ ~ by some ( ~ - 4 ) disjoint smooth lines in ~ . The graph f- of adjacency relation of ~ q ~ ,. ,~ ~ and these lines consists of disjoint trees. If any tree contains ~ ~ terminal points, corresponding to some lines, then we'll remove (~-~3 of last lines. Let L ~ , • °~ U ~ denote all the retained lines. The Rokhlin construction provides classes ~+ ¢ ~ ' " '2 ~ f ~ ~-T E H + respectively. It is easy to show (see [24]), that

t~' ~~,odq), ~/3L~#Cs

(5)

In the case p - ~ O every tree in n contains exactly one terminal point corresponding to some line. Then (4), (5) imply

hence

zA'T

= t ----p + + p o

p+.p° ~ g-(H.)=3

(~

what is equivalent to (3b). If see that D = ~_~ ~-p+-equality

and also according to

2~)/8 p--p

(1)

+ p- ~, 0

then it is easy to . Using the Harnack in-

m+ p ~< g - - t - ' / we

obtain ~ ~ (@.¢)/2

= (,,,.~e- 3 , , ~

e')/~' -~- 3 # " ~

~.,'~J/,~

The inequalities (3a), (3c), (3d) can be got analogously. By means of all the above constructions, Viro 28 and author 24 gave complete classification of the degree 8 curves of type /-,~31. ~ < g ~.ll ~ .II "/ ~'~ .).,> . Probably, complete classification of all curves of degree 8 can be obtained by means of those constructions. Also I think that the important problem of real curves topology - the Ragsdale-Petrovsky conjecture (see

mmF_A.X-~ #, t4 J ~ 3 (t~ 'Z''- 2v~'/)/,g -/- "~

(6)

102 can be proved in limits of a given method. It should be mentioned that this approach was used in other cases: (i) Viro and Kharlamov prohibited some topological types of the degree 5 surfaces in ~ P 3 , (ii) Kharlamov proved non-exactness of the Harnack estimate for singularities (see sec. .3.1). 2. Constructions 2. I. Viro's method. The Viro method is a basic method of real algebraic curves construction at present [25, 27J. Its algorithm is simple and consists in the following. Given the set of real polynomials

with Newton polygons ~q~ '" " 2 ~ # . Assume that: (i) b e ~ is a convex polygon, (ii) the non-empty intersection ~h~ ~ z ~ e is a common vertex or a common edge, (iii) there is the convex continuous function ~: ~--~ ~ linear on each Z ~ and nonlinear on each union ~ U 2 ~ e (it is said to be a regular subdivision of Zk ), (iv) curves ~¢I ,° ,# ~--N are non-singular in ~ 2 \ ~Dc~ = O ~ , (v) the sections of each polynomial ~ on edges of ~ are non-degenerate (i.e. they haven't multiple factors different from % ~ ~ ). Any curve F ~ should be lifted from plane upon the toric manifold defined by polygon ~ o Let ~ be a union of Zh~ and polygons symmetric to ~ a with respect to axes and origin. Then ~ can be naturally projected from the toric manifold to ~ so that an intersection of ~ ~ with amy quadrant is projected to the corresponding polygon. This image is called the chart of polynomial ~ The basic statement ~25, 27S is as follows: the ch~rt of polynomial

A jxi f. with Newton polygon ~

"-'-,+0,

is a topological gluing of charts of

F/~, . . . ~ F,,,V The same method gives a construction of smoothings of singular points determined by their Newton diagram K25], Almost all the curves types discovered over the last ten years ~7, 14, 15, 23, 25, 27S were constructed by Viro method. It should be noted that constructions of Chevallier [5, 6] and Korchagin [13] keep within limits of Viro method.

103

2.2. ~e~ular trian_g~lations of Newton p~l~gon. Consider the particular case of Viro method. We'll call a real curve a triangular curve if it is defined by threenomial with non-degenerate Newton triangle. The chart of triangular curve is determined by si&-nes of its threenomial coefficients. Let /h be the Newton triangle of the degree ~Q curve ~ t h vertexes ( 0 ] 0 ~ ( ~

0~9

(O)~) . Then by means of Viro method we can assign to each couple (regular triangulation of A , function ~ ~ 2 _ _ ~ ~ {~ ) some isotopic type of non-singular curve of degree ~ . Further on the curve having such an isotopic type we'll call T-curve. Apparently, there are curves different from T-curves. At that time all the Harnkck curves are T-curves (see [5, 13]). Moreover, "almost all" non-singular curves are T-curves. We'll show that by the following reasoning of SoYu.Orevkov. Let us introduce affine coordinates in the space of curves of degree ~ , and then substitute them for logarithmic coordinates [25, 27] in each octant, Then in each octant a suitable polyhedron < ~ similar to Newton polyhedron of discriminant cuts out the discriminant hypersurface the part of chart corresponding to a given octant. According to [8 ] the components of complement to discriminant containing vertexes of polyhedron consist of T-curves. Let ~ mean the union of those components, then

vo.e (

D

/ roe

-->

if we'll e n l a r g e ~ infinitely. On the other side the class of T-curves is quite visible. I think the following conjectures are accessible combinatorical problems. Conjecture I. T-curves satisfy the inequality (6). C0n~ecture 2 (on cancellation). Any ovals arrangement obtained by elimination of one empty oval of T-curve can be realized by some curve of a given degree. That follows from C.pn~ecture 3. Any T-curve is isotopic to a suitable non-singular curve of a given degree, which is obtained by smoothing nodal curve having in ~ z one connected component and, possibly, isolated points. 2.3. Two modifications of the Viro method. The first modification enlarges the application range of the Viro method to smoo-

104

thing singular points. We call a singular point quasiordinary if (i) at this point the sections of Taylor series on the Newton diagram edges are non-degenerate, or (ii)this point will satisfy the above condition after a suitable local analytic diffeomorphism. For any singular point Z we put (i) ~ C ~ ) = ~ C a ) -- ~ , if ~ is zero-modal, (ii) ~ ( ~ ) is equal to the sum of orders of ~ and its infinitely near points in full resolution of ~ except nodes of complete inverse image of a given curve, if the modality of ~ is positive. Proposition L 1 ~22]. If F is irreducible and

Z

eca) <: 5 dc~ 7 ,

then it is possible to realize independently the smoothings of arbitrary quasiordinary points Z~ ~M~F isotopic to their prescribed Viro smoothings by means of small variation of coefficients of F , while all other singularities are retained. Application examples of this statement are constructions (a) o~ the 8th degree curve of type ~ ] ~ ~ ~22~, (b) of certain affine sextic curves [15], (c) of quartic curves with a prescribed position of inflexion points [I0~. The second modification allows to eliminate the condition (iv) from sec. 2.1 in certain cases. Take all notations of sec. 2.1 and introduce the oriented graph /- of adjoining the polygons ZAI),.~ / ~ N without cycles. Proposition 2 ~20]. If F f ~ . . . ) FA/ are irreducible in , and

where ~K runs through all edges of ~ K not corresponding to arcs of ~- coming in /~K Then there exists the curve F with Newton polygon /~ , whose chart is the gluing of charts of ~ .+.# ~ , and whose singularities collection in ~ 2 \ ~ S : O} is the union of sungularities collections of ~I# '''~ F~ in ~\ ~=O~. In particular, it is always possible to glue (even reducible) curves ~I#,..) F ~ with only nodes in This statement allows to construct real curves with prescribed sungularities collection. For example, there are curves of

105 degree ;";4 with arbitrary number of real cusps from zero to 3. Algebraic curves and smoothin6s of sin6ular points Let Z ~ ~ P Z be a singular point of a real curve F , the closed ball ~ ~ ~ p z centred at ~ be small enough, and a real curve C-

be sufficiently close to ~--- . Then the set ~ / ~

~

is

called a perturbation of the point ~ E F . An isotopy of the point Z perturbations means the equivariant isotopy in 3 \ { El}; where Z / E ~ D ~ p z \~ is a fixed point. An isotopy of the point ;z

smoothings (non-singular perturbations) means the conti-

nuous isotopy in ~ D ~ p ~ \ ~/~ 3. I. The classification problem for singular points perturbations is like the classification problem for algebraic curves. In fact, all methods of getting restrictions to topology of real curves can be extended to perturbations of singular points. Namely, we have for the non-singular curve ~ of degree ~ and the smoothing ~ of the singular point ~ with Milnor number ~ : (i) The Harnack inequality for ovals number

~

(~z_ 5~.

1 . (-0 ~) / ~

has local analogous one

-o- ~ ( , A -

~o +

(7)

[16]

0/~-,

where ~ o is a number of real local branches centred at more precisely

c8) ~

-~ ( j ~ - z * 3 ) / ~ ,

, or

(9)

where "7_. is a number of all the local branches centred at ~ and ~ is a number of connected components of union of ~ D

,

with real local branches links. (ii) The Petrovsky inequality for even

I~.z(RF±)-'II 4 (5~z_ ~-~)/~

(~o)

has local analogous one [3 _-1 -

A=~;

where

~•4'A-I ~ is a Hodge number on the space of vanishing echomology corresponding to the eigenvalue I = { of monodromy operat or. (iii) Congruences for M- and (M-1)-curves of even degree

106 correspond smoothings in (9))

to analogous local congruences (that means ~ = ~ - ~ - 3 ~ / ~

[12] for M- and (M-1)or ~ Z + f3/.~-

where ~ C ~ ) is the Brown invariant of some quadratic form ~z~ with the range ~ / ~ , which depends only on a singular point type. (iv) For complex orientations of the type I curves (i.e. ~'- \ ~ F is not connected) there are Rokhlin's formulae

~ .+ o--- ~r+ ÷ 2 0"7- 17 ~ ] = ( ~ ~- 0 / ~ ,

~ -- 4 { ~ J z . ) ~

where I']( i~ ~- ) is a number of negative (resp. positive) ovals pairs, and br( q~-* ) is a number of ovals equipped with negative (resp. positive) orientation with respect to one-sided branch. Por the type I smoothings there are analogous formulae

o- + ~. C n - - r 7 " )

o-

( ~ ' : ~ n f R p 2) -

+- . 7

~-÷

{-o--{~J-~+[eJJ

(~M~P

~) =

=

a

~.,

where ~ ru_us through all the non-closed components of ~ b , and 1)- -~ /Zt~ ~ means a number of ovals equipped with positive or negative orientation with respect to line /~i . Here the numbers c~/ ~ depend on the singular point Z type, a non-closed branches position and an orientation of ~ ~pa . However it should be noted that there is the considerable difference: the inequality (7) is exact, but (8), (9) are not exact. That was showed by Kharlamov for singularity ;:XLxf- ~ g "= O 3.2. The Viro method establishes an immediate correspondence between the set ~S ( E ) of isotopic types of the semi-quasihomogeneous singular point ~ smoothings and the set ~C ( ~ ) of isotopic types of curves with Newton polygon lying under the point Newton diagram, and ~fnose number of local real branches centred at non-proper straight line is equal to the number of real branches centred at ~ We'll say that 2 holds the property A, if i 5 { ~ ~- ~ ( ~ > , and holds the property B~ if r~ { ~ @

ls included into

-~C ( ~

"

Zero-modal singular points, an ordinary 4th order point, point of quadratic contact of 3 smooth branches [25, 27~, an or-

107 dinary 5th order point [19], a point of quadratic contact of 4 smooth branches E19, 217 hold the property A. ~n ordinary 6th order point holds the property B ~15]. In known cases L # # ~ is included into [ C ( ~ ) , although, probably, it is not in general. 4. Certain problems Besides Ragsdale-Petrovsky conjecture (6) and the conjecture on cancellation (see sec. 2.2) the problem of discovering new approaches to investigation of the odd degree curves is important and interesting, because here the results list is rather less then one ~or curves of even degree. For example, Korchagin [ 1 4 ] suggested conjectures

m

3 (v -¢gz/8

analogous to inequalities (10), (6). We conclude with the following Arnold problem: v~ether the discriminant ~) in the space of curves of a given degree is M-variety (i. e. ~ ~ j l~/~) ~

Re£erenc es I. A'Campo N.: Sur la le're partie du 16e probl@me de Hilbert. Seminaire Bourbaki, 31~ ann6e, 1978-79, no. 537. 2. Arnold V.I.: The arrangement of the ovals of real plane algebraic curves, involutions of four-dimensional manifolds and the arithmetic of integral quadratic forms. Function. Anal. Appl. 5, 169-175 (1972) 3. Arnold V.I.: The index of vector field singular point, Petrovsky-Olejnik inequalities and mixed Hodge structures. Punction. Anal. Appl. 12, 1-14 (1979) 4. Arnold V.I., Olejnik O.A.: Topology of real algebraic varieties. Vestnik MGU, ser. I, no. 6, 7-17 (1979) (Russian) 5. Chevallier B.: Sur le courbes maximales de Harnack. C.R. Acad. Sci. Paris, set. I, 300, no. 4, 109-114 (1985) 6. Chevallier B.: Courbes maximales et courbes reduites. C.R. Acad, Sci. Paris, ser. I, 301, no. 2, 35-40 (1985) 7. Chislenko Yu.S.: M-curves of tenth degree. In: Investigations on topology IV (Zap. nauch, semin. L O ~ , vol. 122), Nauka, Leningrad, 1982, pp. 147-162 (Russian) 8. Gel'fand I.M., Zelevinsky A.V., Kapranov M.M.: On discriminant of polynomials in several variables. Function. Anal. Pril. 24, no. I, I-4 (1990)

(Russian)

108

9. Gudkov D.A.: Topology of real projective algebraic varieties. Uspekhi math. nauk 29, no. 4, 3-79 (1974) (Russian) 10. Gudkov D.A.: Plane real projective quartic curves. In: Lect. Notes in Math., Springer, 1988, vol° 1346, pp. 341-347. 11. Kharlamov V°M. : Topologikal types of non-singular surfaces of degree 4 in ~ p 3 . Function. Anal. Appl. 10 (1977) 12. Kharlamov V.M., Viro 0.Ya. : Extensions of the Gudkov-Rokhlin congruence. In: Lect. Notes in Math., Springer, 1988, vol. 1346, pp.687-717. 13. Korchagin A.B.: On the singularities reduction and the classification of non-singular affine sextic curves. Deposit. in VINITI 14.02.86, no. 1107-B, 16p. (Russian) 14. Korchagin A.B.: New M-curves of degrees 8 and 9. Sov. Math. Doklady 39, no. 3, 569-572 ( ~ g ~ 9 ) 15. Korchagin A.B., Shustin E.I.: Affine curves of degree 6 and smoothings of non-degenerate 6th order singular point. Math. USSR Izvestiya 33, no. 16. Risler J.-J.: Inv. Math. 89, no. 1, 17. Rokhlin V.A.:

3, 501-520 (1989) Un analogue local du th~or~me de Harnack. 119-137 (1987) Proof of Gudkov's conjecture. Function.

Anal. Appl 6, 136-138 (1972) 18. Rokhlin V.A.: Complex topological characteristics of real algebraic curves. Uspekhi math. nauk 33, no. 5, 77-89 (1978) (Russian) 19. Shustin E.I.: The Hilbert-Rohn method and smoothings of real algebraic curves singular points. Sov. Math. Doklady 31, no. 2, 282-286 (1985) 20. Shustin E.I.: Gluing of singular algebraic curves. In: Methods of Qualitative Theory, Gorky Univ. Press, Gorky, 1985, pp. 116-128 (Russian) 21. Shustin E.I.: The independence of singular points smoothings and new M-curves of degree 8. Uspekhi math. nauk 40, no. 5, 212 (1985) (Russian) 22. Shustin E.I.: New M-curve of degree 8. }~ath. Zametki 42, no. 2, 180-186 (1987) (Russian) 23. Shustin E.I.: New M- and (M-1)-curves of degree 8. In: Lect. Notes in Math., Springer, 1988, vol. 1346, pp. 487-493. 24. Shustin E.I.: New restrictions to topology of real curves of degree dividing by 8. Izvestiya Acad. Nauk SSSR, ser. math.

109

54, no. 5, 1069-1089 (1990) (Russian) 25. Viro O.Ya.: Gluing of algebraic hypersurfaces, smoothing singularities and construction of curves. In: Proc. Leningrad Int. Topol. Conf., Nauka, Leningrad, 1983, pp. 149-197 (Russian) 26. Viro O.Ya.: Progress in topology of real algebraic varieties over the last 5 years. In: Proc. Intern. Congress of ~,~ath., Warszawa 1983, aug. 16-24, pp. 603-619 (Russian) 27. Viro 0.Ya.: Gluing of plane real algebraic curves and const~uctions of curves of degrees 6 and 7. In: Proc. Leningrad Intern. Topol. Conf., Leningrad 1982/ Lect. Notes in Math., Springer, 1984, vol. 1060, pp. 187-2OO. 28. Viro O.Ya.: Progress in topology of real algebraic varieties over the last 6 years. Uspekhi math. nauk 41, no. 3, 45-67 (1986) (Russian) 29. Wilson G.: Hilbert's sixteenth problem. Topology 17, no. 1, 53-73 (1978)

Moduli Real

Problems in Algebraic Geometry R. S I L H O L

Universitg Montpellier II URA CNRS 1407

0. I n t r o d u c t i o n .

To quote Haxtshorne, one can say that the guiding problem in algebraic geometry is the classification problem and typically such a problem falls into two parts, (i) a discrete part, that is, finding numerical invariants; (ii) a continuous lJart, this is the moduli problem. To rephrase this, a moduli problem, is the problem of finding a space that classifies up to isomorphism, all objects with a given set of numerical invaxiants (actually the distinction between (i) and (ii) is not always clear, even in classical algebraic geometry - - see for example Catanese [Ca] - - , and this is even more so in real algebraic geometry since a real algebraic space, even irreducible, may have many connected components). In the above informal definition some words axe voluntarly inprecise, in the classical case "space" means algebraic space (but not necessarily algebraic variety), "isomorphism" means isomorphism over an algebraically closed field and "numerical invariants" depend on the problem considered. For example one can consider the problem of constructing, .tv/g = {iso. classes of curves of genus g} Stated this way the problem is too general and does not have a solution, but one can build, .M/~ = {complex iso. classes of complex curves of genus g} and this space is a normal complex space and an irreducible quasi-projective variety. In real algebraic geometry, the corresponding problem is to build the space, Ad~ = {real iso. classes of real curves of genus g}. Typeset by ~4MS-TEX

111

This will be our guideline problem, but one can consider many other situations, vector bundles with given invaxiants over a given variety, varieties with given Hodge numbers, Chern numbers ... (see section 5 for comments and references). This change has many consequences. For example, as is noted in [Si2], one can not ask for the space to be algebraic but rather semi-algebraic (see [Seg~Si]) and in general only real analytic.

1. C o m p l e x m o d u l i v e r s u s real m o d u l i .

Historically moduli problems in real algebraic geometry axe quite old, and one can say that they originate with the famous 1876 paper of Haxnack, not only because his result can be interpreted as giving the lower bound g + 2 <

for the number of connected components of A/[~, but also because the paper of Harnack ends with explicit references to the moduli problem in terms of periods of abelian integrals on a real curve. But the story gets really started with F. Klein and G. Weichhold and, slightly later, A. Comessatti. Implicitly this first approach is to consider what one can formulate in modern terms as .M~:(R), (see below for a definition) the real part of the complex moduli space, rather than Ad~ (but for obvious reasons this is not always completely clear). This approach has beared many fruits, but before discussing one of these results, I would like to point out here the incovinences of this approach. To introduce this let me define .M~:(I;I) and introduce a notation. Let C be a complex curve of genus g, then the correspondence C F-. C ~, that to a curve assigns its complex conjugate, induces an anti-holomorphic involution on Ad[: and hence a real structure. Thus it makes sense to consider the set of real points of .M[:, which we denote by .ME(R ). Now an obvious relation between Ad~(R) and 3d~ is given by

the map that maps the real isomorphy class. Obviously sequel ~o as a map from 3d~ (1) ~o is not injective. For an defined by,

isomorphy class of a real curve to its complex Im(~o) C Ad~:(R) and we will consider in the to 34~(N). The problem is that, explicit example one can look at the two curves

y2 = l='(x) y2 = - P ( x )

112

where, P(x) is a real polynomial with no real roots. These two curves are complex isomorphic, under the isomorphism (x, y) ~-~ (x,iy) but one has real points while the other has empty real part, and hence the curves can not be real isomorphic. (2) ~ i8 not 3urjective. This came somewhat as a surprise, and I strongly believe that people at the time of Klein were not conscious of this problem. The first examples of this fact are due to C. Earle (1971) [Ea] and, independently, to G. Shimura (1972) [Sh]. The examples given by both authors are based on the same idea, the existence of curves C, with an antiholomorphic automorphism ¢ of order 4 but with no antiholomorphic involution. Such a curve will be isomorphic to its complex conjugate, by the antiholomorphic automorphism (and hence will define a point in Ad~:(R)), but will not be defined over R, since a necessary condition for this, is the existence of an antiholomorphic involution, and hence will not define a point in M ~ . The explicit example of this type given by G. Shimura is the following. Consider the curve defined by, m

y2 = aox m + ( E akxm+k + (--1)kakxm-k) k=l

where m is odd, a0 E R and ak E C \ R for h ~ 0. This curve is complex isomorphic to its complex conjugate via the isomorphism (x, y) ~ ( - x -1, ix-my). But if a0, the ak's and the ak's are chosen algebraically independent over Q, C will have Id and -Id: (x, y) H (x, -y)) (the hyperelliptic involution) as only holomorphic automorphism and there will be no antiholomorphic involution. (3) In my mind this is the worst objection, .h4~: is not of any use to study deformations. As an example for this problem one can consider the case of curves of genus 1 . . £ 4 ~ = C, via the j invariant, and Ad~=(R) = R. On the other hand real curves of genus 1 may have a real part consisting of 0, 1 or 2 components. A real elliptic curve E has one real component if j ( E ) < 1728 two (or zero) i f j ( E ) > 1728. The case j ( E ) = 1728 corresponds to the curves, y2 = x 3 q_ X

and

y2 = x 3 _ x

which have respectively one and two real components. But obviously it is impossible to deforme, while staying in the category of smooth real curves, a curve with one component into one with two. These remarks are quite general, the same type of objections appear in many other situations, and motivate specific constructions for real moduli spaces. People at the time of F. Klein were certainly conscious of (1) and (3), but not, as said above, of (2). F. Klein [K1] and G. Weichhold [We] were nevertheless able to give the exact number of components of .&4~,

2(3g + 3 ) i f g is odd ~(3g + 4) if g is even

113

A. Comessatti [Co] found the same type of results for .A~, the moduli space of principally polarized real abelian varieties of dimension n. 2. C o n s t r u c t i o n o f .h4~. Various constructions of .M~ have been proposed. The first, up to m y knowledge is due to C. Earle (1971) [Ea]. Other constructions are due to S. Natanzon (1980) [Na], B.H. Gross and J. Harris (1981) [Gro&Ha], M. Sepp'£1£ and - - (1986) [Se&Si] and possibly still others I do not know of. I do not have the time here to go into the details of these different constructions, but I will outline the basic ideas of the construction of Earle. First fix an orientation reversing involution a on a topological, compact, connected, orientable surface ~g of genus g. It has been known since Klein, that such an involution is characterised, up to conjugation by an homeomorphism, by two invariants: k the number of connected components of the fixed part, Y].~, of ~g under a; # (equal to 1 or 2) the number of components of ~g \ ~ (these invariants are the ones used by Klein and Weichhold to prove the result alluded to in section 1 ). Let ~g(a) be the set of complex structures on ~g that make a antiholomorphic. Now let Homeo ° and Homeo + be the group of homeomorphisms of ~g homotopic to zero (resp. orientation preserving), and let Homeo°a(a) (resp. Homeo+(a) ) be the subgroup of elements that commute with a (the normalizer). Then, the group Homeo+(a) acts on 2:g(a) by pull-back and

is the set of real isomorphy classes of curves of genus g such that, # ( C ( R ) ) = k, # ( C ( C ) \ C(R) -----/a. Moreover this space is connected, and has a natural structure of stratified analytic space (this is obtained by considering the space

7 ; = xg(o)l Ho, eo which is an equivalent of the Teichmfiller space in this case, where connectedness and the existence of a real analytic structure are easier to prove - 9"a, the ordinary Teichm/iller space, is a ball and ~Y~ is just the fix part of ~Yg under the action of a). From this it is easy to build the moduli space of real genus g curves by letting,

U

(g,k,~)

REMARK: Here one can discuss what the correct numerical invariants are. If one is only interested in smooth structures, then, of course, (g, k,/z) is the correct set. On the other hand if one considers the problem of compactifying the moduli space, then g appears to be the natural invariant one must consider.

114

3. A p p l i c a t i o n s o f R e a l M o d u l i S p a c e s . Many problems in real algebraic geometry can be formulated in terms of moduli. This does not always lead to a solution of the problems though, and often the reverse happens. But let me formulate a few. One important question in the theory of real algebraic surfaces is to determine the number of distinct possible topological types of a surface of degree n in P~I- A partial answer to this problem could be given in the following way: complex surfaces of degree n have specific numerical invariants (essentially its Hodge numbers and Chern numbers). Call/z this set of invariants (think of this as the equivalent of the genus of a curve). Let .A4~ be the moduli space of real surfaces such that a minimal complexification has invariants/z (this makes sense if the surfaces we are looking at are not ruled, which is the case if we start with a surface of degree > 4). Then knowing the number of connected components of M ~ will give bounds, possibly stronger than the ones we know today, for the number of distinct topological types. This would give in particular an answer to the Ragsdale-Petrovski-Viro conjecture for surfaces (see Viro [Vi]). In its simplest instance this conjecture can be formulated in the following way, let X be a smooth and projective real algebraic surface such that, X ( C ) (the complex part) is simply connected (for example X is a surface in p3 or more generally a complete intersection), then the conjecture states that dimHl(X(R), l/2) < dimHl'l(x) . By analogy with the situation of double coverings ramified along a curve, one can also formulate the following conjecture, # X ( R ) = dimH0(X(R)) < l ( d i m H l a ( X ) ) + 1 . In a completely different setting let me give an d priori not obvious application of the knowledge of moduli spaces. From the results of S. Novikov, T. Shiota and E. Arbarello and C. De Concini one knows that an abelian variety is the Jacobian of an algebraic curve if, and only if, one can construct from the Theta function associated to a realisation of the abelian variety a solution of the famous K.-P. equations (K.d.V. in t h e case of hyperelliptic curves). Some authors belive that if you start with a real curve then you obtain in this way a real solution to the K.-P. equations. As noted by B. Dubrovin and S. Natanzon [Du~Na] this turns out to be false, such a solution is real if and only if # = 2, i.e. the curve is a dividing curve. Let me indicate how this can be shown using real moduli spaces. Let C be a complex curve of genus g, let 71,... ,72~ be a basis of H i ( C , / ) and let w l , . . . , Wg be a basis of H°(C, ~21) the space of holomorphic differentials. The %

g × 2g matrix _ - - w i -) i,j

erao

o'

0oo

oorm l zo

(globally) the choices made, in such a way that the period matrix is of the form (~/, lg) with ~2 symmetric and C~m(~) positive definite.

115

In the real case one can normalize still further. One can normalize in such a way that 2~e(12) has interger coefficients (in fact a curve is real if and only if one can normalize in this way - - see for example [Se&Si]). Now let M be the reduction mod.2 of 2~e(f~). Then under the above normalization (to simplify we assume here that # ( C ( R ) ) ¢ ~), k = # ( C ( R ) ) = g + 1 - rank M and/~ = # ( C ( C ) \ C(R)) = 2 if the bilinear form defined by M is of type II (or even, or orthosymmetric, or ...) that is, such that

~nMn ~ 0(rood.2), for all n E Z g. Conversely # = # ( C ( C ) \ C(R)) = 1 if the bilinear form is of type I (or odd, or diasymmetric, or ...). The important point here, is that, by the characterisation of the connected components of the space .h4~t one can reformulate the above result by saying that M = 2~e(~2)mod.2 is constant, of given rank and type, on the connected components of A/t~. Now the Theta function expressed in terms of a given period matrix ~2 is defined by,

e(z,f~) = ~

exp(ritnf~n + Zritnz).

nEZa

From this expression it is easy to see that O(z, f~) is a real function in z if and only if 2~e(f~) is the matrix of a form of type II (one way is obvious, the other comes from the fact that the above expression is a developement in Fourier series of the Theta function). This proves the result of Dubrovin and Natanzon.

4. Compactifications of Moduli Spaces. Real moduli spaces are, as indicated in section 1, the correct setting for studying deformations. But since the study of deformations cannot, of course, be limited to smooth structures, One needs to enlarge somewhat the spaces and this leads to the compactification problem. There ave many ways to do this but there are a certain number of standard procedures. In the case of A//~, for example, the standard procedure is to compactify by, ~

= {complex iso. classes of complex stable genus g curves},

where an, eventually singular, curve C is said to be stable if, (i) the only singularities of C are ordinary quadratic or nodes (i.e. analytically isomorphic - - over C - - to xy = 0 at 0); (ii) Aut(C) is finite.

116

In the real case one can try to do the same and compactify .M~ by, ~t

= {real iso. classes of real stable genus g curves}.

This has been done by M. Sepp£1~ in [Se]. The main difficulty with this approach is that, whereas in the complex case there is only one type of ordinary quadaratic singularity, in the real case there are 3, (i) singularities of type xy = 0; (ii) singularities of type x z + y2 = (z + iy)(x - iv) = 0; (iii) a pair of complex conjugate nodes. The construction of ~ , given by M. Seppiil~, takes into account all three types of nodes. The same type of problem appear in compactifying other moduli spaces, in particular for the moduli space of principally polarized abe]Jan varieties (see [Siz] for an account). A s a n application of this construction, or rather of these constructions, I can give the following, (1) (this small application can actually be proved, in this case, in a more elem e n t a r y way but it can be generalized to situations where no such elementary proof exists - - i.e. not using the compactification of moduli spaces). Let, y2 = 4 x 3 _ p ( t ) x

- q(t)

be a family of el]jptic curves, non singular if t ¢ 0, with a singular, but semi-stable, fibre at t = 0. Then the fibre at t = 0 is of the form,

::Iiilliii]:: o (where - - is a real line and - - - a complex line and there are rn, even or odd, components) if and only if, lira ~ ,-.0

q(t)

> 0

(in such a situation limp(t) must be positive). On the other h a n d the fibre is of the form

117

m components.

if and only if, lira X / ~

,-0

q(O

< 0

1 (2) Let ~ = ~ M + i Y be the period matrix of a real curve normalized as in 3. Then a real genus 2 curve C has empty real part if and only if M 0 1 rood.2, and det Y >_ 4' This can be proved using the compactification of the moduli space because a curve with empty real part that degenerates to a semi-stable singular curve can only degenerate to one with isolated real points (see [Si:]).

5. O t h e r M o d u l i , a n d S o m e O p e n Q u e s t i o n s .

Apart from M ~ other moduli spaces in real algebraic geometry have been constructed, here are some, (1) Real abelian varieties (Comessatti [Co] and - - [Sil]); (2) Real K3 surfaces (Nikulin [Ni], see also [Sil]); (3) Real elliptic surfaces (Seller [Sei]);

(4) Some surfaces of general type (Balli¢o [BAD; (5) Stable rank r vector bundles over P2(R) (Ballico [BaD.

The first open problem is to build still more moduli spaces, especially in the cases when the corresponding complex moduli space has been constructed. Among these the most reachable seems to be the moduli space for so-called numerical quintics (i.e. surfaces such that a minimal smooth complexification has same numerical invariants as surfaces of degree 5 in p3 __ the complex moduli space for these surfaces has been constructed by Horikawa [Ho]). This

118

space is of special importance since it provides the first non-trivial case for the Ragsdale-Petrovski-Viro conjecture for surfaces (see section 3). The second open problem is to obtain a better understanding of the topology and the geometry of the moduli spaces already built and of their compactifications. In this direction some questions which do not have a complex equivalent appear, for example one can prove that the connected components of the moduli space of real K3 surfaces (there are 66 of these) are not all homeomorphic. The third problem is, for the moment, just a reformulation of a well known and well studied problem, the construction of the real moduli space for degree d hypersurfaces in pn. Of course such a problem will probably be solved in some special cases without a moduli problematic, but on the other hand a close analysis of the solutions found, up to now~ for curves of degree 7 and degree 8 and 9 in p2 (partial solutions in the last two cases) shows that these problems would benefit from a more global approach that can eventually be given by a formulation in terms of moduli, or in terms of deformations. The fourth problem, concerns compactifications. There is of course the problem of compactifying the moduli spaces already constructed but not yet compactified (e.g. K3 surfaces) but there is also the problem of studying other types of compactifications in the case when one has already been constructed. For example one can consider torodial compactifications of Jt~ (the real moduli space for principally polarized abelian varieties of dimension n) and study how such constructions can be generalized to other situations (K3 surfaces ? .... ?). Closely linked with this problem is the problem of studying real algebraic deformations which is also a theme of great interest. REFERENCES [Ba]. E. Ballico, Real moduli of complex objects: surfaces and bundles, preprint (1990). [Ca]. F. Catanese, Moduli of algebraic surfaces; in, "Theory of Moduli (E. Sernesi Ed.)," Springer Lecture Notes in Maths, Berlin Heidelberg New York, 1988, pp. 1-83. [Co]. A. Comessatti, Sulle variet~ abeliane reale t, Ann. Mat. Pura Appl. 2 (1924), 67-106; e II, 4 (1927-1928), 299-317. [Du&Na]. B.A. Dubrovin and S. Natanzon, Real Thela function solutions of the Kadom~sev Petviashvili equation, Math. USSR Izvestiya 32 (1989), 269-288. [Ea]. C. Earle, On moduli of closed Riemann surfaces with symmetries; in, "Advances in the Theory of Riemann Surfaces," Annals of Mathematics Studies 66, Priceton Univ. Press, Princeton New Jersey, 1971, pp. 119-130. [Gro&Ha]. B.H. Gross and J. Harris, Real algebraic curves;, Ann. scient. Ec. Norm. Sup. 14 (1981), 157-182. [Ha]. A. Harnack, Ober die Vieltheiligkeit algebraischen Kurven, Math. Ann. 10 (1876), 189-198. [Ho]. E. Horikawa, On deformations of quintic surfaces, Inventiones Math. 31 (1975), 43-85. [Elf. F. Klein, "Uber Riemanns Theorie der algbraischen Funktionen und ihrer Integrale. --Eine Erg£nzung der gewhhnlichenDarstellungen.," B.G. Teubner, Leipzig, 1882. [Na]. S. Natanzon, Moduli spaces of real curves, Trans. MoscowMath. Soc. (1980), 233-272. [Ni]. V.V. Nikulin, Involutions of integral quadratic forms and their applications to real algebraic geometry, Math; USSR Izvestiya 22 (1984), 99-172. [Sei]. W. Seiler, Global moduli for polarized elliptic surfaces, Compositio Math. 62 (1987), 187-213. [Se]. M. Sepp/ilg, Moduli space of stable real algebraic curves; preprint 1990, to appear in Ann. Set. E.N.S..

119

[Se&Si]. M. Sepp/~l£ and R. Silhol, Moduli spaces for real algebraic curves and real abelian varieties, Math. Z. 201 (1989), 151-165. [Sh]. G. Shimura, On the field of rationality for an abelian variety, Nagoya Math. J. 45

(1972), 16T-178. [Sil]. R. Silhol, "Real Algebraic Surfaces," Springer Lecture Notes in Maths. 1392, Berlin Heidelberg New York, 1989. [Si2]. R. Silhol, Compactifieations of Moduli spaces in real algebraic geometry, to appear in Inventiones MaLh.. [Vi]. O. Viro, Curves of degree 7, curves of degree 8, and the Ragsdale conjecture, Soviet Math. Dokl. 22 (1980), 566-570. [We]. G. Weichhold, "~lber symmetrische Riemannsche Fl£chen und die PeriodizitKtsmodulen der zugehfrigen Abelschen Normalintegrale erster Gattung," Leipziger Dissertation, 1883.

Universit~ Montpellier II, D~pt. de Math~matiques, Place E. Bataillon 34095 Montpellier Cedex 5 France

Constructing Strange Real Algebraic Sets by S. Akbulut* and H. King** Recall that algebraic sets are characterized by resolution tower structures [AK1], [AK2]. A topological space with such a structure is called a T R space for short. In low dimensions existence of resolution tower structures can often be detected by combinatorial invariants of the underlying topological spaces. For example, let X be a locally cone-like Euler stratified space (i.e., a locally conelike stratified set so that the Euler characteristics of the links of all strata are even). Then for every component of the codimension one stratum U of X we define ~ x ( U ) to be the number of points in the link of U. For every component of the codimension two stratum W of X we define a i ( W ) to be the number of vertices v of the link L of W so that ~L(V) = i rood 8, where i = 0 , . . . , 7. Then define

co(W) el(W) ,~(W) e3(W)

= = = =

Oto(W) mod 2 a6(W) mod 2 ( ~ 0 ( W ) + ~ 4 ( W ) ) / 2 rood 2 (a2(W) + a6(W))/2 mod 2

and define e x ( W ) = ( e o ( W ) , e l ( W ) , e 2 ( W ) , e 3 ( W ) ) E Z 4. It is an easy exercise to show that e x ( W ) is well defined, i.e., a0 + o~4 , and a2 + a6 are even. For every component V of the codimension three stratum of X and every e E Z 4 we define fl~(V) E Z2 to be the number of vertices v of the link L of V with eL(V) --- e. T h e o r e m . ( [ A K 2 ] ) For any compact topologieM space X , the following are equivalent: a) X is homeomorphic to a reed algebraic set of dimension <_ 3. b) X is a T R space of dimension <_ 3. c) X is a locadly cone-Eke Euler stratit~ed space of dimension <_ 3 and for all codimension three s~rata x of X we have:

fllllO ('~) ~0100 (~) fllO00 (X) ~ . o o (~)

"[- fl1111 (X) + /~0~0~(~) -~ ,~100t (.T) + ~.o~ (~)

= = = =

0

0 0 o

Hence whether or not a stratified space of dimension less then four is homeomorphic to an algebraic set can be decided by purely combinatorial data. In [AK1], [AK2] a similar characterization, with homeomorphism replaced by a stratified set isomorphism, is also given. (One needs all fl~'s to be 0 and if all strata have trivial normal bundle then this suffices.) This gives some surprising examples of real algebraic sets. For example, in [CK] this was used to give an example of a real algebraic set which is topologically imbedded in * Supported in part by the N.S.F. ** Supported in part by the Graduate Research Board, University of Maryland

121

R 4 but cannot be an algebraic subset of R 4. T h e purpose of this paper is to indicate a construction of such algebraic sets which, for these special cases, is simpler than the general construction given in [ A K 2 ] . It falls short of giving explicit equations, but comes much closer to an explicit description. The construction

of the Example

Let us now proceed with the construction, which follows the spirit, but differs quite a bit f r o m the letter, of t h a t in [ A K 2 ] . We shall construct below a compact three dimensional manifold M with b o u n d a r y and closed subsets A, B0 and B1 so that: 1) A is a spine of M , in fact there is a diffeomorphism h: OM x (0, 1] --+ M - A so that h(x, 1) = x for all x E OM and h - l ( B i ) = (OM N Bi) x (0, 1]. 2) B0 and B1 are disjoint. 3) If you tal¢e OM, crush OM 91 Bo to a point and then crush OM 91 B1 to a point, you obtain a two dimensional sphere. 4) Everything can be made algebraic. In particular, there is a c o m p a c t nonsingular three dimensional real algebraic set V C R n, algebraic subsets C, Do and D1 and an imbedding g: M --* V so that C = g( A ), Bi = g-a( Di) and Do 91 D1 = ~. By Corollary 2.5.14 of [ A K 2 ] we m a y as well assume that the V in 4) is projectively closed (i.e., it is Zariskl closed in projective space), since any c o m p a c t real algebraic set is isomorphic t o a projectively closed algebraic set. Alternatively, using the fact we see below t h a t A and Bi are unions of codimension one submanifolds in general position, this is a consequence of T h e o r e m 2.10 of [ A K 4 ] . So there is an overt polynomial r: R n -* R with V = r - l ( 0 ) . Overtness means that if r has degree d and rd is the sum of the monomials of r of degree d, then r ~ l ( 0 ) = {0}. Pick polynomials qi: R n ~ R so t h a t qi(x) >_ 0 for all x and so q~-l(O) = C U Di. For example, qi could be the s u m of squares of generators of the ideal of polynomials vanishing on C U Di. Let q = q0 + ql and pick an integer e bigger t h a n the degrees of q0 and ql. Note that C = q-1(0). Let a: V --* l:t n+2 be the m a p

or(x) = ( qo(x)ql(x)x , qo(x) , q,(x) ) and let Z C R n+2 be the algebraic set

z = {(y, t0,ti) I y = 0 and tot1 = 0 } . We claim t h a t a ( V ) U Z is a real algebraic set and near 0 it has strange behavior which was remarked u p o n in [ C K ] , see their question (Ld). In particular, the line segment { y = 0 , t0 = 0 , tl < 0} is open i n a ( V ) O Z b u t { y = 0 , t0 = 0 , tl > 0} is contained i n a part of a ( V ) U Z which is a topologicM manifold. See Figure 1, which is a fairly truthful representation. In particular, a ( V ) lies over the first quadrant and Z - a ( V ) consists of two half lines. To see our claim that a ( V ) U Z is a real algebraic set, let r~: R '~+2 --* R and q~: R n+2 -+ R be the polynomials

,"(u, to,

= (tot ) r(v/tot

)

122

to

tl

f F i g u r e 1: a ( V ) U Z

q~(y, to, tl ) = (tot1) e ( tl - qi(y /totl ) ) Let W be t h e algebraic set W --- r ' - l ( 0 ) N q~-l(0) n q'1-1(0). T h e n c e r t a i n l y a ( V ) C W a n d Z C W . B u t in fact a ( V ) U Z = W. To see this, pick a n y (y, to,tl) E W. If tot1 = 0 then 0 = # ( y , to,tl) = rd(y) so y = 0 by overtness of r a n d hence (y, to,tl) C Z. B u t if tot1 # 0 t h e n s e t t i n g x = y/(totl) we have r(x) = 0 so x E V a n d (y,to,tl) = a(x). Now let us investigate t h e t o p o l o g y of W n e a r 0. We t h i n k of W as t h e quotient space of V U Z v i a t h e m a p a . S u p p o s e a ( x ) = a(x') for s o m e x , x ' E V. I f x • C U D o U D 1 t h e n q 0 ( x ) q l ( x ) # 0. B u t = and q0( )ql( )x = = so x = x ' . If x E Di - C t h e n qi(x) = 0 a n d ql-i(x) 7~ 0 so qi(x') = 0 a n d ql-~(x') ~ 0 so x' e Di - C also and q(x) = q(x'). If x E C t h e n qo(x) = ql(x) -- 0 so qo(x') = ql(x') = 0 a n d x t C C also. So t h e m a p a crushes C to a p o i n t a n d m a p s Di to t h e half line y = O,

ti = O, t l - i :> O. For small enough e > 0 we have a h o m e o m o r p h i s m f : q-l(e) × (0, e] ---* q - l ( ( 0 , e]) so t h a t q f ( x , t ) = t a n d f - l ( D i ) -- ( D i n q-l(e)) x (0, e]. T h i s follows from Thorn's first isotopy l e m m a , a l t h o u g h in fact C a n d Di are unions of c o d i m e n s i o n one s u b m a n i f o l d s in general p o s i t i o n so if one w a n t e d , one could t a k e f to b e a diffeomorphism. A n y w a y we have a n i n d u c e d h o m e o m o r p h i s m i f : a ( q - l ( e ) ) × (0, e] ~ a ( q - l ( ( 0 , el)) w h e r e f f ( a ( x ) , t) = a ( f ( x , t)). T h u s n e a r 0, a ( V ) is t h e cone on a(q -1 (e)). C o n s e q u e n t l y n e a r 0, W is the cone on the disjoint union of a(q-~(e)) a n d two points. B u t a(q-l(e)) is o b t a i n e d f r o m q-l(e) b y crushing q-~(e) V~Do to one p o i n t a n d q-~(e) VI D1 to another. But this space must be the two sphere. T h e reason is t h a t the two p r o d u c t s t r u c t u r e s f a n d gh give an invertible c o b o r d i s m from t h e triple ( q - l ( e ) , q-~ (e)gtDo, q-1 (e)ND~ ) to (OM, OMV1Bo, OMVIB1). But b e c a u s e t h e d i m e n s i o n s are so small t h e c o b o r d i s m is trivial. So t h e r e is a h o m e o m o r p h i s m b e t w e e n t h e two triples. This algebraic set W has t h e following curious p r o p e r t y . Consider one of the lines in Z , say y = 0, to = 0. T h e n if t 1 ' ( 0 it iS locally o p e n in W . B u t if t l > 0, t h e n W is a t h r e e d i m e n s i o n a l topological m a n i f o l d nearby. T h u s it has the curious b e h a v i o r r e m a r k e d u p o n in [ C K ] . One could m o d i f y this e x a m p l e a bit to o b t a i n t h e specific space m e n t i o n e d in [ C K ] . Let W ' C R ~+a b e t h e union of W x 0 a n d t h e p l a n e {(y, to,tl,t2) I Y -- 0 a n d to = 0}. T h e n n e a r 0, W ~ is t h e cone on t h e disjoint union of a p o i n t a n d t h e wedge of a 2 - s p h e r e a n d a circle. To o b t a i n a real algebraic set h o m e o m o r p h i c to t h e s u s p e n s i o n of this, one

123

can use the Lemma below. L e m m a . Let X be a read Mgebraic set and pick xo E X . It is well known that xo has a neighborhood in X homeomorphic to the cone on some compactum L. Then there is a r e d algebraic set Y homeomorphic to the suspension of L. Proof." We may suppose X C P*-" and x0 = O. For small enough e > 0 we know that X n B( is homeomorphic to the cone on X fl OB~ where B~ is the ball around 0 of radius e and OBe is its boundary sphere. We then take Y = { ( x , t ) e X x R [ t~ + [xI2 -- ~ }" Clearly Y is the suspension of X N OB¢. But L and X n (.3B~ are invertibly cobordant (see [K]) so their products with l:t are homeomorphic and hence their suspensions are homeomorphic also. • T h e c o n s t r u c t i o n o f M , A a n d t h e Bi's We promised above the construction of the manifold M and its subsets A and Bi. We start out with part of the boundary of M. Let E be a two dimensional disc with three handles attached as in Figure 2. For convenience we do not smooth out corners in the figures although in reality they are smooth. Let El, E~ and E3 be the curves shown in Figure 2. Note that the quotient space E / ( E 1 U E2 U E3) is a disc. For another description, note that E is obtained from a disc by blowing up three times and the E~'s ave the strict transforms of the exceptional divisors.

Figure 2: E Next we consider the manifold F -- E x [0, 1]. We show it in Figure 3, although it does not quite imbed in 1~3. In F we have an immersed surface H = (El UE2 UE3) × [0,1]. The manifold F will be part of M and H will be part of B0. Our next step is to add a handle to F as in Figure 4. In particular, letting J = [-1,1], we pick disjoint imbeddings Ti: J x J --* E x 1 for i = - 1 , 1 so that T - l ( 0 , 0 ) is the point (El n E3) x 1 and ~1(0,0) is the point (E2 N E3) x 1, so that ~CI(E3 x 1) = 0 x J, ~11(E2 × 1) = J x 0 and ~_-~(E1 x 1) = J × 0. Furthermore the imbeddings are oriented

124

Figure 3: F as shown by R 3 as we do fori= -l, l This G' will

the arrows in Figure 3. This gives the handle a twist if we try to imbed it in in Figure 4. We let F ' = F U J x J × Y where we identify ( i , s , t ) with ~oi(s,t) and(s,t) E YxJ. Welet H'= HUYxOxYUJ xJxOand G'=OxJxY. be part of A.

Figure 4: F ' Note t h a t the the u p p e r p a r t of the b o u n d a r y of H ' , i.e., t h a t part which is not in E x 0, consists of two disjoint circles and each of these circles has nontrivial normal bundle in OF' and intersects G' transversely in two points. Our next step is to a t t a c h a thickened cylinder to these two circles. In particular, let K be a Moebius b a n d with central circle K ' and let K " be two transverse arcs as in Figure 5. T h e n we can pick disjoint imbeddings 8i: K --* OF' for i = - 1 , 1 so t h a t 8~-1(H ') = g ' a n d 871(G ') -- K " . We now let F " -- F ' U J × K where we identify ( i , x ) w i t h S i ( x ) f o r i = - l , l a n d x E K . We let H " = H ' U J × K ' and G" = G ' U J x K " U O × K . Our construction has the following property. There is a diffeomorphism f : E x (0, 1] F " - G" so t h a t I ( E x 1) = E x 0 a n d / ( ( E l U E2 U E3) x (0, 1]) = H " - G". For example,

125

I~iiiiii[iiiiii!~iliiii}iiiiiiiiii!!iiiiiiiiil

iii!iii!iiiii::i!ili~!i!~:i!il Figure 5: K

f can be obtained by integrating a vector field tangent to H " and to OF" - E x 0 and pointing inward on E × 0. We now let M be obtained by doubling F " along OF~_ = CI(OF" - E × 0). In other words, M = F " x {0, 1}UOF.~ x [0, 1]. We let A = G" x {0, 1}UOG" x [0, 1] and Bi = H " x i. Making M, A and the Bi's algebraic In order to facilitate making M, A and the Bi's algebraic, we modify this example slightly by blowing up F " with center the double point arc of H " . After doing so, G" and H " become unions of proper codimension one submanifolds of F " in general position. (Before doing this, H " was only immersed, not imbedded.) After doubling F " along aF_~ to obtain M , we see that A and the Bi's are still unions of proper codimension one submanifolds in general position. We may now double M along its boundary to obtain a closed smooth manifold M ~. Inside M ~ are B~, the doubles of Bi and A ~ which is a copy of A in one of the copies of M in M q Then A ~ and the B}'s are unions of codimension one submanifolds in general position. Hence by Corollary 2.8.10 of [AK2], or more directly by Theorem 2.10 of [AK4], we may assume that M ~ is a nonsingular real algebraic set and A' and the B~'s are algebraic subsets of M q Consequently we have made M , A and the Bi's algebraic as in 4) above. However, if one tried to construct explicit polynomial equations for M by following the construction in [AK2] or [AK4], many steps would be done which are not necessary for this particular example. Thus we indicate here some simplifications. First, it is only necessary to make F ' , G" and H " algebraic and then an explicit construction allows us to make M, A and the Bi's algebraic. In particular, suppose we have a three dimensional real algebraic set X and algebraic subsets W, Y and Z of X and an imbedding I : F " ~ X so that f - ] ( y ) = G", f - l ( Z ) = H " and f ( g " n G " ) = W C y n z . It makes the construction easier if we also assume that Y and Z are finite unions of nonsingular real algebraic sets, or more generally that they are almost nonsingular in the sense of [AKb]. We may suppose by Theorem 2.5.13 of [AK2] that W is projectively closed. Pick an overt polynomial p : X --~ R so that W = p - l ( 0 ) and suppose that p has degree d. Let X ' = { (x,t) E X x R I p(x) 2 + t 2d = e2d } for a small e > 0. Note that X ' is projectively closed. Topologically, X ~ is the double of a small neighborhood of W in X. LetY~=X IAY×R,Z ~=X ~NZ×RandW t=W×e. Since G" N H " is a spine of F ' , i.e., F ' - G ' N H " = OF" × (0,1] we may as well assume that the image of the imbedding f lies in a small neighborhood of W, in particular that f ( F ' ) C p - l ( [ - e d , ca]). We may then

126 lift t h e i m b e d d i n g I to f ' : F " ~ X ' by setting f ' ( x ) = ( f ( x ) , + 2~/e2d - p f ( x ) 2 ). Let r: X --+ P,. a n d s: X --* R b e the sums of squares of g e n e r a t o r s of the ideal of p o l y n o m i a l s vanishing on Y a n d Z respectively. We now blow up X ~ v i a t h e ideal (r, s, t + e). In o t h e r words let X " b e t h e Zariski closure of { (x,t, [u,v,w]) e X ' x R P 2 I t 7~ - e , wr(x) = u ( t + e ) , ws(x) = v(t+e) a n d u s ( x ) = v r ( x ) }. Let Y " , Z " a n d W " b e t h e strict t r a n s f o r m s of Y~, Z t a n d W ~. T h e n if Y a n d Z are almost n o n s i n g u l a r we have W " = Y ' C I Z ' . If they are not a l m o s t nonsingular, one m u s t do a more careful blowing u p to have this. A g a i n we m a y lift t h e i m b e d d i n g f f to X ' . So we m a y as well replace X , Y, Z and W by X ' , Y", Z" and W ' . W h a t we have gained is t h a t we now have X p r o j e c t i v e l y closed a n d also

YNZ=W. So Z = p - l ( 0 ) for some overt p o l y n o m i a l p. Say p has degree d. (Of course p is different from t h e p in t h e preceeding p a r a g r a p h . ) T h e n for small e > 0 let V = { (x, t) E X x R I P(X) 2 + t2d = e2d } and C = V VI Y x R , Do = Z x ( - e ) a n d D1 = Z x e. T h e n V is p r o j e c t i v e l y closed a n d satisfies 4) above. T h e p o i n t is t h a t F " - H " is diffeomorphic to the p r o d u c t OF"+x (0, 1] v i a a diffeomorphism preserving G ' . B u t for small e > 0, t h e set p-1 ([_ed, 0) U (0, ed]) is also a p r o d u c t with (0, 1]. C o n s e q u e n t l y b e c a u s e t h e dimensions are so low t h a t invertible c o b o r d i s m s are trivial, we know c~F~ i m b e d s in p-l(=Eed) in a w a y which takes OG" to p-l(=l=ed) Cl Y. C o n s e q u e n t l y we see F" d o u b l e d along OFJ~ in V. B u t this is j u s t M . A c e r t a i n a m o u n t of s t a r i n g at F i g u r e 4 convinces one t h a t F " can b e i m b e d d e d in 1%4. This is also seen from the fact t h a t G ' N H " is a spine of F ' , i.e., F ' - G " N H " = OF" x (0, 1]. So F " is j u s t a thickening of G" n H" which is the curve in F i g u r e 6.

F i g u r e 6: G" N H " If we h a d modified F " by blowing up t h e d o u b l e points of H ' , t h e n G" ig H " would b e a little different, t h e double p o i n t of the figure eight would b e r e p l a c e d b y a circle, b u t F " would still i m b e d in R 4. One could then m a k e F ' , G" a n d H " a l g e b r a i c in the following explicit way. Note G" VI H " is t h r e e circles intersecting in a c e r t a i n way, so it is easy to c o n s t r u c t a p r o j e c t i v e l y closed real algebraic set W in 1%4 h o m e o m o r p h i c to G " VI H ' . In fact we m a y let W be t h e union of t h e t h r e e circles x = w = 0, y2 + z 2 = 1 a n d y = w = 0, x 2 + z 2 = 1 a n d z = w = 0, x 2 + 2 y 2 = 1. I m b e d F " in 1%4 so t h a t Wis G'NH'. Let U b e a small n e i g h b o r h o o d of F " i n R 4 a n d let f l : U - - * R P 4 b e d m a p which is t r a n s v e r s e to R P a so t h a t f l - l ( R P a ) = F ' . For e x a m p l e , t h e r e s t r i c t i o n of fl to F " could b e t h e m a p which takes a p o i n t to t h e line p e r p e n d i c u l a r to F " at t h a t point. E x t e n d t h i s m a p to U using the fact t h a t the n o r m a l b u n d l e of 1%p3 in R P 4 is the canonical line b u n d l e over R P ~. Suppose H " a n d G" are m a d e u p of k c o d i m e n s i o n one

127

submanifolds C 1 , . . , Ck, say H" Ui=l ci and G" Ui=,n+l k Ci. Let 7i: U --~ R P 4 for i = 1 , . . . , k be maps transverse to R P a so that F " fl 7~-I(RP a) = Ci. For example 7i could send a point in Ci to the line tangent to F " and perpendicular to Ci. By doing the constructions carefully one may make sure that the restriction of/3 and each ")'i to g is a rational function. Now approximate/3 and the 71's by rational functions using Lemma 2.4 of [AK4] or Theorem 2.8.3 of [AK2]. One then obtains a real algebraic set V (in some higher dimensional affine space), an open set 0 C V, a diffeomorphism qo: 0 --* U and rational functions r: W ~ R and qi: W ---r t t whose restrictions to O approximate/3 o ~o and 7i o q0. Furthermore V contains W and r and qi coincide with /3 and "/i on W. We may then set X = / 3 - 1 ( R p 3 ) , Y = X fl U/k=m+l ")'~-a(Rp3) and Z = X f'l uim_l 7~-X(RPB). By transversality we know that near W; X, Y and Z look like F", G" and H " . Then by the construction we did above we are done. Note that a variation on this example provides a real algebraic set homeomorphic to the suspension of the disjoint union of a point and the connected sum of three RP2's. The only difference is that we replace ~ with the map .

-----

ra

=

o~(x) = ( qo(x)x, qo(x), q1(x) ) and replace Z by the algebraic set z = { (y, c0, t l ) I y = 0 and to = 0 }.

[AK1] S. Akbulut and H. King, The topological classification of 3-dimensional algebraic sets, M.S.R.I. preprint 12912-85 (1985). [AK2] S. Akbulut and H. King, Topology of real algebraic sets, M.S.I~.I. Publications Vol. 25, Springer-Verlag (1992). [AK3] S. Akbulut and H. King, The topology of real algebraic sets, L'Enseignement Math., 29 (1983), 221-261. [AK4] S. Akbulut and H. King, The topology of real algebraic sets with isolated singularities, Annals of math., Vol. 113 (1981), 425-446. [AK5] S. Akbulut and H. King, On approximating submanifolds by algebraic sets and a solution to the Nash conjecture, Inventiones Math., Vol. 107 (1992), 87-98. [CK] M. Coste and K. Kurdyka, On the link of a stratum in a real algebraic set, Topology, Vol. (1991). [K] H. King, Real analytic germs and their varieties at isolated singularities, Inv. Math., Vol. 37 (1976), 193-199. [S] D. Sullivan, Invariants of analytic spaces, Proc. Liverpool Singularities I, Lecture Notes 192, Springer-Verlag (1971), 165-168.

MORE

ON BASIC SEMIALGEBRAIC

SETS

Carlos Andradas*

Departarnento de Atgebra Jesfis M. Ruiz*

Departamento de Geometr[a y Topolog[a Facultad de Ciencias Matemdticas, Universidad Complutense de Madrid 28040 Madrid, Spain

Introduction Let X C I t " be a real algebraic variety, and let T'(X) denote the ring of polynomial functions on X. Recall that a subset S C X is called seraialgebraic if there exist polynomials f ij, gi C 79(X) such that

S = U :p{x E X : fil(x) > O,...,flr,(X) > O, gi(x) = 0}. i----1

As is well known, if S is open the gi's in this expression can be omitted. Recall also that an open semiaigebraic set is called basic open if furthermore p = 1. These basic open sets have attracted a lot of interest in recent times, see :Br2,3!, [Sch], [Mh], [AnBrRzl], culminating in the beautiful theorem that a basic open set S always has a description of the form

S = {x C X : f , ( x ) > 0,..., f , ( x ) > O} where s < dim(X). On the other hand the problem of characterizing basic open sets in geometric terms seems to be wide open. First of all, there is an immediate necessary condition for a semialgebraic open set S to be basic; namely, that S does not meet the Zariski closure of its usual boundary. If this is the case, a theorem of BrScker says that S is basic if and only if its intersection S N Y with any irreducible subvariety Y of X is generically basic, which means that S n Y \ Z is basic for some proper subvariety Z C Y. This brings the problem to the birational setting, or in other words to the fields K:(Y) of rational functions of the Y's. Then using the so-called tilda operation, which relates semialgebraic subsets of Y and constructible subsets of the space of orderings of the field K:(Y), the theory of fans can be used to decide whether S N Y is or not generically basic. For the details of all of this we refer to [Br3] or the forthcoming book [AnBrRz2]. The purpose of this paper is to improve the theorem of BrScker quoted above in the following way:

* PartiM1y supported by a grant Del Amo, Unlversidad Complutense de Madrid, 1990, and DGICYT, PB 89-0379-C02-02

129

1. Let S C X be an open semialgebraic set which does not meet the Zariski closure of its boundary. Then S is basic if and only if S fq Y is generically basic for any irreducible algebradc surface Y C X . Theorem

This theorem shows that the obstructions to basicness appear in the smallest possible dimension, because as is well known everything is basic in dimension 1. T h e result will follow f r o m a more precise statement characterizing generically basic sets t h r o u g h their intersections with irreducible surfaces. Of course the main tool in the proofs is the theory of fans mentioned above, and more specifically, the Theorem 1.2 below: an approximation theorem which shows t h a t for the m a t t e r we are discussing it is enough to consider fans built up from real prime divisors. Since we have not even said w h a t a fan is, let us at least describe a fan built up from a real prime divisor. First we start with a prime divisor V of the field K:(X), t h a t is, V is a discrete rank one valuation ring o f / C ( X ) whose residue field k is a finitely generated extension of I t of transcendence degree d - 1 where d = d i m ( X ) . Let t be a uniformizer of V. T h e n every element f E V can be written in the form f = ut n, where n is the value of f and u is a unit of V, so that its residue class ~ in k is not zero. Now, we pick two orderings ~'1 and r2 in k. (In particular, we are assuming that k is formally real, and that is why V is called a real prime divisor.) It is easy to llft rl and r2 to get four orderings al,a3 and a2,a4 in V, and so in the field/C(X). Namely, for an f = utn as above, put a l ( f ) = 7"1(~),

an(f) = (-1)nrl(~)

a2(f) = r2(~),

a4(f) = (-1)nv2(~)

(for an ordering c~ and an element x we let a ( x ) denote the sign of x in a). In other words, the units of V have the sign of their residue classes, and the uniformizer is declared either positive or negative. It follows from the construction t h a t a4 = al • a2 • a3, which is in fact the definition of a 4-element fan F = {or1 , a2, an, a4 } (orderings are multiplied as signatures). Now we can formulate rigorously our vague assertion that the tilda operation makes the connection between our initial problem and fans. To start with, given a semialgebraic set S C X we consider the set S C Specr(IC(X)) defined by any formula t h a t also defines S. T h e n we will prove: 2. Let X C I t " be an irreducible algebraic variety, and S C X a semiMgebraic set. Then S is generically basic if and only if # ( S fl F ) 7~ 3 for every 4-element fan F built up from a t e a / p r i m e divisor.

Theorem

This improves the following theorem of BrScker:

130

T h e o r e m 3. Let X C It" be an irreducible algebraic variety, and S C X a semialgebraic

set. Then S is generically basic if and only if # ( S f3 F) ~ 3 for every 4-element fan F. W h a t is new in our statement is the very special type of fans which are enough to check generic basicness. To give some hint of how this helps in our special problem, suppose S is not generically basic. Then, by the if part of Theorem 2, there are a real prime divisor V of ]C(X), two orderings rl and r2 in its residue field k, and four Iiftings a l , a 3 and a2,cr4, such that a l , a a , a 2 E S and a4 ¢ S.Here is the clue for the proof of Theorem 1: by a suitable aplication of Bertini's theorem we can decrease the transcendence degree of the residue field k, till in the end we get transcendence degree 1. Hence k is the function field of a real algebraic curve, and we get a homomorphism Y --* k[[t]] (where t is a uniformizer of V). After some work with M. Artin's approximation theorem we may suppose the homomorphism is actually into the ring of algebraic power series Y --~ k[[t]]~g. Furthermore, we can suppose that V contains the ring T~(X) of polynomial functions on X , and so get a homomorphism :P(X) --* k[[t]]~ag. This already gives the surface we were looking for: it is the zero set of the kernel of the homomorphism above. Finally, the four orderings ai we had in V extend easily to k[[t]], then restrict to n(p). Thus we obtain a 4-element fan in n(p) = / C ( Y ) which shows that the intersection S f3 Y is not generically basic (only if part of Theorem 2). The paper is organized as follows. Section 1 contains all results on fans in function fields needed in Section 2 to prove Theorems 1 and 2. We finish in Section 3 with a counterexample showing why our result fails in both the Nash and the analytic categories. This work started while the authors where participating in the Special Year 1990-91 on Quadratic Forms and Real Algebraic Geometry organized by Profs. T.Y. Lain and R. Robson in the University of California, Berkeley. We are glad to thank the RAGSQUADers, who listened to our first presentation of these ideas. We also want to thank Profs. J. Bochnak, M. Coste and P. Milman, who helped us to understand the Nash and analytic cases; in particular, we completed the construction of the counterexample which ends the paper during the conference in La Turballe, using a surface found by E. Bierstone and P. Milman.

131

1. T h e a p p r o x i m a t i o n

theorem

for fans

Let K be a field and V a valuation ring of K with residue field k. Consider an ordering a of K . We say t h a t a and V are compatible if the maximal ideal m y of V is convex with respect to a. T h e n a induces in the obvious way an ordering r in k, and we write a --* r. Now consider a fan F = {al, a2, a3, a4 } of K , that is, a set of four orderings of K such that the p r o d u c t of three of t h e m gives always the fourth. We say t h a t F and V are compatible if every al is compatible with V; we say that F trivializes along V if the ai's induce in k no more t h a n two distinct orderings. T h e n the situation looks like ffl

~3

\/

if2

~4

\/

T h e key result concerning the relation between fans and valuations is the following ([Brl]):

Theorem

1.1. Let F be a fan of the t~eld K. Then F trivializes along a valuation ring

of K . Henceforth we will think of a 4-element fan as a tuple (al, a2, a3,0"4) rather t h a n a set {al, a2, a3, a4 }. This way fans become points in the p r o d u c t E of four copies of the space of orderings Specr(K). Since ~ carries naturally the p r o d u c t topology of the Harrison topology, we can make statements in terms of that topology. Formally we should identify points in ~ up to permutations, but such a formalism is not needed for our purposes. From now on we suppose that K is a function field over R , and denote by n its

dimension, t h a t is, its transcendence degree over R. We recall t h a t V is eLprime divisor if it is a discrete rank one valuation ring of K and its residue field k is a function field over 1~ of dimension n - 1. W i t h this terminology, we have:

Theorem

1.2. The set of fans of K which trivialize along prime divisors is dense in the

set of all fans. Proof: Let X be a compact model of K , that is, a compact irreducible algebraic variety X C R n whose field of rational functions is K . T h e n every polynomial is b o u n d e d on X , from which it follows that every real valuation ring of K contains the ring P ( X ) of polynomial functions of X . Let F = (al,a2,a3,a4) E E b e a f a n o f K and U = U I × U2 × U 3 x U 4 an open neighborhood of F , with Ui = {fil > 0 , . . . , fir, > 0}, fij E 7v(Z); after shrinking the Ui's, we m a y assume they are pairwise disjoint. By T h e o r e m 1.1 F trivializes along a valuation ring V of K : the ai's are compatible with V and induce two orderings in the

132

residue field k of V according to the picture 0.1

0.3

0.2

%/

0.4

%/

(possibly vl = v:). Now we apply resolution of singularities so that after finitely many blowings-up X is non-singular and all the fij's are normal crossings. Let p C ~ ( X ) be the center of V in 7~(X). Then A = 7~(X)p is a local regular ring of dimension say d, and there exists a regular system of parameters x l , . . . , xd such that for all i,j

fij

=

~ux7 ' i ' ' ' "

~,i,,

where the uij are units of A and the a~jk are non-negative integers. In this situation the residue field n(p) of A is a subfield of the residue field k of V, and rl, r2 restrict to two orderings in n(p) that we still denote T1, v2. Then notice that the signs of the elements flj in the ordering ap are completely determined by the signs of the parameters xt in ap and the signs of the units (or more properly of their residue classes) in vq, where p = q mod2. Reordering the parameters we may suppose that al(xt) = aa(xt) for I < r and el (zt) # aa(xt) for l > r. Note that r < d since otherwise al and aa would coincide over the f l j ' s , which is imposible because these elements define a neighborhood of 0.1 which does not contain aa. On the other hand, it might very well happen that r = 0. Recalling that 0.1 "0.2 "0.3 "0"4 ~- 1, we see that also 0.2(xt) = 0.4(xt) for all l < r and a2(xt) ~ 0.4(x/) for all / > r. Consider the diagram A = T ' ( X ) p --~ B = A [ x ~ , . . . , x , - , x , - + l / x d , . . . ,Xd--1/Xd, Xd](xl . . . . . . . . Xd) -'~ W = B(x,~ ) C K

l

l

i

Bt(xd)

c

kw

1 /¢(~)

~

~' = K , ( p ) [ X r + I l X d , . . .

,Xd--1/Xd]

Let us analyse the ingredients of this figure. First, the local ring B is regular of dimension r + 1 with regular system of parameters { x l , . . . , x r , xd}, and dominates A. Furthermore, its residue field is generated over the residue field of A, which is g(p), by the residue classes x r + l / x a , . . . , Xd-1/xd, which are algebraically independent over to(p). Second, the ring W is a discrete rank one valuation ring of K , and it is in fact a prime divisor. Indeed, its residue field k w is by construction a finitely generated extension of R. Moreover, k w is the quotient field of B/(xd), which is a regular local ring of dimension r and whose residue field k t is that of B, and has transcendence degree d - 1 - r over x(p). As the latter has transcendence degree e = d i m ( T ' ( X ) / p ) over R., we see that the transcendence degree of k w over R is > r + ( d l-r) +e = d+e1. But

133

d + e = ht(p) + d i m ( P ( X ) / p ) = d i m ( P ( X ) ) = n, and we find that k w is a function field over l:t of dimension n - 1. Third, we have the following coincidences of signs:

O'I(Xl) ~---o'3(Xl),l < r;

0"l(.TI/Xd) = o'3(xt/xd),r < l < d;

O'l(Xd) # 6r3(Xd)

o'~(zdxd) = o,,(:~dx,d,," < t < d;

o'~(Xd) # ",,(=d)

and since al • a2 " cr3 • a4 = 1,

o'~(zt) = o'4(xt), t _< ,-;

Finally, taking all of this into account, we can extend and lift orderings through the preceding diagram, starting from t~(p) with ri, i = 1, 2, as follows: • Extend ri to an ordering r~ in k' such that r ' ( x t / x a ) = ai(xt/Xd) for r < l < d. This extension is possible since k' is a purely transcendental extension of ~(p) with { x r + l / x r t , . . . , x a - 1 / x a } as a transcendence basis. • Lift r ' to an ordering 7i in k w such that 71(xt) = a i ( x t ) for 1 < l < r. This lifting follows easily after remarking that the completion of B / ( x a ) is a ring of formal power series k'[[g:l,..., g:r]], where { £ 1 , . . . , xr} is the regular system of parameters of B / ( X d ) induced by { x l , . . . , x ~ } . • Lift 7i to two orderings al' and ~'/+2 of I¢ such that o)(=d) = ~A~d). Trds can be done as before using the completion kw[[xd]] of W = B(=d) , or as was described in the introduction for a discrete rank one valuation ring like W. We claim that the four orderings of K just constructed form a 4-element fan F ' which trivializes along the prime divisor W, and that F ' is in the neighborhood U of F fixed at the beginning. Indeed, they form a fan because they are built up from the real prime divisor W, as explained in the introduction, which implies also that F ' trivializes along W. Furthermore, ~ ( u / ) = ~p(ui), ~(=t) = ~p(~) for all p, i, I, and so

for all i , j . Hence a~ E Up for allp, and F' = (a~,a~,a~,a~) E U.

II

134

2. C h a r a c t e r i z a t i o n s

o f basic s e t s

We start by deducing Theorem 2 of the introduction as an immediate consequence of Theorem 1.2: P r o o f of Theorem 2: It is immediate that in case S is generically basic, # ( S N F ) ~ 3 for any fan F, built up or not from a real prime divisor. Conversely, by Theorem 3, if S is not generically basic, there is a fan F such that #(;~ A F ) = 3, and the problem is to replace F by another fan F ' , built up from a real prime divisor. To do it, let hj be the polynomial functions involved in a fixed description of S, and a l , a2, a3, 0"4 the orderings of F. Then consider the neighborhood Ui = {fil > 0 , . . . , fir > 0} of ai where flj = hj if hj(gi) > O, fij = - h i if hj(ai) < 0. Then U = U1 x U2 x (]3 × U4 is a neighborhood of F in the sense of Theorem 1.2, and there is a fan F ' in U which trivializes along a prime divisor, which t t means that F ' = ( a l , a 2 , a3, ' a4) ' is built up from that real prime divisor. By construction the signs of a~ and ai at hj coincide, and so a~ E S if and only if ai E 2~. Consequently, # ( S N F ' ) = # ( S I"1F ) = 3, and we are done. I

Now we are ready to state and prove the characterization of generically basic sets through surfaces:

T h e o r e m 2.1. Let X C R " be an irreducible algebraic variety and S C X a semialgebraic set. Let Z C X be an algebraic set ~ X containing both the singular locus of X and the boundary of S, OS = S \ S °. Then S is generically basic if and only if S M Y is generically basic for any irreducible surface Y C X not contained in Z.

Proof: Assume first that S is generically basic, and let Y C X be an irreducible surface not contained in Z; denote by p the ideal of Y. Since Y is not contained in the singular locus of X the localization P ( X ) p is a regular local ring, and there is a discrete valuation ring V of the quotient field L:(X) of 7~(X) dominating P ( X ) p whose residue field is k = q f ( P ( X ) / p ) = 1C(Y). Now suppose that S N Y is not generically basic. Then there is a fan (rl,v2,r3,v4) of k such that rl,r2,v3 E S and r4 ~ S. Using V one easily lifts this fan to another (al,a2,aa,a4) of K:(X), with a~ -~ rl, and we claim that al,o'2,a3 E S and o'4 ~ S. First, since Z does not contain Y and Z contains OS, ~'i E S \ Z C S ° and consequently al E S, for i = 1,2,3. Second, since r4 E Y \ Z U S , we see that v4 ~ S and a4 ~ S. Whence, S is not generically basic, as claimed. Note that for this implication we do not need fans trivializing along prime divisors. Now assume S is not generically basic. By Theorem 1.2 there is a 4-element fan F of the field K = K:(X) with # ( S N F ) = 3, and F trivializes along a prime divisor V of K; say F m {ffl,ff2, O'3,O'4}, O*1,O"3 ---+ T 1 and a2,a4 --* r2, where r~ and v2 are two orderings in the residue field k of V. Substituting X by its one-point compactification we can assume that

135

X is compact, and so V contains the ring P ( X ) of polynomial functions of X. The residue field k of V is a function field over R: there is an irreducible algebraic variety W C R m whose field of rational functions KJ(W) is k, or in other words, k is the quotient field of the ring 79(W) of polynomial functions on W. Now let H stand for a generic irreducible hyperplane section of W, p for the ideal of H in 79(W) and k ~ for the field of rational functions on H; note that k' is the residue field of p, that is, the quotient field of 79(W)/p. With alI of this we have the following diagram p(x)

c v

~ ,f," =

Jc(w)I[t]] u

T'(W),[[t]] -~/C(H)[[t]] where t is a uniformizer of V and the homomorphism qa is the obvious extension of the canonical mapping 79(W)p -~ KJ(H). Since the ring P ( X ) is an algebra finitely generated over R we can pick finitely many generators fi E 79(X) and add to them the equations involved in a description of the semialgebraic set S and an equation of Z. All these functions fi are in V, and so they have, in its completion r~" =/C(W)[[t]], power expansions fi = fi(t) = ~ t ( g i t / h i l ) t t, where gil, hit E 7:~(W). As our hyperplane section H is generic, we can suppose that no gil, hit vanishes on H (although there are infinitely many gig hit's, their number is countable, and working over the reals we can use Baire's theorem). In particular hit ¢ P implies that the fi(t)'s are well defined elements of 7~(W)p[[t]]. Finally, since the f i ' s generate 79(X) we get ¢(79(X)) C 79(W)p[[t]l and consequently we have the m a p ¢ =

p(x) -. P(H)[[G

Moreover git ~ P implies that the coefficients of the fi(t)'s are units in 7)(W)p and so ¢ ( f i ) = ~,(fi(t)) = ~ t ( g i t / h l t ) t t is a non-zero element of/C(H)[[t]] (here - - stands for residue class mod p). Now we complete the choice of the generic hyperplane section H . We have the two orderings a l , a a in V =/C(W)[[t]], liftings of the same ordering rl in/C(W). It is easy to find an open neighborhood G1 of rl such that for any ordering v~ E G1 its two liftings a 1', rr3' have at the fi's the same signs as a l , a3, where the liftings are chosen so that a~(t) = ai(t). Analogously, we find a neigborhood G2 of r2. This implies that for any two orderings r~ ~ G1 and r~ E G2 the fan F ' built up from them and the prime divisor V verifies # ( S fl F ' ) = 3. Now, G1, G2 will also denote two open semialgebraic subsets of W corresponding to the neigborhoods just constructed. These semialgebraic sets are Zariski dense in W, which guarantees that we can choose the generic hyperplane section H to meet both of them. This implies that there are r~ E G1 and r~ E G2 which make the ideal p of H convex. In other words, r~ and v~ induce two orderings 7t and 72 in the residue field of p, which is KJ(H). Then we lift 71 and 72 to four orderings gl,g3 and g2,6~ which form a fan F " of KJ(H)[[t]] with the conditions gi(t) = ai(t).

136

After this preparation, we have the following diagram P(X)

/ ~(w)p[p]] ~

..L

p(w)p

--~

\ ~(p)[p]]

l

~(p )

~

o~ ~

\/

T~

~

\/

~

, ~ ,

~3 ~

\/

~1

~4

\/

~2

Now consider the kernel q of the homomorphism ¢. Its zero set is an algebraic set Y C X with "P(Y) = T ' ( X ) / q , and dim(Y) = dim(X) - ht(q). Furthermore, the fan F " = {~1, ~2,/~3, ~4} induces a fan F* in/C(Y) such that # ( S A F*) - 3, because by construction the signs of the ~i's at the ~b(f/)'s coincide with those of the cr~'s. Consequently, the semialgebraic set S N Y is not generically basic. Furthermore, since among the f~'s there is an equation of Z, and no fi is in p, Y is not contained in Z. Hence it only remains to show that we can impose the further condition dim(Y) < dim(X) and from that the proof will end by induction. Now consider any other homomorphism ¢ ' : P ( X ) ~ ~(q)[[t]] which approximates ¢ to an order ~,, that is, such that ¢ ' ( f ) _--- ~b(f) m o d t v for every f e P(X). If we look at the requirements that !b : "P(X) --* n(q)[[t]] fulfils, it is clear that for u large enough ¢~ will fulfil them too. Then consider the ideal n = ¢ - I ( t ) and the localization A = P ( X ) , . The homomorphism ¢ extends to the henselization A h, which is a quotient of a ring n ( n ) [ [ x l , . . . , Xr]]atg of algebraic power series. Using this remark and M. Artin's approximation theorem we can arbitrarily approximate ¢ by ¢ ' : A h --~ n(q)[[t]]~g (see [Tg] III §5). Hence, substituting ¢ by ¢ ' we may merely suppose ¢ ( T ' ( X ) ) C t~(q)[[t]]~ag. It follows that ¢ induces an embedding T'(Y) ¢--* ~(q)[[t]]aig, which extends to the quotient fields K:(Y) ~ n(q)((t))~g. Counting transcendence degrees over R we find dim(Y) --- tr.deg.[/C(Y): l:t] < tr.deg.[n(q)((t))~lg : R] ---1 q- tr.deg.[~(q) : R] = 1 + d i m ( H ) < 1 + dim(W) = dim(X) as wanted.

|

We finish this section with the following:

Proof of Theorem 1: It is clear that if S is basic open any intersection S n Y with an irreducible subvariety Y C X is also basic, and so generically basic. Conversely, suppose S is not basic open. Then by Br6cker's theorem there is an irreducible subvariety X t C X such that S fl X ~ is not generically basic. Then, by Theorem 2.1 there is an irreducible surface Y C X t such that S n Y is not generically basic, and we are done. |

137

3. T h e N a s h a n d t h e a n a l y t i c c a s e s The approach followed here to study semialgebraic sets works the same in the Nash and the analytic categories (with some compactness assumption in the latter) [Rz],[AnBrRzl]. Even the approximation theorem for fans proved in Section 1 could be proved in both the Nash and the analytic cases. However any attempt to prove Theorem 2.1 or Theorem 1 would fail. The obstruction is that no Bertini type theorem is available in those cases. An easy counterexample is given by the standard torus T in R3: a lot of hyperplane sections of T consist of two circles, and so cannot be irreducible in either the Nash or the analytic category. More difficult is to produce a counterexample to Theorem 1 itself. Here we present one based on a surface used by Bierstone and Milman in dealing with arc-analytic functions, IBM] Ex.l.2(3). E x a m p l e . Let RP3 be the reM projective 3-space with homogeneous coordinates (x0 : xl : x2 : x3), and Y0 C RPa the surface XoX 1 2 2 = x~ + x~. This surface has two singular points, namely a = (1 : 0 : 0 : 0) and b = (0 : 1 : 0 : 0), and Y0 \ {a,b} consists of two connected components whose closures Y1 and ]I2 meet exactly at a and b. Moreover, Y0 is irreducible from the analytic viewpoint, but has the property that any analytic arc through one singular point has to be completely contained in either Y1 or Y2- This implies that any irreducible analytic curve Z C Yo has to be completely contained in either Y1 or

Y2. After these preliminaries consider the semiaigebraic set S C RPa defined as follows. First S will be a subset of the a ~ n e space R a = {x0 ~ 0} C RP3, in which we take coordinates x = x l / x o , y = x 2 / x o , z = x3/xo. Then Y0 N R a is the surface of equation x2 = y4 + z ~, and we have the sets: •

B1 = bail centered at (1, O, 1) E II1 with a very small radius.

•

B 1+ = B 1 [7 {322 ~__ V4 -1- z 4 ) ,

•

B2 = ball centered at ( - 1 , 0, 1) E Y2 with a very small radius.

• B+ = B ~ n { z ~ > v ' + z ' } ,

B 1 = B 1 [7 {X 2 ~_~ V4 "q- Z4) •

B K = B 2 n { z ~
8,-

g

% Figure 1

Now put S = B1 U B + (see F i g u r e 1 ). We claim that this set is not generically basic, but its intersection with any surface is generically basic, which gives the announced counterexample.

138

T h e assertion that S is not generically basic follows easily by constructing a 4-element fan F = {o'1,o'2,0"3,0r4} where at e B+,a2 e B+2 ,cr3 e B : , a 4 e B ~ . Now suppose t h a t there is a surface Y C R P 3 such t h a t S M Y is not generically basic. T h e n there is a fan F = {al, a2, a3, cra} in the field K of meromorphic functions on Y such t h a t a l , c~2,a3 E 5' and a4 ¢ S. As S = B1 U B + and B I , R P a \ B1, B2, B + are basic, we necessarily have, up to reordering, al, a3 E B1, a2 E B +, a4 E B~'. Moreover, there is a valuation ring V of K along which F trivializes, so that al, a3 -4 vl and a2, a4 ~ v2 where vl, ~'2 are orderings in the residue field h of V (it cannot be a l , a2 ~ rl, because this would imply ~'a is adherent to b o t h B1 and B2, which is impossible if our choice of the radii was small enough). Now, since Y is compact, V contains the ring (.9(Y) of analytic functions on Y; let p stand for the center of V in O ( Y ) and Z for the zero set of p. T h e field L of meromorphic functions on Z is the residue field of p and is contained in h. There are two possibilities: * ht(p) = 2. T h e n Z is a point z and the four orderings in the fan specialize to it. Since aa E B1 and a2 E B2 the point z is adherent to b o t h Bi's, which is impossible. • ht(p) = 1. T h e n Z is an irreducible curve and r2 restricts to an ordering in the field of meromorphic functions of Z, ordering still denoted by v2. Since a2 E B +, a4 E B ~ and a2,a4 -* r2 we conclude r2 C B + M B 2 C Y0. It follows that Z C Y0, and, Z being an irreducible curve, the special choice of Y0 guarantees that Z is completely contained in either Y1 or Y2, say Y1. Since cr2 E B2 and a2 ~ r2, this v2 is adherent to B2, and for a small enough choice of the radius of B2 this implies that v2 is not adherent to Y1However, r2 E Z C Y1, a contradiction. T h e argument is analogous if Z C Y2. Whence, S n Y is generically basic for any surface Y C RP3.

I

This example shows that Theorems 1 and 2.1 fail in the Nash axld analytic cases. However it is very easy to find an irreducible algebraic surface Y such that S M Y is not generically basic from the algebraic viewpoint. Namely, take any plane Y : y = ~, with e very smM1. T h e intersection of such a plane with our distinguished surface Y0 is the curve y --- ~, x 2 = z 4 + e2: It is irreducible from the algebraic viewpoint, and gives rise to a 4-element fan in the plane Y which shows t h a t S C1 Y is not generically basic. Of course the key point here is t h a t such an algebraic fan is not anMytic, in the sense that it does not extend from the field of rational functions on Y to either the field of meromorphic functions or t h a t of meromorphic Nash functions. This m a t t e r will be discussed in detail in the forthcoming paper [AnRz], and is the first step to understand the connection between the algebraic and the analytic categories in the real case.

139

References

[AnBrRzl] C. Andradas, L. BrScker, J.M. Ruiz: Minimal generation of basic open semianalytic sets, Invent. math. 92 (1988) 409-430. [AnBrRz2] C. Andradas, L. BrScker, J.M. Ruiz: Real algebra and analytic geometry, to appear. [AnRz] C. Andradas, J.M. Ruiz: Analytic fans versus algebraic fans, in preparation. IBM] E. Bierstone, P. Milman: Arc-analytic functions, Invent. math. 101 (1990) 411-424. [BCR] J. Bochnak, M. Coste, M.-F. Roy: G6om6trie alg~brique r~elle, Ergeb. Math. 1~, Springer-Verlag, Berlin-Heidelberg-New York, 1987. [Brl] L. BrScker: Characterization of fans and hereditarily pythagorean fields, Math. Z. 151 (1976) 149-163. [Br2] L. BrScker: Minimale Erzeugung von Positivbereichen, Geom. Dedicata 16 (1984) 335-350. [Br3] L. BrScker: On basic semialgebraic sets, to appear in Exposit. Math. [Mh] L. Mah~: Une d6mostration ~l~mentaire du th~or~me de BrScker-Scheiderer, C. R. Acad. Sci. Paris 309, Serle I (1989) 613-616. [Rz] J.M. Ruiz: On the real spectrum of a ring of global analytic functions, PubL Inst. Recherche Math. Rennes 4 (1986) 84-95. [Sch] C. Scheiderer: Stability index of real varieties, Invent. math. 97 (1989) 467-483. [Tg] J.-C. Tougeron: Id6aux de fonctions diff~rentiables, Ergeb. Math. 71, Springer-Verlag, Berlin-Heidelberg-New York, 1972.

MIRROR

PROPERTY

FOR

NONSINGULAR

MIXED

C O N F I G U R A T I O N S O F ONE LINE AND K POINTS IN R 3

Alberto Borobia* Dpto. Matem~ticas Facultad de Ciencias, U.N.E.D. 28040 Madrid, SPAIN ABSTRAC: Given h,k~O, M(h,k) denotes the set of nonsingular mixed configurations

of h lines and k points in ~3 We will say that f6M(h,k) is mirror if it is isotopic to its mirror image in any plane. The following problem has been proposed by Viro and Drobotukhina [3]: given h,k>_O, does there exist some mirror configuration on M(h,k)? A satisfactory answer is given when h=l: fcM(1,k) is mirror if and only if k<3.

O. Introduction By a nonsingular

mixed configuration of lines and points in tR3 we understand a

collection of lines and points in general position in ~3 this means that tile points not lie on tile lines, that no lille of ~3 contains 3 of tile points, and that no plane of ~3 contains 4 of the points or one of the lines and 2 of the points or 2 of tile lines. Given h,k>O, M(h,k) will denote the topological space whose points are the nonsingular mixed configurations of h lines and k points in R3. The topology induced by ~3 in M(h,k) is the obvious one. We will call each connected component of M(h,k) a camera. An isotopy will be a continuous function y:[0,1]----~M(h,k), that is, y is a path joining two points that lie in tile same camera. Therefore, if f, g~M(h,k) lie in tile same camera we will say that f and g are isotopic.

f~M(h,k) is said to he mirror or to have the mirror proper O, if f is isotopic to its mirror image in any plane. Otherwise f is said to be nonmirror. Letf~M(h,k) be a mirror configuration. Given any reflection o in a plane of ~3 consider an isotopy T:[0,1]--~M(h,k) such that T(0)=o(]) and T(1)=f. The composition of o with T acts over f permuting its h lines and ils k points. Any permutation of the lines and the points of f obtained in this way will be called a mirror permutation o f f Given f6M(h,k), if "f'~M(h',k') is composed of h'
141 Let f c M(h,k) be a mirror configuration, ¢p a mirror permutation of f and f'~ M(h',k') a subconfiguration of f. If cp(f')=f" then it is easy to see that f ' is mirror and that ~p'=q0Jf, is a mirror permutation of f'. In particular the configurations of M(h,O) and M(0,k) composed respectively of the lines and the points o f f are mirror. Let h,k>.>.Oand s:M(h,k)---~. We will say that s is preserved under isotopies if it is constant in each camera. And that s is antisymmetric if s(o(/))=-s(D when off) is a mirror inaage of f. Therefore, if f~M(h,k) is mirror and s:M(h,k)-------~ is preserved under isotopies and antisymmetric then s(J)=0. In this work I am principally concerned about the following problem: given k>0, does there exist some mirror configuration on M(l,k)? I will prove that fcM(l,k) is mirror if and only if/c~3. A big part of this paper is contained in Viro's paper [2]. Namely: linking coefficients were introduced into the subject; it contains the construction that I will employ in order to define adjacent points in f~M(l,k); part of tile results of lemmas 2 and 5 are inmediate consecuence of following Viro's lemma: M(O,k) does not contain mirror configurations if k'~=6(mod8) or if k'~=3(mod4) and k>7; moreover, the arguments that I use here on the proof of lemma 2 and on part (iii) of the proof of lemma 5 are slight modifications of Viro's arguments in order to show that lemma. The study of configurations in general position was introduced by O. Ya. Viro in [2] under the influence of V. M. Kharlanlov's account of their conection with the problem of classifying nonsingular real algebraic surfaces of degree 4 (see [1]). The mirror property can be used to prove in an elementary way lllal certain of these surfaces are not amphicheiral. Other questions related to configurations in general position can be found in [3], and in their references. I would like to thank Professor O. Ya. Viro for introducing me to this problem, and Professor H. R. Morton and I. Itemberg for his helpful comments. This work was done while the author was visiting the University of Liverpool during July and August of 1991. I thank very much the Department of Mathemathics for ils hospitality.

1. Mirror configurations in M(I,k) Lemma 1.-f~M(l,k) is mirror if k=0,1,2 or 3. Proof: (i) It is easy to see that if k<2 then M(l,k) has only one camera, hence any

fcM(l,k) with k_<2 is mirror. (ii) Given f=Jl;pl,p2,p3]cM(1,3), consider the plane rtr that contains to PI' P2 and p:¢ in this plane draw the lines /i2, /13 and /23, where /ij denotes the line containing the points Pi and pj. Tile plane rtf is divided in tilree differellt kinds of regions (denoted by different letters in figure 1). It is easy to see tllat f, gcM(l,3) are isotopic if and only if the line of g inlersects the same kind of region on rt that the line of f on ~. From this we deduce that all the g

configurations of M(I,3) are mirror

f

142

[l,12}6M(2,0 ) and {ll,l ~_} the same lines endowed linking coefficient lk(l'~,l*) equal to +t or -1 depending Let

has

with some orientations;

l*l,2.1*l

if it is isotopic (by means of

an isotopy consisting of configuralions of M(2,0) with oriented lines) to those lines in figure 2 or figure 3 respectively. The sign of the linking coefficient changes if we reverse the orientation of one of the lines, or if we consider a mirror image of

111,121.* *

I

Let f=[ll,l,I]~M(3,0 ) (we will call to any configuration of M(3,0) a uiple). The linking coefficient of f, Ik(/), is defined as follows. Let [It,l,l 3}_ be the same lines endowed with some orientations, then lk(])-Ik(l*l*).lk(l*l*).lk(l*l*). It is well defined I' 2 l I ..I z ~ a since it does not depend on the orientations chosen. The function lk:M(3,0)

>11,-11 is

preserved under isotopies and antisynametric in M(3,0). Lemnm 2.-f¢M(l,k) Proof: Let k

f~M(l,k)

is nonmirrorif

k-4, 5, 6 or 7(rood8).

with k>4. Define sq)~Z to be the sum of the linking coefficients of

k-2

the (~)'(2)/2 triples that consist of the line of f and 2 lines determined by two disjoint pairs of points of f. Clearly, s is preserved under isotopies and antisymmetric in M(l,k). If k=4, 5, 6 or 7(modS) then s(]) is an odd number (since it is a sum of an odd number of 5:1's) and therefore f is nonmirror

Let f=ll;pl,p2,..,pk}~M(1,k )

with k>3. Consider tile bundle of halfplanes S I with

boundary 1. As f is in general position, then any halfplane of S I can contain at most one point of f. Let S t denote the halfplane of S 1 that contains Pi" We will say that pi and pj are

adjacent in f

if some component of ~3kl Sik.)Sj } does not contain points of f. Construct a

bijection ~ between the k points of f and the vertices of a regular polygon Pk of k edges such that adjacent points in f correspond to adjacent vertices of P (an example with k=6 is given in figure 4).

143

>M(I,k) be any isotopy with ~/(0)=f. Then Pl and pj are adjacent in f if and only if their images are adjacents in y(1). Suppose the opposite. Then, as T is a continuous transformation, for some t~[0,1] T(t) would be a configuration that is not in general position. In the same way, if o" is any reflection in a plane of ~3 then p and On the other hand, let y:[0,1]

q are adjacents in f if and only if o(p) and o(q) are adjacents in a(j).

,0,~

YIr','l,vi

vs 3

3.

J~qttce /-t

In what follows we will suppose that f=ll;p,,..,pkleM(1,k ) with k>8 is mirror and q~ is the mirror permutation of f obtained from the composition of one reflection cr with one isotopy T:[O,l]---oM(h,k) such that T(0)=o'(]) and T(l)=f. From tile preceding paragraph we deduce that q) transforms adjacent points into adjacent points. Then q~ induces (by means of ~) the action over Pk of some ~:~D2k. where D2k denote the dihedral group of 2k elements. Let C k denote the cyclic group of k elements, then Lemma 3.- xc CkCD2k. Proof: Suppose xc D2k~Ck. Any non-orieutable element o f D2k acts on Pk as a reflection; then x acts on the set of vertices of Pk as a permutation with 0 or 2 vertices fixed if k is

even, or 1 vertex fixed if k is odd. The remaining vertices of P are divided into disjoint transpositions. As k>8 we have more than 2 transpositions. Let f" be composed of / and 4 points of f whose images by ~ are the vertices corresponding to two transpositions. Then q)(f')=f' and f'~M(l,4) would be mirror. But this is not possible by lemma 2 Corollary 4.- If the lhw I is endou'ed with some orientation then (p reverses it. Proof:

Construct one loop q3 passing through all the points of f and such that any

halfplane of S l intersects ~ in one single point. We can extend the action of cr and ~' over lt.~ in such a way that this properties will be preserved in the images of the loop during the isotopy, and that if we denote by r I to the composition of o" with T then rl(d?)=qb. Consider l and ~ endowed with some orientations. As 11 is composed of one reflection and 1 and ~ are linked, rl must reverse the orientation on l or on ~. By lemma 3, rl doesn't reverse the orientation of ~ • Now, we can prove tile following Lemma 5.- f~ M(l,k) is nonmirror if k>_8.

144

Proof: Suppose f=ll;pcp2,..,pk}cM(1,k ) with k>_8 is mirror. Let to be a mirror permutation o f f . Then, from lemma 3 we deduce that to permutes the set of points o f f in such a way that [pCp2,..,pk I will be divided in disjoint cycles, all of the same length t. We will see that no value for t is possible. (i) t=l, 2 or 3: consider some f'~M(l,6) composed of the line I and the points of 6, 3 or 2 cycles respectively. Then q~(f')=f" and f ' would be mirror. But this is not possible by lemma 2. For t_>4 we will consider some f'~ M(1,0 composed of tile line I and the t points of one single cycle. (ii) t=4, 5 or 6: to(f')=f' and therefore f'cM(1,t) would be mirror, but this is not possible by lemma 2. (iii) t->7 and odd: we will show that M(l,t) doesn't contain mirror configurations. Let g~M(l,i). If i>7 define s(g) to be tile sum of tile linking coefficients of tile i triples that consist of 3 lines determined by three consecutive pairs of adjacent points in g. Then s is preserved under isotopies and antisymmetric in M(l,i). If i is odd, then s(g) is odd and g is nonmirror. (iv) t->8 and even: let p be a point of f', then f'=[/;p,q0(p),..,q]l(p)} and q~(p)=p. We will denote by 1' to tile line determined by p and tot~(p) endowed with some orientation, and by ~ ( f ) to the line determined by toi(p) and toi(totn_(p))=~÷t,/2-)(p) endowed with the orientation induced by ~ and /'. Then, r and to~a(l') denote the same line with opposite orientations. We know that to is composed of one reflection, therefore if the line t of f is endowed with some orientation we will have Ik(l',/) ~ -lk(tp(/'),to(l)) Co~. 4 Ik(tp(/'),/). Hence, applying t/2 times to we obtain lk(r,i)=lk(tp~P--(r),l) which is a contradiction By lemmas 1, 2 and 5 we have tile following: Theorem 6.-f~M(l,k) is mirror if and only if k<3 •

Note: Since this paper was completed, the i)rol~lem addressed in it (tile existence of mirror configurations on M(h,k)) has been soved by this author for an extense quantity of cases by using differents methods (see "Mirror property for nonsingular mixed configurations of lines and points in R3", in preparation). Bibliography [1] V. M. Kharlanlov, Non-amphicheiral smfaces of degree 4 in ~P'~, Topology and Geometry (Rokhlin Seminar), Lecture Notes in Math., vo1.1346, Springer, 1988, pp. 349-356. [2] O. Ya. Viro, Topological pt'oblems concerning lines and points of three-dimensional space, Soviet Math. Dokl. 32 (1985), no.2, pp. 528-531. [3] O, Ya. Viro and Yu. V, Drobotukhina, Configurations of skew lines, Leningrad Math. J. 1 (1990), no.4, pp. 1027-1050.

Families of Semialgebraic Sets and Limits by Ludwig BrScker

1

Introduction

and notations

T h e motivation for this article is the question, to what extent a semialgebraic set S C N '~ can be approximated by algebraic sets W. More generally, let U C llU~ be semialgebraic open and let S C U be semiMgebraic and closed in U. Do there exist algebraic sets W C R ~ such that W 0 U is arbitrarily close to S? Some comments are needed: 1. T h e question makes no sense if S itself is of the form S = W O U where W C R ~ is algebraic. 2. T h e question makes no sense unless the algebraic sets W are required

to belong to a fixed finite - p a r a m e t e r family. Before we get more precise, let us consider an example: Let n = 2 and U = Q2 = { z e R 2lx~ < 1,x] < 1} the open unit cube. We shall also use the short notation

v = { x t < 1,x

< 1}

Let S C Q2 be the right angle

s = Q~ n ({x~ = o, x2 > o} u {x2 = o, x~ > o}) So we want to approximate S in U by Mgebraic curves W which belong to a fixed family. For instance, we may require that deg(W) <_ d where d is fixed. Clearly, S cannot be a p p r o x i m a t e d in U by curves of degree d _< 2 and one can also see that for d = 3 this is not possible. But an approximation of S

146

in U by algebraic curves of degree 4 is possible. For instance, for W we can take a couple of very fiat ellipses which are very close to S.

By a small perturbation W can also be made irreducible. However, this kind of approximation might be considered to be not good, since it is two fold or of multiplicity 2. Of course, it has to be defined precisely, what multiplicity means, and a major part of this paper and [Br] is concerned with this. We are mainly interested in multiplicities mod. 2 which have a simple topological meaning. Let us stay in our example: Let deg(W) _< 4, W C U~(S) = {x e R 2 ]d(x, S) < 6} and 6 sufficiently small. We consider W as a semialgebraic chain (after omitting isolated points and intersecting with a sufficiently large ball if W is not bounded). Oonsider another semialgebraic 1-chain T C Q2 such that 9T N U6(S) = @. Then the intersection number T o W is defined [Do,VII § 4] and the claim is that it is even. This does not only hold for the family of curves of degree _~ 4 but for any finite parameter family of curves, if 6 is (depending on the family) sufficiently small. The main goal of this article is to show this kind of non-approximability of semialgebraic sets by algebraic sets in any dimension. For the precise statement see Th. 4.8. I am grateful to M. Coste, who gave me a hint how to simplify and clarify my earlier proof of that theorem. In § 1 we shall introduce families of semialgebraic sets and we consider Hausdorff limits for sequences of semialgebraic sets in a family. The main result is that these Hausdorff limits are again semialgebraic. This is done by interpreting these limits as reductions of semialgebraic sets defined over a real closed real valued field R. Now this kind of reduction is well understood [Br]. Let me quote the main results: Let R be a real closed field with real valuation ring V and residue field R. For mE V let ~E ~ the residue class. Also for m = ( x l , . . . ,m,~) let ~ =

147

(~1,... ,~,~). This defines the reduction map V~ ~ R-~ ; z ~ g For S C V '~ we denote by S the image of S under the reduction map. . If S C V '~ is definable (in the language of valued real closed fields [Ch-Di]), then S is semialgebraic. ,

3.

If S is semialgebraic, then d i m ( ~ )

_< dim(S).

The reduction map extends to the real spectrum. If S C V" is semialgebraic and a e ~ with dim(a) = dim(S), then the number #s(a) of elements/3 e S which are mapped to a, is finite. Moreover, the map

a ~-~ #s(a) is locally constant on the subspace of elements a ~

of

dimension equal to dim(S). Here " .~ " denotes the usual correspondence between semialgebralc sets and constructible sets in real spectra [B-C-R, 7.2][Kn-Schei, III§ 5]. The numbers #~(S) are algebraic analogues of the visual multiplicities we observed (but did not define) in the example. 4. The multiplicities allow to define the reduction of semialgebraic pchains which leads to a reduction of homology groups. Also intersection numbers of p- and q-chains, p + q = n, are well behaved under the reduction (here we use coefficients in Z/2Z). The latter point leads to the direct topological interpretation of multiplicities we considered in our example. For all this work, we shall use the following notations. semialgebraic where R is a real closed field. Then

Let S C R ~ be

Cl(S) = closure of S, Int(S) = interior of S, Bd(S) = boundary of S, Ctz(S) = Zariski-closure of S. dim(S) = dimension of S. dim=(S) -- local dimension of S at x. S* = {x e S[ dim=(S) = dim(S)}. We provide R" with the euclidean distance d. Then we also set, for e > 0 :

u,(s) B,(s) s,(s)

= { x ~ a " t a(=, s) < ~} = {= e R" I d(=,s) _< ~} = {= e R- l d(=, s) = e}

The real spectrum of a ring A [B-C-R, chap.7], [Kn-Schei, Kap. III ], will be denoted by Spar(A) and the maximal real spectrum by Mar(A).

148

2

L i m i t s of s e m i a l g e b r a i c sets

Let R be a real closed field. Consider R "+k = R ~ × R k. Let ~r : R "~+k --+ R k be the canonical projection. Let S C R ~ × R k be semialgebraic. So S = {(x,y) E / ~ x R k l ( b ( x , y ) } . S is considered as a family of semialgebraic sets, S = US~ where y e ~-(S) and S~ = 7r-l(y) n S

for y e R k. R k (or lr(S) if we want S~ to be non empty)

is called the parameter space. We write ST for ~r-l(T) N S if T C R k is semialgebraic. We shall also consider the underlying real spectra of R[Y] and R[X, Y] for X = ( X ~ , . . . , X~) and Y = ( Y ~ , . . . , Yk). We use the tilda operator, which associates to every semialgebraic set S the constructible set S in the corresponding real spectrum. We are looking for limits of semialgebraic sets S,, depending on the parameter y. There are various ways, how they occur. Let us consider some of t h e m and the relationships between them. S e m i a l g e b r a i c l i m i t s 2.1 Let T C ~r(S) be semialgebraic, and let yo a C l ( T ) = closure of T. Then we put lim ST := Cl(~r-l(T

T..--~yo

\ {y0}) n S) N ~r-'(yo)

So this limit is obviously semialgebraic in the linear space ~r-X(y0). A b s t r a c t l i m i t s 2.2 N

Let a c r e ( S ) and let R~ be the corresponding real closure. Then the fiber S~ C R~ is defined as follows. Take a formula @(X, Y) for S and set

& :=

R,, I

By model completeness this definition does not depend on the particular description • for S. Now consider a specialisation a --+ fl in Sper(A) for

A = R[Y]. This yields a real valuation ring V of R~ where we assume R~ to be algebraic over k(sul)p(a)) , namely V = convex hull of (A/supp(a)),upp(#)

149

in R~ See [B-C-R, 10.2]. We refer to this valuation of R,, as the valuation of a -+/3. We set

natural

lim S~ := S~ N V"

a---,fl

where "bar" denotes the reduction map of section 0 with respect to V. H a u s d o r f f l i m i t s 2.3 For simplicity we assume that R = R. For any subset M C N '~ and e > 0 let

U(e, M) be the open e-neighbourhood of M. Moreover, for any two subsets M, N C N '~ we set

d(M, N) := inf{¢ I M C U(e, N), N C U(e, M)} (Hausdorff distance). Now let C be the collection of all closed sets C C R'*. For e, pElt~, ¢, p > 0, we set

= {c'

c I d(C' n BAo), c n B,(o)) <

For each C e C we take the sets

U~.p(C),c, p > 0, as a basis of neighbourhoods,

thus defining a topology on C. Now let {y,~} be a sequence in the parameter

CI(S~,) converges in C, we write lirnl~,}..,~0 S~, for the limit and call it Hausdorff limit. (We space, tending to y0. If the corresponding sequence

hope that, in the sequel, the running index n will not be confused with the dimension of R'L) Let us consider the relations between these limits. From the geometrical point of view we are interested in Hausdorff limits. By [Br, § 1] we get information on abstract limits. The semialgebraic limits are somehow between them. First let us visualize the map V --, R where (R=, V) is given as in 2.2 and R is the residue field. For this we feel free to consider the easy case that R = R and fl = Y0 is an ordinary point in R k, that is, supp(fl) is a maximal ideal. Let ~o be the prime filter corresponding to a. Recall that R= is the field of all Nash functions, living on some T e ~o, T open in supp(~o), modulo the obvious equivalence relation defined by restrictions. Then V C R= is the ring of all elements which have a bounded representant. Now let I" be the graph of the Nash function f on T, so T C R k and r c R x N k. So F is a family of semialgebraic sets. However, for Y ~ T the fiber Pv consists of the single element f(y).

Nevertheless limr-~vo I'T is not a single point

but a closed semialgebraic subset of R in general. For instance, let k = 2,

150

T = {X~ > 0, X~ < X ~ } , y o = (0,0) and f = Y / X . [-1, = {x

ThenlimT_.uoPT =

_ _ _

Note t h a t limT-.u0 FT is bounded and non empty, if f is bounded. So it is compact. Now having chosen one representation T, f for a e R,~, a e V we look at all smaller T' • ~o (ultrafilter of a), T' open in supp(~o). Then Q lim FT, =: F T T'"4Y°

is compact and non empty. We claim L e m m a 2.4 F consists of a single point and this coincides with the residue

class "~ of a. Proof:

Assume, there are Ca, c2 e F. Take neighbourhoods U~(c~) such t h a t

2~ < ICx - c21. Now let

and Zi = ~r(~). Then Zi is open in T and Zx N Z2 = 0. Thus at most one of the Zi belongs to ~o. Contradiction. So F = {c} for c e R and we have to show t h a t c = ~. For this it is enough t h a t there are representations f , T of a for which f - c is arbitrarily smM1, which is clear by construction.

[]

R e m a r k 2.5 If we work over an arbitrary real closed field R we can make

a similar argument, but then ~ defines only an ultrafilter ¢ on the residue field R~, which corresponds to a point in Mar(Rg[X]). This agrees with the fact that R---~ does not coincide with RO in general, but -R~ is an archimedean extension of R~. Now let us look at the relation between semialgebraic limits and abstract limits. We consider the same situation as before: R = R, a --*/3 = y0. Let ~o be the ultrafilter corresponding to a. Then we get P r o p o s i t i o n 2.6 limS,=

~'-'Yo

N TE~

lim ST

T---*~o

151

Note t h a t a. fortiori it is not clear that the right hand side is semialgebraic, and in fact, if we do not work over the ordinary reals, it is only defined over an extension of R~0.

Proof:

Let a = (al,... , an)E Sa and let T E ~o. T h e n the al can be repre-

sented by Nazh function fi, living on an open set T ' e ~ , T ' C T and T ~ is contained in the variety of supp(a). Moreover ( f l ( Y ) , . . . , f,~(Y)) ~ S, for all y e T ~. Hence from 2.4 we get

ST, C lim ST

= ( ~ 1 , . . . ,gn) C tim T ' "* yo

T -"p y o

Conversely, assume that cE ~

lim ST,

T ""*y o

c ~ lim S~ c~"'~ y o

TE,~

Note that both sets are closed. Take a neighbourhood U~(c) C

R~ such that

U2~(c) does not intersect lina~_.~o S~. Similary as before consider

r :=

s I•

y

y0}

and Z = 7r(Y). T h e n yo E Cl(Z) by construction. If Z N T = 0 for some T e ~o we get immediately a contradiction. So let's assume Z e ~o. B y Hardt's theorem [B-C-R, 9.3], [Ha] we have a decomposition: Z = Z~ U . . . U Z r where the Z, are open in their Zariski closure and Nash sections g~ : Z~ --~ Y for i = 1 , . . . ,r.

So replacing Z by one of the Z~ we m a y even assume that

there is a Nash section f = ( f l , . . . , f,~) : Z --* Y. this defines an element a E~

with a E S~ for which, by construction and 2.4, we get g E Cl(U~(c)).

Contradiction.

[]

Now let us see, how Hansdorff limits fit into this. Again assume t h a t R = ~.

Let {y,~} be a sequence tending to yo where Yn, Y0 E R k, and assume

that lim(y.}_.~o S~o exists. Let ~' be the filter of all semialgebrMc sets T C R k which contain almost all y,.

Let ~o D ~o' be any ultrafilter and a the

corresponding element in Sper(A). T h e n a --* fl = yo. W i t h these notations we have

Proposition

2.7 lira

{~,~}--'~o

S~. = liln S~ a--,yo

152

Proof: First observe, that any T e ~ contains a subsequence of {y,,}. Thus, if c • lim(u.}_.u0 S~., so c • limT-~y0 ST, from which by 2.6 we get c • lirn~.~0 S~. Conversely, assume c E limS~,

c~

lim

S~.

Again choose a neighbourhood U,(c) of c in 7r-l(y,,) such that U~,(c) N lim{~,,}_.uo Su. = 0, and consider Y = {(x, y) • S I x • U(e, c)} and Z = 7r(Y). Apparently, Z ~ ~o. Hence we find T • ~o with Z A T = 0. Then c ~ limT-.uo S r a~ld, by 2.6, c ~ lin~_.vo S~. Contradiction.

[]

C o r o l l a r y 2.8 Every Hausdorff limit So = lirnl~.I-~uo Su, is semialgebraic.

Moreover, dim So < liminf(dim(Su.)). Proof: Apply [Br, Th. 1.7] and [Br, Prop. 2.5]. One should compare this with the limits considered in [v.d.D.].

3

Continuity

For all this section we assume that R = R. Let S C R '~ × R k be a semialgebraic family. First we consider 1 - parameter families, that is, k = 1. P r o p o s i t i o n 3.1 Let S be a 1 - parameter family. Then for all sequences {yn} -* yo, Vn y~ < yo, the Hausdorff limit lirn{~,}.~° S~. exists and the

following limits coincide. 1. limT--.~o ST for all T such that T =]Yo - e, yo[ in a neighbourhood of yo. 2. lilrk~-~o S~ for the left hand generization of yo. 3. lirn/~.}_~0 S~. for all sequences {y,,} ---* Yo, Yn y~ < yo.

Proof: First we observe that the limit in 1) is independent of T subject to the condition on T. So the coincidence of the limits 1) and 2) follows from 2.6. For 3) we fix an interval T --]z, Y0[, z < Yo and a sequence {y,} --* Y0, Y,~• T. Let e,p > 0. It remains to show that d(S~. NBp(O),goClBp(O)) < e f o r sufficiently large n where d is the Hausdorff distance and So = limT--,~o ST. Let

U = U~(SonB~(O)) be the e-neighbourhood of SoNBp(O) in N ~. Since Bp(O) is bounded, it is clear t h a t S~,, cl B,~(0) C U for n > no. Now assume t h a t there

153

are infinitely m a n y m • N such that (So A Sp(0)) C U(e, S~., ClB,(0)). In particular, there is a point zm • SoABp(O) such that z,,~ ~ U(e, S~,~NBp(O)). Since So fq Bp(0) is compact, we m a y assume that {z,~} --* Zo. Let U = Ue/v.(zo) in ll~'~ x N. By construction, U n Sy.~ = 0 for all m with :,~ - Zo < e/2. So now T1 := {y • T [ S ~ n U = 0} is infinite and yo • Bd(T~).

On the other hand,

T1 is semialgebraic. So we m a y assume Tx = T. But then z0 ~ limT-~0 ST. Contradiction.

[]

Let us consider further properties of a family S. P r o p o s i t i o n a n d D e f i n i t i o n 3.2 For a semialgebraic family S C R '~ × R k

the following properties are equivalent: a) For any sequence {(xn, y~)} .in S with {(x,~, y~)} -~ (Xo, y0) and y,~ # Yo

one has (Xo, Yo)• Cl(S o) (The vertical closure of S is closed.) b) For any semialgebraic set T C ]~k, Yo • CI(T), one has limT-~o ST C Cl( Svo ).

b') Same as b), but with the further condition that dim(T) = 1. c) lim~:k.~o S~> C CI(S~o). d) For all a • Sper(R[y]), a ~ yo one has li,r~..vo S, C Cl(S~o). d') Same as d), but with the further condition that dim(supp(a)) = 1. e) lin~.}_.y 0 Sv. C Cl(S~o) whenever the left hand side exists.

If these conditions hold, the family S is called semicontinuous. Proof: Obviously, a) ~ b) ---, c) ~ b), b) ---* b') and d) ~ d'). Moreover, by 2.6 : b) ~ d), by 3.1 and the curve selection lemma [B-C-R, 2.5]: b') ~ d') and e) ~ b'). Finally, by 2.7: d) ~ e). So it remains to show b') ~ a): Let {(x,~,y~)} (xo,yo) and assume that (xo, yo) ~ Svo. By the curve selection lemma (loc. cit) we find a semialgebraic arc-~, : [0, e [ ~ S with 0'(0) = (xo, yo) and O'(t) 7r-~(yo) for t > 0. Let T := 7r o 7- Then dim(T) = 1 and by construction (x0, y0) E lim ST , T --~ y o

which contradicts b').

[]

154

R e m a r k 3.3 In fact~ the semicontinuity is a condition at the points yo e R k. "Natural" families will be semicontinuous, but by brute force one can replace

fibres by bad ones at certain points to make it non-semicontinuous. From property c) above we learn, that for an arbitrary semialgebraic family S the subset T C N k of points at which S is not semicontinuous, is semialgebraic. If S is semicontinuous, then ~r(S) is closed. It should be clear, what we do now and how we do it. P r o p o s i t i o n a n d D e f i n i t i o n 3.4 For a semialgebraic family S C R'* x N ~ the following properties are equivalent:

a) For any semialgebraic set T C ~r(S), Yo e Cl(T) one has limT-~o ST = C l( S~o).

a') Same as a), but with the further condition that dim(T) = 1. b) For all a e Sper(R[Y]), a ---+ yo one has lim~_~o s~ = Cl(S~o).

b') Same as b), but with the further condition that dim(supp(a)) = 1. c) Fo~ any sequence {y,} with y~ ~ ~ ( S ) a~d {y.) -~ yo ~ s one has {s~.} -,

Cl( S~o) (in

the Hausdorff topology).

If these conditions hold, the family S is called continuous. Proof: Obviously a) ~ a') and b) --* b'). Moreover, by 2.6: a) --* b) and by 3.1 and the curve selection lemma (loc. cit): a') --* b'). b') --+ c): Let {y,~} -* Yo. Assume also that y~ # yo. Suppose {S~.} does not tend to Cl(Svo ). Then there exists e, p > 0 and infinitely many m e N such that d(Sv.. N Bp(O), Svo N Bp(O)) >_ ~. If (Sv= N Ba(O)) (~ U,(Svo N Ba(O)) for infinitely many m, we find a curve 3' : [0, p[---~ R '~ × R k, 7(]0, p[) C S, 7(0) = ze~r-l(yo) \ (U,(Svo) A Bp(O)) and 7(t) ~ lr-l(yo) for t > 0. Then ~r(3') defines an element a ~ Sper(N[y]) such that a --+ yo. For this, by 3.1, we get z e lim~.~ ° S~, which contradicts b'). So assume that (S~o n Bp(0)) U~(S~., N Bp(0)) for infinitely many m. As in the proof of 3.1 we find Zo e Cl(S~o ) n Bp(O) and a ball U around Zo in ]R~ × IRk such that U N S~.~ = 0 for infinitely many m. Therefore we find a curve 8 : [0, p[---* ~r(S), 8(0) = yo and fi(t) # yo for t > 0, such that S~(t)NU = 0 for all t e]0, p[. Let a be the element in Sper(IR[y]) defined by 5. Then a --* y0 but, by 3.1, z0 ~ lir~-~0 S~,

155

which again contradicts b'). c) --+ a): Let T C ~r(S) be semialgebraic with yo E e l ( T ) .

Then we find

a sequence {y~} --+ Yo, Y,~e T \ {yo}. Clearly Cl(Suo ) = lin~,_.uo S,. C limT-~vo ST. The converse inclusion follows from 3.2. []

R e m a r k 3.5 Again, continuity is defined at points. One might ask for the

size of the set D C re(S) of all points, where S is not continuous. A first information is given by the following partition into continuous parts. There is a decomposition re(S) -- T1,... ,T~ into disjoint semialgebraic sets such that the following hold: a) T~ is open in its Zariski closure. b) SIT~ "is continuous. This is an immediate consequence of Hardt's theorem [Ha]. In particular, D is nowhere dense in re(S). May be that D is also semialgebraic, but I didn't see anything which makes it evident.

4

L i m i t s of c h a i n s

Let R be a real closed field. chains 4.1 A semialgebraic q-chain S in R ~ is a bounded closed semialgebraic set S C R '~ of dimension q such that S = S*, that means, also the local dimension dim,(S) = q for all x E S. The sum of two q-chains S and T is defined to be 0 if S = T or C I ( S A T ) (symmetric difference of S and T) if S ¢ T. The boundary OS of a q-chain S is defined via any triangulation of S but it does not depend on the particular triangulation. For semialgebraic sets A C X C R n the semialgebraic q-chains give rise to a definition of homology groups Hq(X/A, Z/2Z). For R = R these coincide with the usual ones and in general they coincide with those defined in [D-Kn] or

[B-C-R, 11.7] (see [Br, § 4]).

E x a m p l e 4.2 Let S C R '~ be algebraic of dimension q such that S* is

bounded and closed. called algebraic.

Then S* is a semialgebraic q-cycle. Such a cycle is

156 Reduction

o f c h a i n s 4.3

A s s u m e t h a t the real closed field R a d m i t s a real valuation v with residue field R and valuation ring 1/. As before, for a semiMgebrMc set S C V '~ let S := { x ' e R ' ~ ] 3 x e S

: x' = 5}.

If S is a q-chain, we d e f i n e r ( S ) , the

reduction of S, as follows: If d i m ( S ) < q then r(S) := 0. If d i m ( S ) = q, there exist semiMgebraic q-chains T 1 , . . . , T,,~ C ~

such t h a t S* = T1 + . . . + T,n

and the multiplicity #s(a) is constant = ri for all a e ~bi such t h a t d i m ( a ) = q

(see § 1 or [Br, § 4]). N o w r(S) := riT1 +... + r~T,, T h e n r COlrlnlutes with 0 [Br, Th. 4.4] and thus for semiMgebraic sets A C X C V '~, the reduction r defines a b o m o m o r p h i s m

r: H,(X/A, Z / 2 7 / , ) ~ H , ( X / A , Z / 2 Z ) We want to u n d e r s t a n d these multiplicities in the context of limits in a purely geometric way, at least modulo 2. For this we need another tool. Intersection

numbers

4.4

Let r/ be a p - - c h a i n and ~ a q-chain in N '~ w i t h p + q

= n. T h e n r/ and

are called admissible if lr]t cl 10~1 = 0 and [~] N IOr/] = O. In this case, an intersection n u m b e r 7/o ¢"e Z is defined [Do, V I I § 4], [S-T, § 731. We denote by 77 02 T the intersection n u m b e r rood.2. If R is replaced by any real closed field, S 02 T is still defined for admissible semialgebraic p-chains S and qchains T [Br, § 5]. For an isolated point x e S N T we denote by i2(x, S, T) the local intersection n u m b e r mod. 2 at x. If x ~ S N T we set i2(x, S,T) = O. Now let us r e t u r n to families S C N '~ x IRI' of semialgebraic sets.

Such a

family S is called a family of q-chains if S~ is a q-chain for all y e ~r(S). It is easily seen, using H a r d t ' s t h e o r e m [Ha], [B-C-R, 9.3] or similar a r g u m e n t s , that

OS :=

U

OS~

v e ~(s)

is semiMgebraic, so it is a subfamily of S. Now assume t h a t S C ~'~ × ]~k is a family of p-chains, T C IR'~ × N k a family of q-chains, p + q = n and ~r(S) = 7r(T). Let y~ e ~r(S) = ~r(T) for n c N and {y,~} --~ yo. We assume f u r t h e r m o r e , t h a t all Hausdorff limits lira S w =: S0, lim OSw =:(OS)o {v-}-~o {v.}-~yo lira % . = : T o , lira 0 % , = : ( 0 T ) 0 {~.}-~uo {u.}-~uo

157

exist and are bounded. By 2.7 there exists an element a•(~-(S)) ~ with a ~ Yo such that (with respect to the natural valuation v of the specialization a ~ y0), So = lim S~ = S~, ~'-~YO

(OS)o =

lira

(OS)~ = (OS):

cx --~ y 0

To = lim T~ = T~,

(OT)o = lira (OT),, = (OT)~ . ~ "* YO

In this situation we have P r o p o s i t i o n 4.5 Assume that So N (OT)o = 0 and To (3 (OS)o = O. Then a) S~ and T~ are admissible. b) r ( S , ) and r(T~) are admissible. c) There exists no • N such that Sv. and Tv. are admissible and Sy. o2 T~. = S~ o~. T~ for n >_ no.

d) So o: T . = r(So) o: Proof: a): Obvious. b): This follows from the fact that 0 commutes with v [Br, Th. 4.4]. c): Obviously, Su. and Tu. are admissible for sufficiently large n, and from the general theory of intersection numbers [Do, VII § 4] it is also clear t h a t S~. o2 T~ becomes constant. This follows also from b) and the construction below. First of all, by the compactness of (~r(S)) ~ with respect to the constructible topology, there is a semialgebraic set D C r ( S ) = ~-(T) such that a e / ) and S#, T# are admissible for all fl E/). Now for y • D the statement, t h a t S~ 02 Ty takes a certain value, is elementary. Therefore, there is a semialgebralc partition: D = T1 U . . . U T,~ such that Sv o2 Tu is constant for y • T~. But a • T~ for exactly one i • { 1 , . . . , m}, say a • ~bx. By construction, y~ • T1 for infinitely m a n y n and for these we have S,j o~ Ty, = S~ o2 T~

d): This is just stated by the intersection - reduction formula [Br, Th. 5.3]. []

As a consequence, we get a geometric description of multiplicities mod. 2. for this, let the family S of p-chains be given as before, and also the sequence

158

{Y,~} -~ Y0, the limits So and (OS)o and the valuation a which leads to the reduction of chains:

so

r(So) a

C o r o l l a r y 4.6 Let x •-S~ and let T C ~'~ be a q-chain such that O T N S , = ~, T n (-:d-S)~ = ~ and T N ~ = {x}. Then T and S~, are admissible for almost all n and T o2 S~, is stationary, Moreover a) I f x ~ r(S~), then T o: S~. = 0 for almost all n. b) I f x • r(S~), then T 02 S~, = T 02 S*~ for almost all n.

We conclude this work by a result on the non approximability of proper semialgebraic sets by algebraic ones. We consider a relative situation. So let us fix an open semialgebraic set U C N n. D e f i n i t i o n 4.7 A semialgebraic set B C. U is essentially algebraic in U, if there is an algebraic subset W C N '~ such that B* = ( W n U)*. If S is a p-chain and • 7~ S n U is essentially algebraic in U, then OS n U = (k Any p-dimensional semialgebraic set B C U admits a unique maximal essentiatiy algebraic subset of pure dimension p. This subset will be denoted by B ~ (but note that B" does not only depend on B but also on U). T h e o r e m 4.8 Let U C R '~ bc semialgebraic and open. Let S C R'* x R k be a semialgebraic family for which the following conditions hold: -

or(S) is bounded.

- 3 9 is a p-chain for all y e or(S). - Sv N U is essentially algebraic in U for all y • or(S). Let B C U be semialgebraic such that - B is closed in U.

- dim(B) _< p. Then there exists e > 0 such. that - for a l l 5 w i t h O < 5 < e , - for all q-chains T C U \ B ~ where p + q = n and OT N Us(B) = 0.

159

-

for all y ~ ~r(S) with (S,

r3

U)

C

U~(B)

one has T o: S~ = O. This looks a little complicated. So let us first consider an example. Let n = 2 and Q = Q= the unit cube:

Q = {xcR2i[x, I <_ 1}. We take U = Int(Q). Let S' C R 2 x]t{ ~ be the family of all curves of degree < 3. So R k is ths space of the coefficients. By normalizing we can achieve t h a t ~r(S') is bounded.

Let S C S' such t h a t

Su = (S~ n Q)*. T h e n S is a family of 1-chains such t h a t S~ n U is essentially algebraic in U for all y. We set

B := U n ( { X ~ = O} u { X , = 0,.',:~. > 0}) Here B a = U n {X2 = 0}. Now, of course, for an arbitrarily smM1 6 > 0 we find y e ~r(S) such t h a t (S~ n U) C U~(B), but t h e n the T h e o r e m says, t h a t for any 1-chain T in U with T N B ~ = 0 and OTA U~(B) = 0 one has

T o 2 S ~ = O.

Us(B)

~---.e

It m i g h t be very difficult to find the smallest possible ¢ for which the T h e o r e m holds, even for the above example.

proof of the theorem: 1) Let U ' : = U \ B

a. We set for ~ > 0 H~(6) := H~(U'/(U' \ U~(B)),Z/2Z) and

H~(O) :=

H~(U'I(U' \ B), Z/2Z)

Then

~I~(o) = lira H~(6) ,S>O

160 We need a little more: First of all, Ha(O ) and all Hq(d~) are finite Z / 2 Z vectorspaces. Moreover, by Hardt's theorem [Ha], [Br-C-R, 9.34] there exists e > 0 such that ~r : U' D (U~(B) n U') H ~ is a trivial fibration for 6e]0, e]. In fact, for 0 < 81 < 6~ < e the inclusion

(u', u' \ U6 (B)) c (u', u' \ U,,(B)) is a retraction, hence the canonical maps

Hq(,)

Ha(65) Ha(*1) nq(0)

are isomorphisms. 2) Assume t h a t the statement of the Theorem is wrong. Then there exists a convergent sequence {y,~} --+ Yo, Y,~e lr(S) and a sequence {T,~} of semialgebraic q-chains with the following properties: i)

(S~. n U) C U~(B).

ii)

T~cU'=U\B

iii)

S ~ 02 T~ = 1.

~and0T~nU¼(B)=0.

We choose e according to part 1) of the proof. Since Sv, o2T~ depends only on the class of T~ in Hq (¼) we may assume that n > ~ and all T~ are restrictions of a single semialgebraic q-chain T C U' with OT Q U,(B) = 0. Now pick a e ~r(S) such that each element of the ultrafilter associated to a contains infinitely m a n y y~ (compare the proof of 2.7). Then S,,QU C B and S~ o5 T = 1. On the other hand, by 4.9 below, S, Cl (U x Nk)~ is essentially algebraic in (U x Rk),. Hence by [Br,Cor. 3.8] we get r ( S , ) n U C B ". Since T N B ~ = ~ and T = r T we get r T o5 rS~ = O. But this is a contradiction to the intersection-reduction formula [Br, Th. 5.3] saying S~ohT = rS~,o2rT. []

For the L e m m a below we use the following notation: Let S C R ~ be semialgebraic and let x l , . . . , x,~ e R ~. Then S is called essentially algebraic a~ x l , . . . ,x~, if there is an open set U containing z l , . . . ,z~ such t h a t S is essentially algebraic in U. L e m m a 4.9 Let U C N" be semialgebraic and open and let B C U be semi-

algebraic. Then the statement "B is essentially algebraic in U" is elementary. Proof:

This is an immediate consequence of the following facts.

161

a) B is essentially algebraic in U if and only if B is essentially algebraic at any two points xl, x2 e U. b) Let dim(B) < n - 1. Then B is essentially algebraic at x l , . . . ,x,~ if and only if for all y e S '~-I and all ¢ > 0 there exists z e U~(y) such that vz(B) is essentially algebraic at r ( x l ) , . . . , ~r(x,~), where r , is the orthogonal projection: R ~ -~ (z) ±. c) The statement is true if dim(B) = n - 1. Here a) is clear and b) holds, since the image of an algebraic set under an injective polynomial map is essentially algebraic. For c) first one finds among the polynomials, which describe B, equations for an algebraic set W D B with dim(W) = dim(B). Since ]~[X], X = X~,... ,A~ is factoriM, the degree of equations for the Zariski closure of B can be bounded. []

References [B-C-R]

Bochnak, J., Coste, M., Roy, M.F.: G6om~trie Alg~brique R6elle. Springer, Berlin Heidelberg New York, 1987

[Brt

Brhcker, L.: On the reduction of semialgebraic sets by reM valuations. To appear.

[Ch-Di]

Cherlin, G., Dickmann, M.: Real closed rings II. Model theory. AnnMs of pure and applied Logic 25, 213-231 (1983)

[D-Kn]

Dells, H., Knebusch, M.: On the homology of algebraic varieties over real closed fields. J. reine angew. Math. 335, 122-163 (1982)

[Do]

Dold, A.: Lectures on Algebraic Topology. Springer, Berlin Heidelberg New York 1972

[Ha]

Hardt, R.: Semi-algebraic local triviality in semi-algebraic mappings. Amer. J. Math. 102, 291-302 (1980)

[Kn-Schei] Knebusch, M., Scheiderer, C.: Einffihrung in die reelle Algebra. Vieweg, Braunschweig Wiesbaden 1989

t62

[v.d.D.]

van den Dries, L.: Tarskis Problem and Pfafilan functions. Logic Colloquium 1984, ed. I.B. Paris, A.J. Wilkie, G.M. Wilmers, North Holland 59-90 (1986)

Ludwig BrScker Mathematisches Institut Einsteinstr. 62 4400 Mfinster Germany

A H o p f F i x e d P o i n t T h e o r e m for S e m i - A l g e b r a l c M a p s G. W. BRUMFIEL*

Introduction. This note is a sequel to [2], in which we proved the Brouwer fixed point theorem for the real spectrum compactification of R n, that is, for PV' = S p e c a ( R [ x l . . . xn]), the real spectrum of the polynomial ring in n variables over the real numbers. Here, we fix a real closed field K and a semi-algebraic subset X C K'*. Let ) ( C Pi~ = Speca(K[xl . . . x n ] ) be the constructible subset corresponding to X C K n. Any continuous semi-algebraic map ~0 : X -* X extends to a continuous map Q : X ---* X. Define the Lefschetz number of ~ to be t r ( ~ , ) = ~(--1) I trace(~p, : H i ( X , Q ) --* H i ( X , Q ) ) . Homology of semi-algebraic sets X c K n is defined in [3J. The Hopf fixed point theorem basically states that if tr(q0.) ~ 0 then q3 : .~" -* X has a fixed point. Before giving a precise statement, we make a few remarks. It seems reasonable to investigate topological properties of Fix(Q), the fixed point set of Q. For example, we can show that Fix(Q) C_ X is always closed. I am indebted to Claus Scheiderer for the proof of this fact given below. His proof replaces a more cumbersome argument in an earlier version of this note, which only worked under the added hypothesis that X C K '~ was locally closed. Note that Fix(q3) is obviously closed in the Tychonoff, or constructible topology on X , because the Tychonoff topology is compact and Hausdorff and q3 is Tychonoff continuous. A Tychonoff closed subset is closed in the real spectrum topology, precisely when it is closed under specialization a ~ fl [1, Proposition 2.11]. We will show below that if a, fl E )~, a --* fl, and Q(a) = a, then q3(fl) = ft. Thus we obtain P r o p o s i t i o n 1. I f X C K'* is any semi-algebraic set and T: X continuous semi-algebraic map, then Fix(q3) is closed in f(.

--* X is any

The classical Hopf fixed point theorem concerns maps ~ : Y --~ Y, where Y C R n is a finite closed simplicial complex. If t r ( ~ . ) ¢ 0 then ~o has a fixed point in Y. A key step in our general Hopf fixed point theorem is to extend this result to arbitrary real closed ground fields K. P r o p o s i t i o n 2. Suppose Y C K n is a complete semi-Mgebraic set, that is, Y is dosed and bounded in K n. Suppose ~ : Y --. Y is a continuous semi-Mgebraic map with tr(qo.) 7~ O. Then ~ has a t~xed point in Y . * Research supported by NSF grant DMS 85-06816 at Stanford University

164

The a s s u m p t i o n that Y is complete is equivalent to assuming t h a t Y is a closed finite simplicial complex over K. In fact, we use triangulations in a crucial way in our proofs. Specifically, we exploit the fact that any semi-algebraic set X __ K " can be triangulated and a d m i ~ strong deformation retractions onto complete subspaces Y C X . Define F r i n g ~ X ) C X to be all those points which do not belong to any constructible C C X , where C C X is a complete subspace. Here is a precise s t a t e m e n t of our main theorem. T h e o r e m . i f ~ : X -+ X is any continuous s e m L a l g e b r a l c m a p with t r ( ~ . ) ~ 0 then e i t h e r ~ : X ~ X has a fixed point or ~ : X --+ X has a fixed point in

inge(

). Moreover,

:

always has a closed fixed point,

nd, if either

X C_ K n is locally dosed or if ~he ground field K is R, then ~ : f ( --~ X always has a dosed fixed point in Fringe(.~) U X . The proof of the Brouwer fixed point theorem for A m in [2] was s o m e w h a t unsatisfactory because at one stage a transcendental a r g u m e n t was used to construct certain retractions from R '~ onto closed disks in R". In fact, the details of that transcendental argument, which a m o u n t e d to constructing a suitable gradient vector field and integrating, were not included. The argument in this p a p e r is purely algebraic. Of course, if X C K " is any contractible space, then tr(~p.) = 1 for any : X ~ X , so the H o p f fixed point theorem certainly implies the Brouwer fixed point theorem. Finally, we point out that it would be desirable to have an even m o r e general Hopf-Lefschetz fixed point formula for ~3 : X -~ )(, which expresses tr(~p.) in terms of local d a t a near Fix(~) C ~7. However, this looks rather messy. For example, if K contains an infinitesimal e relative to Q c K , consider the m a p ~ : I ~ I given b y ~ ( z ) = z + ex(1 - x), where I = [0, 1] C K. T h e n ~ : _T~ .T has a very complicated fixed point set. Somehow, only the two b o u n d a r y fixed points 0 and 1 should be relevant, b u t for each r E K , 0 < r < 1, the Dedekind cuts of I C K defined by the sequences r + ne or r - he, n = 1, 2, 3 . . . . , determine points of I which are fixed by ~. T h e r e are also m a n y other fixed points. E v e n when K = R, if X C R" is not compact, the point-set nature of the fixed points of ~ : )~ -+ -~ in Fringe(A ~) will often rcsemble the b a d fixed point set of the m a p ~ : _T--~ T above. Finding a good generalization of the Hopf-Lefschetz fixed point formula looks challenging. P r o o f o f t h e t h e o r e m . We first point out that if ~ : )~" --* .~ has any fixed point then it has a closed fixed point. Namely, if ~ ( a ) = a and a ~ fl with fl closed in ~7. T h e n ~ ( a ) = a --* ~(fl). Since the specializations of a form a chain, either fl ~ ~(fl) or ~(fl) ~ ft. But, since fl is closed, fl ~ qb(fl) implies fl = ~(fl), and, since dim(~(fl)) _< dim(fl), ~(fl) --* fl also implies fl = ~(fl). Next, consider the fringe of )~, defined as Fringe()~) = )~ - -

U cc_x

C complete

C"

165

Since each constructible C is Tychonoff open, Fringe(,Y) is Tychonoff closed, hence compact. If a E Fringe(.~) and ~ ( a ) # a then 9~(a) 7/* a since dim(93(a)) ~ dim(a). Thus, there exists a polynomial f E K[xl,...xn] with

f(a) < 0 and f(~(a)) >_O. T h e set tS'a = .{fl E )( [f ( f l ) , < 0 and f(c~(fl))>_ 0} is a constructible set containing a, with the property that fl E V a implies qb(fl) ~ VaIl qb : ..Y ~ ]~" has no fixed points in Fr.ingeO~'), we can find finitely m a n y such constructible sets ~ with Fringe(,Y) C U ~ . Let ~ ---- l)i A X , a semi-algebraic set in X. Choose a semi-algebraic triangulation of X such that all the Vi are subcomplexes. Recall that a semi-algebraic triangulation is simply a semi-algebraic h o m e o m o r p h i s m between X and some union of open simplices of a closed finite simplicial complex Z C K " . Boundary faces of simplexes of X need not belong to X and also, of course, the I'~ are not necessarily closed subcomplexes of X. Next, we take two barycentric subdivisions Z" of Z. As in [3], there is now a canonical strong deformation retraction, r : X ---* Y, of X onto a closed subcomplex Y of Z ' . T h e picture below illustrates this construction when X is a single closed 2-simplex with one vertex and two edges removed. X c Z = A 2/\

Y c X C Z ~ x

T h e general formulation can be found in [3]. For our purposes, the crucial property of r : X ~ Y is that if a is any open simplex of the original triangulation of X then r(a) C a. Thus, if a C Vi then r(a) C I,~ so r(Vi) C 17/. Also, if r b # Id then a m u s t contain fringe points of )( so cr C Vd, some j. Thus X - Y C UVi. Consider the composition r~0 : X --* X ~ Y. By Proposition 2, r~qy : Y --* Y has a fixed point y E Y since t r ( ( r ~ I y ) . ) = t r ( ~ , ) # 0. If ~(y) ~ Y, say 9~(y) e ~ . T h e n r~(y) E ~ , and it is impossible that y = rqo(y) because y E ~ implies 9~(y) ~ Vi. Therefore _~(y) E Y and we have thus proved that either 9 : X ~ X has a fixed point or ~ : X --* X has a fixed point in Fringe(X). If X C K " is locally closed then Fringe()~) is closed in )~. This is most easily seen by identifying X with a closed semi-algebraic set X ~ C K n+l. T h e n Fringe(~) corresponds to the set of points of ~ t at infinity relative to K, which is closed in X ~. Therefore, in our theorem, we can conclude that for locally closed X, 9~ : ~ --* )( either has a fixed point in X or a closed fixed point in Fringe(.~).

166

We can also make this same conclusion for arbitrary X C W'. Namely, suppose a e Fringe(~r), ~ ( a ) = a, and a ~ fl with/3 closed in )(. We know 9~(/3) =/3. If fl 6 Fringe(X), then ~ C C for some complete C C X. But when the groundfield is R, the only closed points in a complete C are actually points of C, hence fl E C and ~o(fl) = fl is a fixed point of q0 in X. It seems plausible that this stronger conclusion, that ~5 : ~7 -* X either has a fixed point in X or a closed fixed point in Fringe(~7), holds for arbitrary X C K ' . However, if X C K " is not locally closed and K ~ R, there will often exist a E Fringe(X) with a -* fl, /3 closed, and/3 6 Fringe(X) U Z . Conceivably, the only closed fixed points/3 of ~ : )~ ~ )( could be of this type. P r o o f o f P r o p o s i t i o n 1. We introduce the ring C(X) of continuous K-valued semi-algebraic functions on X C K " . If a E X corresponds to a homomorphism ev(a): K[xl... x,] ---* k(a), where k(a) is a real closed field, algebraic over the image of K[xl... xn], then every f E C(X) can also be evaluated at a. T h a t is, ev(a) extends to ev(a): C(Z) ~ k(a). In this way, we obtain a continuous inclusion X C SpecR(C(X)), which, in fact, is a homeomorphism if X is locally closed in K", [4]. But, in any case, the map ~o:X --* X induces a ring homomorphism ~o*: C(X) --+ C(X), which, in turn, induces ~o.: SpecR(C(X)) --+ SpecR(C(X)) such that the diagram below commutes.

2

C SpecR(C(X))

ot X"

C

SpecR(C(X))

It thus suffices to prove that if ~ . ( a ) = a and a ---* fl in SpecR(C(X)), then = 8.

The assertion ~ . ( a ) = a is equivalent to the existence of an order preserving K-endomorphism i: k(a) --. k(a) such ~hat the following diagram commutes ,~(~) c(x)

,

c(x)

,

Since k(a) is real closed and has finite transcendence degre over K, the inclusion i is an isomorphism. Thus, i induces a permutation of the convex valuation rings in k(a) which contain K, and since these valuation rings form a finite chain under inclusion, each such valuation ring maps isomorphicaily onto itself under i. On the other hand, the specializations of a in SpecR(C(X)) correspond, by the real place existence theorem, to the prime ideals of C(X) which are the inverse images under ev(a) of the maximal ideals of those convex valuation rings in k(a) which contain

167

the image of C ( X ) . From the commutative diagram above, we see that for each such prime ideal p, ( ~ . ) - l ( p ) = p, hence a , : SpecR(C(X)) -+ SpecR(C(X)) is the identity map on the specializations of a, as desired. P r o o f o f P r o p o s i t i o n 2. We have a continuous semi-aigebraic map ~o : Y -+ Y, where Y C K n is a closed finite simplicial complex. The graph P(~o) C Y × Y C K'* × K n is defined in terms of finitely m a n y polynomial inequalities with coefficients in K. We know that if K C R and t r ( ~ . ) ~ 0 then T has a fixed point in Y, because the classical proof of the Hopf fixed point theorem works for any Archimedean ground field K. We will deduce the proposition for arbitrary K by transfer from the real algebraic nttmbers, using Tarski's principle. We may assume that Y itself is defined over Z, since we may as well assume the vertices of Y are vertices of some standard simplex A N C K N+I. We regard the coefficients of the polynomials used to define F(~) C Y × Y as parameters and we regard the statement that F(~p) is the graph of a continuous function ~0 : Y --+ Y as an elementary statement about these parameters. Also, the desired conclusion that T has a fixed point is an elementary statement about these parazneters. For example, a fixed point of ~ is just a point of F(~) ¢3 A(Y), where A ( y ) C Y x Y is the diagonal. The crucial point is that the hypothesis tr(~p.) ~ 0 is also an elementary statement about the parameters defining ~. Assuming this, the truth of the proposition is clear, because if the propositon were false, there would exist in K some values of the coefficient parameters for which (a) F(~o) C Y x Y is the graph of a continuous semi-algebraic function, (b) t r ( ~ . ) ~ 0, and (c) F(~0) n A ( y ) = 0. By Tarski's principle, there would also exist in the real algebraic numbers some values of the parameters for which (a), (b), (c) hold, which is a contradiction. The claim that the statement tr(~0.) ~ 0 is an elementary statement amounts to understanding how the homology maps T, : H i ( Y , Q) --+ Hi(Y, Q) can be computed in terms of the polynomial inequalities which define F(~) C Y x Y. We have the cover of Y by (open) semi-algebraic simplices, hi. Since the ~ - 1 ( ~ i ) C Y are also semi-algebraic, we can retriangulate Y so that all the sets ~ - l ( a i ) , as well as the original hi, are unions of (new) simplices. The important point is that this retriangutation can be done constructively. We now have a second closed, finite, simplicial complex Y', semi-algebraically identified with Y. We claim that the original map T: Y' -+ Y is homotopic to a simplicial map ~': Y ' ---+ Y . Specifically, for each vertex v~ E Y', ~(v~) is in some unique open simplex a i of Y. Choose ~'(v~) to be any vertex of this aj. Suppose r ' is an open simplex of Y' with vertices {v~,..., v'm} and suppose a is the simplex of Y with r ' C T - l ( a ) . Then ~o(~') C ~. Thus, the {~'(v})} are vertices of a, hence span a face of a, which shows that our vertex assignment extends to a simplicial map ~o': Y' --+ Y. From the construction it is clear that ~o' is canonically homotopic to ~, since for any y' E Y', both ~0(y') and T'(y') belong to the same closed simplex of Y.

168

The vertex choices made above axe constructive because of the way Y' was obtained from qo: Y --, Y. Thus, qo' induces a constructively obtained chain map t.: C , ( Y ' ) ~ C . ( Y ) . There is also an algebraic subdivision m a p of chain complexes s. : C . ( Y ) -* C.(Y'), since each simplex of Y is a union of simplices of Y'. The chain m a p s, induces a homology isomorphism. Identifying H , ( Y , Q) = H . ( Y ' , Q) via s , , we have tr(w.) = t r ( 8 , t . ) and the Lefschetz number of , , t , : c . ( r ' ) -* C , ( Y ) ~ C . ( Y ' ) can certainly be constructively computed. The above proof of Proposition 2 might seem to just be an adaptation of the classical proof of the Hopf fixed point theorem, using the existence of simultaneous triangulations of a finite collection of semi-algebraic ~ t s , rather than baxycentric subdivisions. If we just wanted to prove that ~ : Y ~ Y had a fixed point whenever tr(w,) # 0 then we could adapt the classicM argument in this way, and it would not be necessary to appeal to the Tarski principle. (In fact, this is essentially what we did in the proof of the main theorem.) However, the example of ~0 : I --~ I discussed in the last paragraph of the introduction shows that there is not such a close connection between fixed points of ~ : Y --* Y and fixed points of W : Y --~ Y. Thus, something additional is needed to derive the strong conclusion of Proposition 2 that ~ : Y ~ Y has a fixed point. In the chain map , . t . : C . ( Y ' ) -~ C . ( Y ) ~ C . ( Y ' ) which we constructed in the course of the proof, we do not claim that all cells of y i are "moved fax away from themselves" if ~ : Y ~ Y has no fixed point. (This would obviously imply t r ( s . t . ) = 0.) Instead, we simply exploit the facts that t r ( s . t . ) = t r ( ~ . ) and that computation of t r ( s , t . ) is constructive to validate the use of Taxski's principle. There is something a little mysterious about this. The classical Brouwer fixed point theorem over K is much easier to prove than the Hopf theorem. T h a t is, if B " C K n+l is the standard unit disk and ~ : B n --~ B n is any continuous semi-algebraic map, then ~ has a fixed point in B " . Namely, if not, one constructs a continuous semi-algebraic retraction r : B " --~ S n-x = OB" in the usual way, by following rays from ~(x) E B n through x C B n until the rays hit OB n. Thus, r I

I Sn-1

= Id, and the existence of the homology functor is enough

to get a contra ction, since

# 0 and

= 0, [3].

Curiously, it isn't clear whether this proof of the Brouwer theorem for B n is adequate to deduce the Hopf theorem for K n, or, more precisely, for/~l~.. The proof of the Hopf theorem depends on a strong deformation retraction K n ~ Y for some finite simplicial complex Y C K n, which is, of course, contractible. W h a t is not clear is whether Y can always be assumed to be semi-algebraically a closed disk. This seems to be closely 1:elated to the problem of semi-algebraic uniqueness of cone neighborhoods of points in semi-algebraic sets. But, in any case, Proposition 2 applies to Y.

169

References

1. E. Becker, "On the reM spectrum of a ring and its application to semi-MgebrMc geometry", Bull. Amer. Math. Soc. 15 (1986)19-60. 2. G.W. Brumfiel, "A semi-algebrMc Brouwer fixed point theorem for reM affine space", Contemporary Mathematic3 74 (1988), Geometry of Group Representations, 77-82. 3. H. Dells arid M. Knebusch, "Homology of algebraic varieties over real closed fields", Journal f~r die reine und angewandte Mathematik, 335 (1982) 122-163. 4. N. Schwartz, "Real closed spaces", Habilitation~schrift, Miinchen, 1984.

On regular open semi-algebraic sets G. W . Brumflel* I n t r o d u c t i o n . Let k be an ordered field with real closure K. If f 6 k[xl"'x,], let U(f) = {~ 6 K " I f ( ~ ) > 0} C K " Given a finite collection of polynomials {fi} C k[zl.., x,], let U{fi} = N U(fi) C It'". The Finiteness Theorem asserts that any semii algebraic subset of K " which is open (in the local sense that it contains a ball around each of its points) is, in fact, a lqnite union of sets of the form U{fl}. [2], [5], [6], [7], [10]. A regular open set V in a topological space is an open set which satisfies V = V*, where bar indicates closure and circle indicates interior. For any open set V, V* is regular. Also, finite intersections of regular open sets are regular. In the semi-algebraic context of the first paragraph above, let V(f) = ~ C K" and V{fi} = N V(fi) C If". By the Tarski principle, the V(fi) are semi-algebraic sets.

i From the second paragraph above, they are also regular open sets. The goal of this note is to prove a regular version of the Finiteness Theorem. Specifically, we show that any open semi-algebraic set U c K " is a finite union of sets of the form V{fi}. In general, U(f) need not be regular. For example, U = U(~x~) = It'" - (0) is not regular, since V ' = K " . Of course,

= Utr(+x,)

and the V ( + x l ) =

are

regular. It is conceivable that some algorithmic proof of the standard Finiteness Theorem always produces regular U(fi) in the process of decomposing a given open U. The argument we give in this note, producing V(fi), is an existence argument only and does not clarify this fine point of whether one can get by with regular U(fi). We pose this as an open question. Actually, we prove a relative form of the regular Finiteness Theorem. T h a t is, we fix a semi-algebraic subset X C K " . The sets U(f) and V(f) above, as well as closures, interiors, open sets, and regular open sets, now refer to the subspace topology on X. It is not true that a regular open set in K " necessarily intersects X in a regular open set in X. For example, i f X = K 1 C K 2 then U = U(x 2 _y2) is regular in K 2 but Uf'IX = K 1 -(0) is not regular in X . The relative regular Finiteness Theorem requires one assumption on X . For example, if X = [0, oo) C K 1 and if V is any set containing an open interval (0, e) C X , then 0 6 V ° (relative to X). Thus, U = (0, co) C X is not a finite union of regular open sets. We will say that X has boundary of codimension greater than 1 if no point of X has a neighborhood semi-algebraically homeomorphic to a neighborhood of a boundary point in a half-space (of any dimension). Note that algebraic sets have this property, for example, by Sullivan's even local Euler characteristic condition, [11]. Also, if X has boundary of codimension greater than 1 and if X ' is obtained from X by deleting a semi-algebraic subset W with the property that for all p 6 W - W, dim(W,p) < dim(X,p), then X ' = X - W * Research supported by NSF grant DMS85-06816 at Stanford University

171

has boundary of codimension greater than 1. Here dim(W,p) and dim(X,p) mean the local dimensions of W and X at p. It is rather easy to see what the boundary condition above says if one makes use of the existence of "triangulations" of semi-algebraic sets X, [8]. This means that X is semialgebraically homeomorphie to some finite union, Y, of open simpliees (of any dimensions) of some closed, linear, finite simplicial complex in affine space over K. Boundary faces of a simplex of Y need not belong to Y. The condition "boundary codimension greater than 1" on X just says that the picture below cannot occur as an open subset of Y. The simplex a must adhere to at least one other simplex/t ~ r with dim(/~) > dim(a).

~//~/,/..

dim(r) = dim(a) + 1

s

Triangulations Y of X can be chosen so that any preassigned finite fasnily of semialgebraic subsets of X correspond to subcomplexes of Y. We will use triangulations in the proof of the rational, relative, regular Finiteness Theorem, which is stated as follows. T h e o r e m : If the semi-algebraic set X C K n has boundary of codimension greater than 1, then every open semi-algebraic subset of X is a finite union of regular open sets

of the form V{gi}, g~E k[=l--. =.].

R e m a r k : This result was "conjectured", at least in a special case, in [3, page 233], and also at the end of [4]. I would like to thank J. M. Ruiz for some correspondence concerning the proof which follows. P r o o f : The proof will use properties of the real spectrum, SpecR(K[xl • • • x,]), which can be found in [1] and [9]. As far as the "rationality" aspect of the theorem is concerned, that is, finding polynomials with coefficients in k rather than K , the point is that SpecR(K[xl-.. x,]) is homeomorphic, in both the constructible and Harrison topologies, to the subset of SpecR(k[xl ... x,]) which extends the given order on k. Let U C X be open and let W -%X - U._.By the Artin-Lang Theorem, X, U, and W correspond to constructible sets X, U, and W C SpecR(K[=I "'" x,]). By the Finiteness Theorem, f f is open in X. Now, U and W are compact in the constructibleN topology. Fix a E U. We will first construct a certain constructible cover of W and extract a finite subcover. This will lead to a regular open neighborhood V{gi} of a, disjoint from W, that is, contained in U. Since these neighborhoods cover U, which is compact, the Theorem follows. Here, if g E k[xl...x,], we abuse notation somewhat and also write U(g) and V(g) for the constructible subsets of SpecR(g[xl... x,]) corresponding to U(g) and V(g) C Kn. This is so that bars, circles, and tildes don't accumulate too much. The point is, the Artin-Lang correspondence preserves closures and interiors, by the Finiteness Theorem, so V(g) = U(g)* holds in SpecR(K[xl--. x,l), as well as in K " . Consider fl E W. Since W is closed, fl 74 a, where -* indicates specialization of points in the real spectrum. Thus, either a ~ fl or a and fl are not compa~ble, that is, neither specializes the other. In the second case, there exists g# E k[X1 ... X,] such that

172

g#(a) > 0 and g#(fl) < 0. In particular, fl E U(-g#) C U(-g#). In the first case, that is, if a ~ fl, then we claim that there exists g# E k[xl... Xn] such that again g#(a) > 0 and fl E U(-g#). (See Lemma 4.7 of [4].) Assume this claim for the moment. Then we have a cover of W h y constructible sets

U(-g#) with a E U(g#) C V(g#). Extract a finite subcover, say W C U U ( - g i ) . Now, obviously, U(-gi) fl V(gi) = ~ , since V(gi) = U(gi) ° and gi > 0 on U(gi). Therefore, a E NY(gi) = V{gi} is disjoint from W, as desired. It remains to prove the following separation lemma, which is an improvement of Lemma 4.7 of [4]. L e m m a : If X C K n has boundary of codimension greater than 1 and if a ~ fl in .~ with a ~ fl, then there exists g E k IX1 ... X , ] with a E U(g) and fl E U(-g). P r o o f : We claim that because of the boundary codimension assumption, there exists 7 E X with 7 "* fl, but with a and 3' not comparable. Then choose g E k[xl . . . x , ] with g(a) > 0 and g(7) < 0. Since -y E U(-g) and "7 "-~ fl, we have fl E ~ C U(-g). To find 7, triangulate X so that X n Supp (a) and X V) Supp (fl) axe subcomplexes. Here, Supp indicates the irreducible varieties in K " corresponding to points of the real spectrum. The triangulation decomposes )[ into disjoint "constructible simplices". Say fl E and a E "~. Certainly a is a proper face of r, since a ~ fl and dim(r) = dim(Supp (a)) > dim(Supp (fl)) = dim(a). Now, find a simplex/~ ~ r , with a also a face of/~ and with dim(#) = dim(r). This may require retriangulating X slightly if the local picture is like one of the following. /

The first barycentric subdivision of a closed simplicial complex which contains X as a union of simplices will induce a suitable new triangulation of X. In any case, here is where we use the boundary codimension greater than 1 hypothesis. It is now possible to find 7 E p with 7 -* fl and dim(Supp (7)) = dim(/J) = dim(r) = dim(Supp (a)). In particular, a and 7 are not comparable. The point is, semi-algebraically h o m e o m o ~ h i c sets correspond to homeomorphic constructibles, so the structure of ~ C U/2 in X is exactly like the corresponding structure for a pair of linear affine simplices. The linear simplex case is the same locally as a pair of positive coordinate orthants, as below.

lfl !!!!l : Then the existence of 7 is just the observation that any order on a field/5 extends to /5(x), with x infinitesimally small and positive, relative to/5.

173

References

1. E. Becker, On the real spectrum of a ring and its application to semi-algebraic geometry, Bull. A.M.S., Vol. 15, No. 1, (1986), 19-60. 2. J. Bochnak, G. Efroymson, Real algebraic geometry and the 17th Hilbert problem, Math. Annalen 251 (1980), 213-242. 3. G. Brumflel, Partially ordered rings and semi-algebraic geometry, Cambridge University Press, 1979. 4. G. Brumflel, The ultraYilter theorem in real algebraic geometry, Rocky Mountain J. Math. 19 (1989), 611-628. 5. M. Coste, Ensembles semi-alg6briques, Proc. ConL on "Gdom6trie Alg6brique R6ele et Formes Quadratiquds", Rennes 1981, Lecture Notes in Mathematics, vol. 959, Springen (1982), 109-138. 6. C. Delzell, A finiteness Theorem for open semi-algebraic sets with applications to Hilbert's 17th problem, Contemporary Math 8 (1982), 79-97. 7. L . v . d . Dries, Some applications of a model theoretic fact to (semi-)algebraic geometry, Indag. Math. 44 (1982), 397-401. 8. H. Hironaka, Triangulation of algebraic sets, Proc. Syrup. in Pure Math. 29 (A.M.S. 1975), 165-185. 9. T.Y. Lam, An introduction to real algebra, Rocky Mountain J. Math 14 (1984), 767-814. 10. T. Recto, Una decomposici6n de un conjunto semi-algebrdico, Actas del V Congresso de la Agrupacion de Matem£tieos de Expresi6n Latina, Madrid (1978). 11. D. Sullivan, Combinatorial invariants of analytic spaces, Proc. of Liverpool Singularities Symposium I, Lecture Notes in Mathematics, vol. 192, Springer 1871, 165-168.

MEROMORPHIC

SUMS OF 2n-th POWERS OF FUNCTIONS WITH COMPACT Ana Castilla

ZERO SET

1. I n t r o d u c t i o n Geometric characterizations of sums of 2 n - t h powers by means of analytic arcs have been thoroughly studied in different contexts: for polynomials, [Pr], for meromorphie functions on a compact analytic surface, [Kz], and finatly for meromorphic germs and meromorphic functions on compact analytic manifolds of arbitrary dimension, [Rz2]. In all the situations they are based on the so called Becker's valuative criterion, [Be]:

T h e o r e m . Let K be a ~eld and f C K. Then f is a s u m o £ 2 n - t h powers of elements of K i f and only i f f is a s u m of squares in K and for any reM valuatlon v of K , we have 2 n l v ( f ).

The aim of this short note is to give a geometric condition for certain meromorphic functions on an arbitrary paracompact analytic manifold to be sum of 2 n - t h powers, and which generalizes the resull of [Rz]. To be more precise, let M be a connected paracompact real analytic manifold, let O ( M ) be the ring of analytic functions on M and let K be the field of meromorphic functions on M, that is, the field of fractions of O ( M ) . T h e n we show

T h e o r e m 1. Let f E O ( M ) be suc~ that its zero set is compact. Then f is a sum of 2 n - t h powers in K i£ and only if t'or every analytic curve c : [ - e , e] ~ M such that f ( c ( t ) ) = at m + . . . , with a ¢ O, it holds a > 0 and 2nlrn.

Our proof is based on Becker's criterion, Ruiz's result for the local case, as well as the attachment to each ordering ~ E S p e c r ( K ) of an ultrafilter U# of closed semianalytie subsets of M , which allows to give some geometric criteria for a function f to be positive in 3. This m e t h o d was used in JAn-Be] and is similar to the one exposed by P. Jaworski in his contribution to this volume. In fact the results he presents were also independently obtained by us, as he kindly points out. Therefore we omit the background concerning the attachment of N#, which can be seen in either of these references, and will appear in my dissertation, [Ca].

175

2. P r o o f o f t h e r e s u l t . Let Oh(M) denote the ring of b o u n d e d analytic functions on M . Let /3 be an order of K , and let W E be the convex hull of R in K . Then, W E is a valuation ring containing Oh(M) and with residue field R . Let m E denote its m a x i m a l ideal, and let S be the family of closed global semianalytic subsets of M . Thus the set /4E = {X • S i X F1 f - 1 ( [ - - 1 , 1]) 7~ 0 for all f • m E VI Ob(M)}. is an ultrafilter a m o n g the filters of sets of the family S, cf. [Jw, P r o p 1]. The main feature we are going to use is the following

L e m m a 1. Let f • Oh(M) and assume that there is X • Lt~ such that from below over X . Then f is a unit in W E.

If[

is bounded

Proof: [Jw]. Before entering in the proof of T h e o r e m 1, we need a technical lemma.

L e m m a 2. Let X C M be a dosed globM semianalytic set, and let f • Ob(M) be such

that fl X > O. Then for each n • N there are u, g • Ob(M) such that Ulx > O, u is bounded from below over X and u f = gn.

Pro@ Let v* : M ~ R be a continuous function such t h a t v*(x) > 0 for all x E M , and V i x = f n - 1 , which exists by Tietze's Theorem. Now let v : M ~ R be an analytic a p p r o x i m a t i o n of v* such that Iv(x) - v*(x)l < l v * ( x ) . We set ul = f ' ~ - l / v and u = u l / ( 1 + u21) • Oh(M). Since for x • X we have t h a t u l ( x ) > 4 f n - l ( x ) / h v * ( x ) = 4/5, we get t h a t u is b o u n d e d f r o m below on X . Now, u f = f'~ ( l / v ( 1 + u~)), and since v(1 + Ul2) is strictly positive, we have t h a t v(1 + u 2) = h '~ for some h • Oh(M). Altogether we get u f = gn, where g = f / h .

Proof of Theorem 1: Necessary condition. If f is a sum of 2n-powers, obviously f is nonnegative on M , so t h a t the only thing to be shown is the condition on the divisibility of the exponent m by 2n. Take any representation of f as sum of 2n-th powers:

f ~ E gi2nl~2n /n ,

withgi, hEO(M),h¢O,

M be an analytic curve. We set x = c(O). If h(c(t)) # O, since hZn(c(t))f(c(t)) = E gi(c(t)) 2n then

and let c : [O,s] ~

2n ord(h(c(t))) + ord(f(c(t))) = 2n ord E

gi(c(t))

176

and easily we get the result. In general, we need to use suitable valuations of the field K associated to the point x. Let m be the m a x i m a l ideal corresponding to x. The curve c defines a sequence

O(M)~, ¢--+O , ( M ) C-~R{t}, where ¢ ( f ) is the expansion at 0 of f o c and O , ( M ) denotes the ring of analytic germs at the point x. Let q = (ker¢) n O ( M ) m = ker(¢ o i). Thus, the h o m o m o r p h i s m ¢ induces an embedding k(q) ~ R({t}), where k(q) is the quotient field of O(M)r,/q and R ( { t } ) denotes the quotient field of R { t } . W e restrict the ordinary valuation of R ( { t } ) to a valuation ~ of k(q). In particular the class i in k(q) verifies 9 ( i ) = o r d ( f o c) = rn. Since O(M)q is regular, there is a valuation w of K with residue field k(q). Finally let v be the composite valuation of 9 and w. We have F~, = rv/r . Since f is a sum of 2n-th powers in K , by Becker's criterion 2n divides b o t h v(f) and w ( f ) , and therefore 2n divides ~ ( f ) = m = ord(f(c(t))).

Su~cient condition: Assume t h a t f ~ ~ I ( 2n. If f(x) < 0 for some x 6 M , then for any curve c: [0, c] --* M with c(0) = x, we have f(c(t)) = at m + . . . with a < 0. Therefore we m a y assume t h a t f is nonnegative over M. Then, by Artin-Lang, see [Jw], [Ca], f is a sum of squares. In particular, it follows from Becker's criterion, t h a t it exists a real valuation w of K such t h a t 2n does not divide w(f). Let fl be an ordering of K compatible with w, and let V = W E ( R , K ) be the convex hull of R in K with respect to ft. Thus, V is contained in the valuation ring of w. In particular Fw = F v / A where v is the valuation defined in V and A is an isolated subgroup of Fv. Therefore, 2n can not divide v(f) . Now, let m E be the m a x i m a l ideal of the ring V and let L/E be the ultrafilter of global closed semianalytic sets defined by

ldE = { X 6 S i X FI f - 1 ( [ - 1 , 1]) # 0 for all f 6 m E F10b(M)}. If

f--1 (0) ~ ~'~E,there

exists X e / 4 E such t h a t f [ x >

0. T h e n we can apply L e m m a

2 to the function f : there exist a unit u in V, and g 6 Oh(M) such t h a t u f = g2n. In particular 2n divides v(f), contradiction. Next, assume that f - l ( 0 ) 6 b/E. Since f - l ( 0 ) is compact, necessarily/~E is the principal ultrafilter defined by a point, say p 6 M . In particular ~9 is centered at p, t h a t is, the m a x i m a l ideal mp of p is convex with respect to }9. Now, we have :

O(M),,

O,(M)

177

Since, V dominates O ( M ) m , we can argue as in [Rz] to find and analytic curve germ c: [0, ~] --* M with c(O) = p and such that f ( c ( t ) ) = at m + . . . , where 2n does not divide m.

3. R e f e r e n c e s .

JAn-Be] C. Andradas, E. Becker: "A note on the Real Spectrum of Analytic functions on an Analytic Manifold of dimension one". Proceedings of the Conference on Real Analytic and Algebraic Geometry (Trento 1988), Lect. Notes in Math. no. 1420, 1-21, Berlin-Heidelberg-New York, Springer, 1990. [Be] E. Becker: "The real holomorphic ring and sums of 2 n - t h powers", in Lect. Notes Math. 959, Berlin-Heidelberg-New York, Springer, 1982. [Ca] A. Castilla: Dissertation, in preparation. [Jw] P. Jaworski: "The 1 7 - t h Hilbert problem for noncompact real analytic mangolds", this volume. [Kz] W. Kucharz: "Sums of 2 n - t h powers of real meromorphic functions", Monatshefte fiir Mathematik. 107 (1989) 131-336. [Pr] A. Prestel: "Model theory applied to some questions about polynomials", Proceedings of the Salzburg Conference, May 29 - June 1, 1986. [Rzl] J. Ruiz: "On Hilbert's 1 7 - t h problem and real NullsteUensatz for global analytic functions", Math. Z. 190, 499-514 (1985). [Rz2] J. Ruiz: "A characterizatization of sums of 2 n - t h powers of global meromorphic functions", Proceedings of the A.M.S. 109, 915-923 (1990).

P~zuDOOR~KOGONALITY OF POWERS OF ~ OF A HO~RPHIC

[~wt]gPING I N T W O

WITH

I,.K ~

COORDINATES VARIABLES

JACOBIAN

Zyn~,ut Charzy6ski a n d P r z e m ~ s l a w S k i b i 6 s k i

Summary.

In the paper the e q u i v a l e n c e

bian of holomorphic mapping a characterization mapping

is showed.

of holomorphic m a p p i n g

There is

obtained

w i t h the constant

This note is d e v o t e d

Jacobian of

the

to

h o i o m o r p h i c mappings with

In section 1 there is given an auxi ~liary infor-

In section 2 it is showed that the c o n s t a n c y

is equivalent coordinates coefficients

to the integral p s e u d o o r t h o g o n a l i t y

of the mapping

(theorem I).

Also

ficients are investigated

of

of

the

of the inverse mapping are derived

3 more general expressions

powers

presenting the coefficients

section are

and auxiliary

later precised.

a

obtained, reduced

to

those re-

(theorem 4).

These

Here a characterization

under consideration by one of its coordinates

be a number fixed for the sequel and

Er

is

in particular,

of the inverse m a p p i n g

in the next section.

notations

the

the mentioned coef-

values of the second one on one of the axes is obtained I. Terms,

the

(theorem 2). In section

some integrals of functions of two variables

the mappings

of for

(lemmas i, 2 and theorem 3). In section 4

some integrals of functions of one variable,

facts are applicated

the Jacobian

formulae

than those r e p r e s e n t i n g

simplification of the formula from the previous

{1.1)

Jaco-

to one of the axes.

Introduction.

namely,

also

and the restriction of the other coordinate

the constant Jacobian. mation.

of

in two v a r i a b l e s with the pseudoorthogon-

ality of powers of its coordinates by one coordinate

of the constancy

infqrmation, r > 0

and

Dy

of the

(theorem 5).

i. Let

0 < ~ < 1

a number

which

lyl

~ <

will be

Let further

{(x,y)

~ ~2

: 0 <

Ixl

< r,

The set (I.i), as one can easily see, p. 299).

0 <

< r,

I

<

is a Reinhardt domain (com~

~}[2],

179

Let be given the pair of plain circles C : x(s) = ~e is,

0 & s ~ 2~,

~ > 0,

D : y(t) = ~e it,

0 ~ ~ ~ 2~,

~ > 0.

(1.2)

This pair, treated as a mapping in two variables is called the bicircle with the radii (~,q) ~ E r

i.e.

2. Let point

(P,Q)

(0,0)

(1.3)

the bicircle

~,n.

Pj

and

shall

its

assume that

E r-

x,y

with the coordinates of the form

Qj

Q(x,y) = y + Q2(x,y) + ...

denote homogeneous polynomials of the degree

It is easily seen that from form (1.3) of the mapping follows that the Jacobian of this mapping at the point to

I.

x,y,

Thus, in a neighbourhood of the point the mapping is invertible.

Denote by

verse mapping. Then in a neighbourhood variables (1.4)

z,w

variables

(U,V)

of

~ CKlZKW l, k,l=0

(0,0)

it

is equal

the holomorphic in-

the point

V(z,w)

the values of

(P(x,y),Q(x,y))

lately mentioned neighbourhood of the variables

x =

(P,Q)

of the variables (0,0)

=

of

the

Z dklZKwl. k,l=0 (0,0)

in the

are contained

in the

in a certain neighbourhood of the point

x,y,

Of course,

(0,0)

j.

the coordinates of this mapping expand in the series

U(z,w) =

Simultaneously,

range

be a holomorphic mapping in a neighbourhood of the

in the variables

and

We

(1.2) is lying in

P(x,y) = x + P2(x,~) • ....

where

s,t,

in this situation,

z,w.

there is

5q Ckl(P(x,y))k(Q(x,y)) I, k, l=0

(1.5) y=

dkl(P(x,y))k(Q(x,y))l. k, i=0

Now, let us notice that for

r

sufficiently small the set

is contained in the neighbourhood of the defintion of Similarly,

for sufficiently small

r

(P,Q).

the following

inequalities

hold Pj(x,y)

(1.6)

I x j= 2

x

I < i,

(l.1)

co Qj(x,y) I x I < i; j=2 Y

180

at the same time we have

1 P(x,y)

~

1

=!

- Pj (x,y) ) 7 j=2 x

x(l +

x

Pj (x,y))g

(-

x g=0

j=2

x

(1.7)

1

Qj(x,y) y(l +

and,

in consequence,

morphic (comp.

in

Er

) Y

the functions

i/P(x,y)

and expand there in Laurent

the first of the series pears

7. j=2

[2], p. 301). Moreover,

tive powers

= ! 7. (_ ~ Qj(x,y))g Y g=0 j=2 Y

1

Q(x,y)

in (1.7) the variable

powers only;

terms of degree not smaller 3. Simultaneously

than

in (I.i)

i/Q(x,y)

it is easily seen from

only and in the second series

in non-negative

and

series centred

in -!

y

(1.7)

appears

in (1.7)

both series

are holoat

(0,0) that

in non-nega-

the variable there

in

x

ap-

appear

the

only.

there exist some single-valued branches

of logarithms log P(X,y) ....... X

_

7. g=l

log Q(x,y) Y

=

Z (-l)g-i g=l g

P j (x,y) ~ ( 7. )g, j=2 x

(-i) g-I g

(1.8)

Similarly,

as above,

the branches

the first one the variable

y

in the second one the variable in both series 1

~ Qj(x,y) ( Z )g. j=2 Y

there appear

(1.8) expand in Laurent series;

appears x

in non-negative

appears

the terms

of

powers only and

in non-negative degree

not

in

powers only;

smaller

than

only. We easily check that the functions (log ~ + is) + log P([eiS'qeit) it ~e

(1.9) (log q + it) + log Q([eiS'neit).... it ne

0 ~ s ~ 2#, 0 & t & 2~,

181 are some branches of the logarithm circle

P(x,y)

(1.2). We shall denote them b y

and

Q(x,y)

log P(x,y)

we see that if one of the variables

and

then

log P(x,y)

is a branch of the logarithm of the function

P(x,y)

along

the circle

C

2wi

or

y

log Q(x,y). fixed,

D

and

bi-

is

or

x

"along"

and its increment along this circle is equal

Analogously for the branch of logarithm

0.

of

to

the function

Q(x,y). 4. By

WT

we shall denot~ the linear W h i t n e y operator (com~ [1])

assigning to any holomorphic function point

(0,0)

the expression

S

in a neighbourhood

SxTy - SyT x.

function in this neighbourhood,

Here

T

fixed for a moment.

of

the

is a holomorphic The iterations

of

Whitney's operator we shall denote as powers.

In particular,

for the

operator

the considered

mapping

(P,Q)

WQ

where

Q

is the coordinate

of

we shall apply the following notations

(i.i0)

W~(S)~ = Sj,

j = 0,i .....

In the sequel as

r

S o = S.

we choose an arbitrary fixed number so small

that in the set

E relation (1.6) and representations (1.5), (1.7), r (1.8) and the corresponding Laurent expansions hold as well as the set

Er

is contained in all neighbourhoods of the variables

in the points

2, 3

and

x,y mentioned

4.

2~ Pseudoorthoqonality.

Let us adopt the constructions, notations

and agreements introduced in the previous section. THEOREM I.

For the functions

(1.3) the following Jacobian condi-

tion (2.1)

PxQy - PyQx = 1

holds if and only if for every integers

k

and

1

the

following

pseudoorthogonality relations hold

(2.2) (2,i) 2 P r o o f. (2.3)

1 (2~i)2

f

1

(f

C

dy)dx = ~ 0 for (k,l)@(0,0),

D (p(x,y))k+l(Q(x,y)) I+I

First,

~ i for (k,I)=(0,0).

let us notice that the following equalities hold Px(X'Y)Q? (x'y) - p y ( X ' Y ) Q x ( X ' [ )

f(_/"

C

D

......

(P(x,y))k+I(Q(x,Y)) I+I

dy)dx

182

=f0

Indeed,

for

(k,1) @ (0,0),

for

(k,l) = (0,0).

assume that

(k,1) @ (0,0).

Consider

the

case

k @ 0.

Passing in the first of the integrals in (2.3) to common integrals with respect to the parameters

s

and

t

by the substitutions

(1.2)

and

applying the integration by parts we have (2.4)

f(f D C

P (x,y) x (P(x,y)) k+l

1

Qy(X,y) (Q(x,y))l+ 1 dx)dy

1 Qyx (x'y) (I+I)Qx(X'Y)Qy(x'Y) C (P(x,y) )k ( (Q(x,y) )l+l (Q(x,y))l+ 2 )dx)dy

= ~ fD( f and

(2.5)

f

Pv(X,Y) )k+l (P(x,y)

(f C D

Qx(X,Y) i+i dy)dx (Q(x,y)) (l+l)Qx(X,y)Qy(X,Y) .............. (Q(x,y)) I+2

I 1 Qxy (x'y) = ~ f (f k ( )i+I C D (P(x,y)) (Q(x,y) Subtracting

(2.4) and (2.5) and changing the order

of

)dy)dx.

the integration

in (2.4) we obtain the first equality of (2.3). In the case

1 @ 0,

we proceed analogously.

Assume that

(k,1) = (0,0).

Applying

again

the integration

parts and taking into account the remark concerning branch of logarithm (2.6)

f D

(f C

(comp. section 1.3),

we have

Px(X,Y) P(x,y)

- log

Qv(X,Y) dx)dy = f (log Q(x,y) D Qy(~,Y) P(~ei0,y)) Q(~,y) dy Qyx(X,y)Q(x,y)

- _F ( f log P(x,y) D C

(2.7)

f (f C D

Py(x[y) P(x 'Y)

Qx(X,y) Q(x,y)

• - log P(x,qe~0))

by

the existence of a

P(~ei2~,y)

- Qy(x,Y)Qx(X,y) dx)dy,

(Q(x,y)) 2

dv)dx = f C

Qx(X,~) Q(x,q)

dx

(log P(x,~ei2nj"

183

-

.F(_F C D

log P(x,y) Qxy(X'Y)Q(x'Y) - Qx(X'Y)QY(x'Y) (Q(x'Y)) 2

dy)dx.

At the same time we have (2.8)

Qy(~,y) Q ( ~ , y } dy =

f

• , , • log Q(~,~el2"j - log Qt~,~el0).

D Subtracting

(2.6) and (2.7), changing the order of integration in (2.6)

and taking into account

(2.8) as well as remarks concerning the incre-

ment of the logarithm of the function

P

or

Q

(comp. section

1.3),

we obtain the second equality from (2.3). The necessity of relation ting,

(2.2) is immediately obtained, substitu-

according to (2.1), the numerator of the integrand in (2.3)

by

i. The sufficiency follows from the following observations. First,

let us notice that according

to the choice of

r

(comp.

section 1.4) in E r the representations (1.5) hold. Consequently, the Jacobian of the mapping can also be represented in the form of the following series (2.9)

Px(X,y)Qy(X,y)

- Py(X,Y)Qx(X,y)

= h,J=0x ehj(P(x,y))h(Q(x,yJ) j

Next, by the assumption that the bicircle Er

(1.2)

is

contained in

and by (2.9) we get

(2.10)

f C

( f D

=

Px(X,y)Qy(x,y) - Py(x,Y)Qx(X,y) )k+l )l+l (P(x,y) (Q(x,y)

Z f(f h,j=0 ehj C D

(P(x,y))

dy)dx

1 k+l-h(Q(x ' l+l-j dy)dxy))

Since, according to (2.2), the integrals on the right-hand side of (2.10) vanish for all (h,j) # (k,l), then by (2.10) and (2.3) we have I~ ekl =

for

(k,l) ~ (0,0),

for

(k,!) = (0,0).

The above and (2.9) give (2.1).

184

THEOREM 2.

For the mapping

the coefficients

Ckl,dkl

(comp.

(P,Q)

fulfilling condition

(1.4)) of the inverse mapping

(2.1), (U,V)

are given by the formulae 1 Ckl = ~ (2~i)2

f(c C D

(P(x,y))k+l(Q(x,y) )l+l dy)dx,

1 dkl = (2~i)2

f(j C D

Y dy)dx. (P(x,y))k+l(Q(x,y)) l+l

x

(2.11)

P r o o f. From the definition and from (1.5) we obtain x --

y

Z k, I=0

=

of

(U,V)

(comp.

section

(1.2)

Ckl(P(x,y))k(Q(x,y))l,

dkl(P(x,y))k(Q(x,y))l, k,l=0

whence, following the analogous pattern as in the proof of the preceding theorem and by this theorem, we obtain (2.11). 3. Generalization and reduction of degree. structions, notations and agreements introduced

Let us adopt the conin section i.

Here we shall assume (2.1).

(1.3) fulfil condition

that

the functions

LEMMA 1. Let k be a positive integer, 1 - an integer, M - a holomorphic function in a neighbourhood of the point (0,0) containing Er .

(3.1)

Then 1 (2~i)2 fC( • D 1

I

(2~i)2

k!

P r o o f. (3.2)

~(f D C

=

M(xry) ........ (P(x,y))k+l(Q(x,y)) l+l dy)dx 1

f(f C D

P(x,y)

Mk(X'Y)

(Q(x,y)

)l+l dy)dx"

Applying the integration by parts we have Px(x,y)

Qy(x,y)M(x,y)

(P(x,y)) k+l

(Q(x,y)) I+I

dx)dy

1 T_r(: D

1

C (P(X,y))

(Qyx(X,y)M(x,Y)

+ Qy(x,y)Mx(X,Y) l+l (Q(x,y))

185

(i + l)Qy(x,Y)Qx(X,y!M(x,y) )dx)dy, (Q(x,y)) 1+2

(3.3)

Py(x,y) "f(-f )k+l C D (P(x,y) I

(Qxy(X,y)M(x,Y)

+ Qx(X,y)My(x,y) l+l (Q(x,y))

1

-f(f D C

= ~

Qx(X,y)M(x,y) )i+i dy)dx (Q(x,y)

(P(x,y))

k

(I + l)Qx(X,y)Qy(x,y)M(x,Y))dx)dy" (Q(x,y)) I+2

Now, let us denote by (3.1). Subtracting (I.i0), we get

Ikl(M)

the integral

on

the left-hand side of

(3.2) and (3.3) and taking account

f(f C D

(2.1)

and

MI(x,Y) i+l dy)dx (Q(x,y))

! Ikl(M o) = ~

of

k (P(x,y))

1 ~ Ik-i I(MI )-

=

Hence, by induction, we obtain (3.1). LEMMA 2.

Let

1

be an integer,

neighbourhood of the point positive integer, v (3.4)

1 (2~i)2

(0,0)

N - a holomorphic function

containing

E r.

Then

for

in

a

every

the following equality holds

f(f ! N(X'7) dy)dx C D P(x,y-----~ (Q(x,y)) i+i 1 2~i

v (_l)h-! h-Z-I ( ~ [

× Qy(~'Y)Nh-I(~'Y) (Q(~,y))l+!

× [log P(x,y)

h-I fD

dy +

- (i+½

(P(~'Y))

1 t~-1.w (2~i) 2 ~ f ( f ( P ( x ' Y )D) V - ! C

+ . . . + v-ll) ]

Nv(x,Y) i+i dy)dx (Q(x,y))

where we assume that 0! = 1 and for v = 1 the expression in the square brackets on the right-hand side of (3.4) reduces to the first summand (for

N

compare also sec. 1.4 (I.i0)).

186

P r o o f. 1°

We shall do it by induction

Integrating by parts and taking into account the remarks con-

cerning the branch of logarithm

(3.5)

Px(X,y) P(x,y)

D-F(~

Qy(x,y)N(x,y) (O(x,y))l+ 1 d x ) d y

• (log P(~el2~,y)

= f D

(comp. section 1.3), we obtain

Qy(~,y)N(~,y) - log P(~ei0,y))

dy (Q(~,y))l+l

(Qyx(X,y)N(x,y) D

+ Qy(x,y)Nx(X,y) l+l ........ (Q(x,y))

(log P(x,y))

f(f

-

C

(I + l)Qy(x,Y)Qx(X,y)N(x,y) (Q(x,y))l+ 2

(3.6)

Py(x,y) P(x,y)

f(f

C

D = f C

Qx(X,y)N(x,y) l+l (Q(x,y))

........dx)dy,

dx)dy

(log P(x,ne i2~) - log P(x,Nei0))

Qx(X'~)N(x'~) l+l

,dx

(Q(x,n)) Qxy(X,y)N(x,y)

-

f(f

C

D

+ Qx(X,y)Nv(X,y) 1+2 (Q(x,y))

(log P(x,y)(

(I + 1)Qx(X,y)Qy(x,y)N(x,y) . . . . 1+2 (Q(x,y)) Subtracting

dy)dx.

(3.5) and (3.6) and taking into account

marks concerning

(2.1)

and

the increment of the branch of the logarithm

the reof

P

(comp. section 1.3) as well as (i. I0), we get

(3.7)

f(_F C D

1 P(x,y)

~ N(x,y) )l+l dy)dx = 2~i f (Q(x,y) D

- f(f log P(x,y) C D which gives 2°

(3.4) for

dy

NI(X,Y) 1+2 dy)dx, (Q(x,y))

u = I.

AssuJne that (3.4) hold for

~.

Applying

by parts to the second integral on the right-hand tain

Qy(~,yJN(~,y) l+l (Q(~,y))

the

integration

side of (3.4), we ob-

187

(3.8)

J-¢s D

CP¢x,y))~-~[log P¢x,y) - ¢~ + ½ +

... +V~_~)]~x(X,y)

C

×

Qy(x,y)Nv(X,Y) (Q(x,y))l+ 1

dy)dx = fD [~(P(~ei2~'Y))V

log P (~ei2~,y)

_ 1 2 (P(~ei2~'Y) )v - (1 + ½ + "'" + v 1- i)i (p(~ei2~,y))v _

! (p(~ei0,y) v log P(~ei0,y) + ~

(P(~ei0,y))

V

÷ (I + ½ + "'" + v -i i) $i (p(~ei0,y))V] Qv(~,Y)Nv(~,Y) [!(P(x,y)) v log P(x,y) (Q(~,y))l+l dy - "/'(-/'D C

-

--~ (P(x,y)

-

(i +

+ ... +

) ~-

%;

× (Qyx(X'y)Nv(x'Y) + Qv(x'y)Nvx(X'Y) (Q(x,y)) !+I (i + l)Qy(x,Y)Qx(X,y)Nv(x,y ) )dx)dy, (Q(x,y)) 1+2

(3.9)

f(f C D

(P(x,y))v-l[log P(x,y) - (i + Qx(X,y)Nv(x,Y) (Q(x,y)) l+l

x Py(x,y)

1 + ... + v - 1 ]

dy)dx

_/" [~1 (P(x, qei2, )) v log P(x, qei2~ ) - 1~(P(x,qei2n)) v C v - (i + 1 + ... + IV_ _ _ ~ ) U 1 (p(x,Dei2n) )v _ !v (P(x'qei0))v log P(x,vei0)) + ~12(P(x,qei0))~

+

(I

+

1

~

+

...

+

l__!__) -1 (p(x, qei0))v] v 1

Q.(x,n)N (x,q) dx i+i (Qlx,q))

- f(./D C

[!v (P(x'y))V log P(x,y) - --12(P(x,y))V v

-

÷

-,- ½

...

-,-

188

x (Qxy(X'y)Nv(X'Y)-

+ QxtX'y)Nvv(x'Y)'-~

(Q(x,y))I+I (I + I ) Q x ( X , Y ) Q v ( X , y ) N

(x,y)

)dx)dy.

(Q(x,y)) 1+2

Subtracting

(3.8)

the i n c r e m e n t

and

(3.9),

taking into a c c o u n t

of a b r a n c h of l o g a r i t h m

(comp.

(2.1)

section

and remarks 1.3) as

well

on as

(i. I0), we o b t a i n

(3.10)

f(.F C D

x

(P(x,y)) v-I

[log P(x,y)

N (x,y) ~ 2~i (Q(x,y))l+ 1 dy)dx .... v

!

- f ( w r ~ (P(x,y)) C D

v

- (i + ½ + "'" + v 1 - i) j

f "D

Qy(~,y)Nv(~,y) i+i (Q(~,y))

(p(~,yj)V

~

(log P(x,y)

dy

1

- (i + 2 + "'" + ~))

Nv+I(x,Y) i+i dy)~x.

(Q(x,y))

Putting

(3.10)

into

The i n d u c t i o n THEOREM

3.

(3.4) we o b t a i n

(3.11)

Let

(0,0) 1

k

be a n o n - n e g a t i v e

M - a holomorphic

containing

f(f

(2~i) 2

for

v + i.

ends the proof.

v - a positive integer, the p o i n t

(3.4)

C

1 2,i

D

E . r

(2~i) 2

function

1 - an integer,

in a n e i g h b o u r h o o d

of

Then

M(x,y) dy)dx (P(x,y))k+l(Q(x,y)) l+l

I k!

v 5h= 1

h-i (-i) (h - i) !

x QY(~'Y)Mk+h-I(~'Y) (Q(~,y)) i+i

+__I

integer,

(-I) ~ k!

× (log P(x,y)

fC( f D

f D

(p(~,y))h-i

i (v - i)'

( P ( x , y ) ) v-I

dy

- (I + ½ + ... +

v

1 -

i))

Mk+~(X,Y) !+! (Q(x,y))

dy) dx.

189 P r o o f. and p u t t i n g 2,.o.

Putting

in (3.4)

in (3.4)

N = Mk,

N = M,

by

(3.1),

we o b t a i n we o b t a i n

(3.11)

(3.11)

for for

k=0, k = i,

.

4. R e d u c t i o n notations

to s i n g l e

and a g r e e m e n t s

THEOREM

4.

v - a positive

Let

in a n e i g h b o u r h o o d (4.1)

1 (2~i)2

introduced

k

integer

(Co,Do)

~o,qo,

(0,0)

1 k:

lying in

I.

integer, ,

1

- an integer,

M - a holomorphic

containing

E r.

function

Then

M(x,y) ....... dy)dx ( P ( x , y ) ) k + l ( Q ( x , y ) ) l+l

v • h=l

h-i (-i) (h - I) :

f D

(p(0,y))h-I o

dy

is an a r b i t r a r y b i c i r c l e

P r o o f. (4.1)

in s e c t i o n

v > 1

of the p o i n t

× Q y ( 0 ' y ) M k + h - i (0'y) l+l (Q(0,y)) where

L e t us a d o p t the constructions,

be a n o n - n e g a t i v e such that

.C (f Co Do

1 = 2~i

integrals.

of the type

(1.2) with the radii

E r. Let us c o n s i d e r

also the circles

besides

appearing

the

circles

appearing

in

in (3.11).

We shall base our r e a s o n i n g

on the four m e n t i o n e d

below

observa-

tions: (i)

The term on the l e f t - h a n d

term on the l e f t - h a n d given in s e c t i o n x,y.

side of

1.2 the i n t e g r a n d

In this s i t u a t i o n

tegrals values

(il)

as one can see,

x

is the sum

and

equal.

(4.1)

Indeed,

the c o n s i d e r e d

of terms c o n t a i n i n g are,

side of

(3.11).

y

of

integrals

side of

1 2.i

x

i k!

v ~ h=l

_l)h-i ( (h - 1)!

Qy(O,y)Mk+h_I(O,Y) (Q(O,y)) l+I

dy

JD

the facts

a L a u r e n t series in

-i

to the inand

their

the announcement. (4.1)

the t e r m

(4.1)

to

are r e d u c e d

in the powers

This gives

The term on the r i g h t - h a n d

is identical with the

according

(P(O'y))h-I

is

identical

with

190

arising

from

(4.1) by substitution

Indeed,

it follows immediately

ly small the rational poles

of the circle

(iii) The difference and term

in (4.1)

let us represent 1 2~i

(4.2)

between

(4.1")

and

(4.1)

v X h=l

f D

~,q

difference

h-i (-I) (h - l)l

× Qy(~'Y)Mk+h-l(~'Y) (Q(~,y))l+l

r

does

Do

sufficientnot

and

have any

D.

the first term on the right-hand

tends to zero as

the investigated I k!

by the circle D.

from the fact that with

integrand

in the closed ring bounded by the circles

of (3.11)

Do

tend to

0,0.

side

Indeed,

in the form )h-i

[(p(~,y)

-(p(0,y)) h-I

x : QY (0,y)Mk+ h !(0,Y) ]dy. (Q(0,y))i+i

According

to the facts given in section

pressions

in the square brackets

series

in

~,y

with the terms divisible

sitive powers of

~.

In this situation

to the sum of integrals gree

-i

respect tends to

with respect to

~ 0

1.2 we easily see that the ex-

in (4.2)

~

by

y,

i.e.

~,q

tend to

tends to

0,0.

powers of

0.

indeed,

(v ~l)! [(P(X'y))V-I

- (i + ½ + 1 + (v- i)~

5.

according

x,y

side of

(3.11)

can be represented

this

sum

tends to

0

in the form

Mk+v(x'Y) P(x,y) (Q(x,y))l+ 1 (log x

... + l__~_f))] [(P(X,y))

v-i

Mk+~(x'Y) (Q(x,y))l+l]

{log ~ + is}.

v > 1

the expressions

in (4.3) are the sums of some Lauren~ series in

with the terms of degree not smaller

the integral

Of course,

dewith

to the facts given in section

One can readily see that by the assumption in the square brackets

the po-

This gives the announcement.

1.2 and 1.3 we see that the integrand

(4.3)

so, containing

to the sum of a power series

(iv) The second term on t h e right-hand as

~

the considered integral reduces

of the terms of the series which have the to

with the positive as

are the sums of some Laurent

of the first summand

from

than

(4.3)

-I.

reduces,

In this situation of

course,

to

191

zero. On the other hand, the integral of the second summand from (4.3) with respect to to

x

y

reduces to the sum of a power series

multiplied by the factor in the brace.

Hence

mediately that the considered integral is of the type tends to

0

as

~,Q

tend to

0,0.

with

it

respect

follows

im-

0(~ log ~)

and

This gives the announcement.

Passing to the demonstration of the assertion of the theorem, see that according to observations

(i),

(ii), equality

(3.11)

can

we be

represented in the form (4.4)

1 (2~i) 2

×

1

1

2~i

k!

~ X h= 1

(-i) h-I (h - 1)!

f (P(0,y)) Do

Qy(0'y)Mk+h-i (0'y) 1 dy + (Q(0,y))l+l 2~f

× f D

-

M(x,y) dy)dx (p(x,y))k+l(Q(x,y)) I+I

f (f CO DO

(p(~,y))h-i

Zv h= 1

(-l)h-i (h- i)'

k!

× (log P(x,y)

[ X

Qv(~'Y)Mk+h-!(~'Y) (Q(~,y))l+l

dy

f(f

C

(Q(0,Y)) l~'i

i

D

(v - 1)! Mk+~(x,Y) l+l dy)dx. (Q(x,y))

- (1 + ½ + "'" + v 1 l) )

we obtain, according to observations

(iii)

and

~,n

tending to

(iv),

the

0,0

required

(4.1).

5. Characterization of a mapping by values axis.

dy]

(P(x,y)) v-i

Passing to the limit on both sides of (4.4) with equality

(h - i)'

h=l

)h-I Qy(0'y)Mk+h-i (0"y)

~I,)~

(2ni)2

(_i) h-I

1 k[

fD (P(0,y)

"

+ ~

h-i

Let us adopt constructions,

of

a

coordinate

on

notations and agreements introduced

in section i. THEOREM 5. 1

The mapping of form (1.3) with the Jacobian equal to

is uniquely characterized by the

first one

P

second coordinate

restricted to the axis of ordinates.

Q

and the

192

P r o o f. or

M(x,y)

represents (P,Q).

= y

Let us notice that with the specification the left-hand

the coefficient

on the other hand,

side of

Ckl

or

Thus,

of

Mk+h_ 1

ordinates.

Q

eral variables.

of

of

M(x,y)

of

on an~

on the second

the by the

coordi-

is uniquely characterized P

restricted

the assertion

by

to the axis of

of the theorem.

for the mappings

in Bulletin

de

Serie: Recherches

Narutowicza

= x

theorem 2,

the inverse mapping

- depends

(P,Q)

It will be published

of the editor:

to

side of (4.1) depends

1 and 2 can be generalized de L6d~,

according

on the axis of ordinates

(I.i0))

immediately

et des Lettres

(the address

P

and by the coordinate

This gives

The theorems Sciences

(comp.

the inverse mapping

the coordinate

dkl

the right-hand

values of the first coordinate definition nate Q.

(4.1),

of

sev-

la Soci~t~

des

sur !es deformations

56, 90-136 56d~, Poland).

Bibliography

[1]

Charzy~ski

Z., Ch~dzydski

J., Skibidski

Keller's Jacobian conjecture 39 (58) (1989), 1-8. [2]

ma6aT B.B.

Bse~eHHe

Institute of Mathematics University of 56d~ ul. S. Banacha 22 90-238 L6d~, Poland

iII,

B KOMnneRCHbl~

Bull.

p.,

A

Soc.

aHanH3,

contribution

Sci. Lettres

MOCRBa

1969.

to

Ldd~,

Trivialit6s en famille Michel Coste et Miloud Reguiat IRMAR (CNRS, U.R.A. 305), Universit6 de Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France

Introduction On s'int6resse ici a des probl6mes de trivialisation de families d'ensembles ou de fonctions semi-alg6briques. On utilise en quelques endroits les techniques du spectre r6el, qui sont bien adapt6es ~t ce genre de probl~mes : elles perrnettent de traduire des r6sultats obtenus de mani6re 616mentaire pour un corps r6el clos quelconque en r6sultats pour les families. C'est ce qu'on peut appeler "cacher les param~tres dans le corps de base". Une premi6re illustration de ce principe concerne la construction de triangulations lipschitziennes de compacts semialg6briques. La lipschitzianit6 ne concerne qu'un sens : du complexe simplicial h l'objet triangul6. On en donne une d6monstration dans la premiere section. On montre dans la deuxi~me section comment en d6duire une version lipschitzienne du th6or6me de trivialisation semi-alg6brique de Hardt [HA], et les classiques bomes m6triques uniformes pour les families. Le m~me r6sultat donne darts la troisibme section des trivialisations semi-alg6briques, au dessus d'un intervalle ]0, ~], de fonctions semi-alg6briques propres, qui peuvent se prolonger par continuit6 en 0. Ceci pourrait aussi s'obtenir en utilisant le th6or6me de triangulation des fonctions semi-alg6briques de Shiota [SH]. Ce demier th6or6me, pass6 ~ la moulinette du spectre r6el dans la quatri6me section, fournit une trivialisation semi-alg6brique des famiUes de fonctions (~ valeurs dans R) semi-alg6briques, copi6 sur le th6or6me de Hardt mentionn6 plus haut. On a ainsi des bornes sur le nombre de types semi-alg6briques (et, a fortiori, topologiques) de fonctions semi-alg6briques, en fonction r6cursive du degr6 des fonctions et de la dimension de l'espace affine de d6part, retrouvant des r6sultats de Benedetti et Shiota [BS] qui g6n6ralisent ceux de Fukuda pour les polyn6mes [FU]. Dans la derni~re section, un contre-exemple inspir6 de Nakai [NA] montre que, par contre, on ne peut pas 6tendre les r6sultats de finitude de types topologiques des applications polynomiales en deux variables de Aoki [AN] aux applications semi-alg6briques.

1. Triangulations lipschitziennes On se propose de montrer comment transformer n'importe quelle triangulation en une triangulation lipschitzienne. On va travailler sur un corps r6el clos quelconque R, ce qui nous sera utile par la suite et qui ne complique pas les d6monstrations. Par contre, il faut 6tre un peu soigneux ?i cause du fait que R n'a aucune raison d'6tre archim6dien. On insiste sur le

194

fait que l'on veut une P-lipschitzianit6 pour un entier P. Ceci ne peut bien stir s'obtenir que si l'ensemble semi-alg6brique h trianguler n'est pas infiniment grand par rapport aux entiers, c.-~-d, que s'il est contenu dans un pav6 I - N , N] n ob N est un entier. Dans ce qui suit, / d6signe l'intervalle [0, 1] de R. Proposition 1. Soit G : I a ~ R ~ une application semi-algdbrique continue. On suppose qu'il existe un entier N E N tel que G ( t a) C [ - N , N] '~, et que G est N-lipschitzienne sur les faces de I a. Alors il existe un homdomorphisme semi-algdbrique H : l a -+ I a, qui est l'identitd sur les faces de I a, et un entier P tel que G o H soit P-lipschitzienne. La d6monstration se fait en appliquant plusieurs lois le lemme suivant. On note pour ce lemme (x, y, z) les coordonn6es dans R d, avec z C R, y C R k et z E R d - k - l , o?.10 ~ k < n. On reprend les notations et les hypoth6ses de la proposition, en supposant que n = 1 (c.-~-d. que G est une fonction h valeurs clans R). L e m m e 2. Supposons, en plus des hypothdses de la proposition, que G est une fonction N-lipschitzienne par rapport aux variables y : Vx, z

IG(z,y',z)-G(x,y,z)l

< N]]y'-yl].

Alors il existe un hom~omorphisme semi-alg~brique H : I d -~ I d qui est l'identitd sur les faces de I n, et un entier Q tel que Go H et H soient Q-lipschitziens par rapport aux variables D~monstration. On gcrit/Td = ./X I d - l , Ob Ie premier facteur correspond ~ la variable x. On "ddmonte" ensuite le c u b e ,rd de la mani~re suivante : I a-1 est divis6 en une partition finie en sous ensembles semi-alg6briques, et pour chacun des sous-ensembles semi-alg~briques A de cette partition, le cylindre I x A est divis6 en tranches au moyen de fonctions semi-alg6briques continues : 0=~0 <~l<...<~p=l:A ~R telles que, sur chaque tranche ](i-a, ~i[, i = 1 , . . . ,p du cylindre I x A, G est une fonction de Nash, et OG/Ox est ou bien constamment major6 par N en valeur absolue, ou bien constamment plus grand que N en valeur absolue. On notera T l e nombre maximal de ces tranches, quand A parcourt la partition semi-algdbrique de I d-1. Consid~rons la fonction semi-algdbrique L : I n --* [0, 1] ddfinie comme suit. On pose toujours L(0, y, z) = 0. Soit (y, z) E A, et supposons L(x, y, z) d6fini pour x E [0, (t(y, z)]. Si, clans la tranche ]~l, (t+t [, OG/Ox est major~ en valeur absolue par N, on pose simplement L ( x , y, z) = L(~t(x, y), y, z) + x - ~t(y, z)

pour ~z(y, z) < x < ~t+~(y, z ) .

Si dans la tranche ]~l, ~t+l [, OG/Ox est plus grand en valeur absolue que N, on pose 1 L ( x , y , z ) -- L ( ~ t ( x , y ) , y , z ) + -~ I G ( x , y , z ) - G ( ~ t ( y , z ) , y , z ) l pour ~t(y, z) _< x _< ~t+l(y,z). On peut remarquer que ~ [ G ( x , y , z ) - G ( ~ t ( y , z ) , y , z ) [ est, dans la tranche, une fonction croissante de x, et minor6e par x - (l(Y, z). On remarque anssi que, pour (y, z) fix6, la

195

fonction x ~ L ( x , y, z) est un hom6omorphisme semi-alg6brique de [ sur [0, L(1, y, z)]. On notera H la bijection semi-alg6brique r6ciproque de la bijection : (z,y,z), Id

, (L(x,y,z)

\L(1,y,z),y,z

)

) I d.

Notons tout de suite que H est l'identit6 sur les faces de [d. En effet, puisque G est N-lipschitzienne sur les faces, on a, pour tout ( y , z ) appartenant h une face de i-d-l, L ( x , y, z) = x. U reste fi 6tablir les propri6t6s annonc6es de H . La premiSre chose est de montrer que L est continu. Ceci suffira h 6tablir que H -1 est continu, et donc que H est un hom6omorphisme : une bijection semi-alg6brique continue d'un ferm6 born6 sur un ferm6 born6 est bicontinue, m~me sur un corps r6el clos quelconque. Soit ¢ > 0, ¢ C R. Fixons (y, z) E [d-1. Alors il existe 5 > 0, 5 E R tel que I ] ( y ' , z ' ) - (y,z)l ] < 5 entra~ne Vx I G ( x , y ' , z ' ) - G ( x , y , z ) l

< ¢.

En effet, m~me sur un corps r6el clos quelconque, une fonction semi-alg6brique continue sur un ferm6 born6 est uniform6ment continue. Soit alors ( y ' , z ' ) E I d-~ v6rifiant II(y', z') - (y, z)ll < ~. L'intervalle I e s t subdivis6 en au plus 2 T - 1 intervalles tels que, quand x varie dans l'int6rieur d'un de ces intervalles, ni (z, y, z) ni (x, y', z') ne sautent de tranche dans leurs cylindres respectifs. Soient x0 = 0 < . . . < Xm < . . . < xq = 1 les bomes de ces intervalles. Pour x E [Xm,Xm+l], on a suivant les cas : [L(x,y',z') - L(x,y,z)l

<_ I L ( x m , y ' , z ' ) - L ( z m , y , z ) l +

I° +

I G ( x , y ' , z ') - G ( x m , y ' , z ' ) l / N - x + xm IG(x,y,z)_G(xm,y,z)l/N_x+x m

IIC( , y', z') - c(xm,y',z')l /N -IC(x,y,z) - G ( x m , y , z ) l / N I. Dans le dernier cas, on a clairement (1)

2¢ IL(x, y', z') - L ( x , y, z)l < ]L(xm, y', z') - L ( X m , y, z)l + -~-.

Le deuxi~me cas se pr6sente quand O G / O x est major6 par N en valeur absolue sur ]xm, x m + l [ × {(y, z)}, et donc IG(x,y',z') - G(xm,y',z')l

< IG(x,y,z) - G(xm,y,z)l

+ 2¢ < N (x - Xm) ~- 2¢.

On aboutit aussi dans ce c a s h la majoration (1). On procede de m e m e darts le troisi~me cas. Finalement, on obtient Vx • I

IL(x,y',z')-L(z,y,z)l

< (2T-1)~-

2¢

.

Si on a choisi 5 suffisamment petit pour que l'on ait aussi, pour tout (y, z) • i-d-l, I x ' - - x I < 5 entra[ne I L ( x ' , y , z ) - L ( x , y , z ) l

< ¢,

196

alors I](x',y',z') -- ( x , y , z ) [ I < 6 entra~ne

IL(x',y',z')-L(x,y,z)I

(2(2~---1)) <

+1

c.

Ceci montre la continuit6 de L, et donc le fait que H est un homdomorphisme. I1 reste h v6rifier que H et G o H sont Q-lipschitziens, pour un entier Q, par rapport aux variables ( x , y ). On voit que, s i x ' > x, alors L( x', y, z ) - L( x, y, z ) > x' - x, e t q u e L(1, y, z ) < T N (on utilise ici l'hypoth6se G ( I ~) C [ - N , N]). Donc H est ( T N)-lipschitzien par rapport x, et G o H est ( T N2)-lipschitzien par rapport h x. Voyons maintenant ce qui se passe pour les variables y. Les calculs faits ci-dessus pour la continuit6 montrent aussi que :

In(x, y', z) - L ( x , y, z)] < 2(2T - 1)lly' - yl[. Posons L(I~ y, z)

L(1, y', z) '

I1 vient

I(' - ~l < L ( 1 , y , z ) I L ( x , y ' , z ) - L ( x , y , z ) l + L ( x , y , z )

I L ( 1 , y , z ) - L(1, y',z)I

< 4 T N ( 2 T - 1)Ib'- ylt, I[H(~,y',z) - H ( ( , y , z ) [ I <_

<_ [IH(Ly',z) - H ( ( , y ' , _< TN

z)l I + NH((,y',z) - H ( L y , z)l [

t~' - ~I + JJ(x, y', z) - (~,y, z)lt

< ( 4 T ~ N ~ ( 2 ~ - 1) + ~)N~' - YH, IIc o H(~, y', z) - G o H(~, v, ~)11 <

<_ [IGoH(Ly' , z ) - G o H ( ~ , y', z )'[ ] + l [ G o

H ~~'¢ , y , ' z )

-

Go

H(~,y,z)][

-< T N ~ I~' - (I + NC(~,y', z) - G ( ~ , y , z)ll < lV (4T~ N~ ( 2 T -

1) + ~ ) l l y ' - vII-

En prenant par exemple Q = 16 T 3 N 3, on a que H et G o H sont Q-lipschitziens par rapport aux variables (x,y). [ ]

Ddmonstration de taproposition. On commence par remarquer que l'on peut 6tendre le lemme au cas off G est ~ valeurs dans R '~ avec n quelconque. Pour cela, on traite successivement chaque coordonnde de G au moyen du lemme. Comme l'hom6omorphisme dans le lemme est lipschitzien, on ne perd pas, en rendant une coordonnde lipschitzienne, la lipschitzianit6 des coordonn6es pr6cddentes. En appliquant d fois le lemme ainsi 6tendu, on transforme G en une application P-lipschitzienne, pour un certain entier P, en toutes les variables. [ ] Nous passons maintenant du cas d'un pav6 ferm6, vu dans la proposition, ~t celui d'un complexe simplicial fini. Nous expliquons d'abord comment "quadranguler" un complexe simplicial fini. Considdrons le simplexe standard

A d=

(to,...,td);

ti=l, i=0

ti>_O

c R d+l.

197

On le divise en d + 1 morceaux

=

,td)

vj

_<

chacun projectivement isomorphe au pav6 ferm6 I d. La formule pour l'isomorphisme de Cod sur

id

est

:

On remarque que la subdivision induite sur les faces de A a est bien la subdivision des simplexes de dimensions infdrieures ~ d : les subdivisions se recollent bien. Si maintenant K est un complexe simplicial fini dans R m, on le quadrangule en transportant, pour chaque simplexe a de dimension d de K , la quadrangulation du simplexe standard A d par un isomorphisme affine de A d sur a. On a alors K comme r6union de boltes projectivement isomorphes ~ des pav6s ferm6s, l'intersection de deux boites 6tant une face (it6r6e) commune, ou vide. On notera/~qu~d le complexe ainsi quadrangul6 ; le k-squelette de Kqu~a est la r6union des boites de dimensions inf6rieures ou 6gales ~ k. Th6or~rne 3. Soit K C R m un complexe simplicial fini, dont les sommets ont des coordonn~es entikres. Soit G : K ~ R '~ une fonction semi-algdbrique continue, avec G ( K ) C [ - N , N ] " pour un certain entier N . Alors il existe un homdomorphisme semi-algdbrique H : K ~ K et un entier P tels que G o H soit P-lipschitzien.

On quadrangule K et on construit H par induction sur les k-squelettes de Kq,~a. kes hom6omorphismes que l'on construit ?t chaque 6tape pr6servent les boites de Kq,~d. Supposons que l'on ait un tel hom6omorphisme Hk tel que G o H k soit Pk-lipschitzien sur le k-squelette de Kq,~a, pour un entier Pk. Alors la proposition permet d'obtenir un hom6omorphisme semi-alg6brique hk+a du k + 1-squelette sur lui-m~me, pr6servant les boites, qui est l'identit6 sur le k-squelette, et tel que G o Hk o hk+l soit Pk+a-lipschitzien, pour un nouvel entier Pk+~. On peut ensuite 6tendre, par une construction de c6ne, Hk o hk+l ~ un hom6omorphisme semi-algdbrique Hk+a de Kquad sur lui-mame, pr6servant les boites. [ ]

Ddmonstration.

a) Vu la faqon dont on l'a construit, on peut ajouter dans le th6or~me que l'hom6omorphisme H est semi-alg6briquement homotope h l'identit6. b) I1 n ' y a pas d'effectivit6 dans le th6or6me, mais on peut en rajouter apr~s coup. Rappelons que pour mesurer la complexit6 d'un objet semi-alg6bfique, on consid~re son degr6, qui est la somme des degr6s des polyn6mes qui interviennent dans sa description. On obtient alors, pour N = 1, une borne pour le degr6 e de H et pour P, en fonctions r6cursives du nombre de sommets s du complexe K , du degr6 d de G e t de r~. En effet, si l'on fait N -- 1 et si l'on fixe s, d, r~, e et P, alors le thdor~me 3 s'6crit comme un 6nonc6 ~ ( s , d, n, e, P ) de la th6orie des corps r6els clos. Etant donn6s s, d, et n, il existe e et P tels que cet 6nonc6 soit un th6or~me de la th6orie des corps r6els clos ; sinon, on obtiendrait par ultraproduit un corps r6el clos o~5 le th6or6me 3 serait en d6faut. Enfin, l'algorithme de d6cision pour la th6orie des corps r~els clos monlxe que ces e et P sont des fonctions r6cursives de s, d, et n. Nous suivrons la m~me d6marche pour le th6ol~me 8. c) Le r6sultat est 6videmment li6 au "Triangulation Lemma" que l'on peut trouver dans [GR]. Ce dernier dit que l'on peut subdiviser une triangulation G : K ~ G ( K ) C R ~, de fa¢on que pour tout k-simplexe A de la subdivision de K , on a un hom6omorphisme semi-alg6brique H a du k-simplexe standard sur A, avec toutes les d6riv6es partielles jusqu'~

Remarques.

198

l'ordre r (donn6 h l'avance) de G o Hzx born6es par 1. Ici on ne parle que de lipschitzianit6, mais on ne subdivise pas la triangulation, et l'hom6omorphisme H qui rend lipschitzien est d6fini globalement, de I~" sur K.

2. Bornes m~triques uniformes Le fait d'avoir montr6 l'existence de triangulations lipschitziennes sur un corps r6el clos quelconque permet d'obtenir des r6sultats m6triques sur les families, l_z passage se fail en "cachant les param~tres dans le corps de base". On obtient une version Iipschitzienne du th6or6me de trivialit6 semi-alg6brique de Hardt [HA]. La lipshitzianit6 est uniforme darts les fibres, mais on ne peut bien stir pas esp6rer de lipschitzianit6 par rapport aux variables de la base. Th6or~me 4. Soil B u n ensemble semi-alggbrique, et soil X un sous-ensemble semialg~brique de R n × 13. On note p la projection de X sur 13. On suppose qu'il existe un entier N tel que, pour tout b C B, la fibre Xb soil contenue dans [ - N , N] n. Alors il existe un entier P, une partition de B e n un hombre fini de sous-ensembles semi-algdbriques 13i, avec pour chaque i un ensemble semi-algdbrique F i e t un homdomorphisme semi-algdbrique hi:F ~ ×B i

~XMp-I(B

i)

tel que p o h i soil la projection s u r B i e t tel que pour tout b E 13i, l"appIication h~ : F i ~ X b soil P-lipschitzienne. Dgmonstration. On suit fid~lement la seconde demonstration du th6or6me de Hardt dans [BCR], 9.3.1. Soil d'abord X l'adh6rence de X dans R n × B. on note comme d'habitude 13 le constructible du spectre r6el correspondant ~t B, et pour un point c~ d e / 3 , k(cx) d6signe T, le corps r6el clos associ6. La fibre X ~ est ferm6e dans [ - N , N ]k(,~)" On a donc une

triangulation P-lipschitzienne, pour un certain entier P, de X~, compatible avec X,~, par un complexe simplicial fini contenu dans k(c~) n, dont les sommets sont h coordonn6es enti~res. Soil Fk(,~) la r6union des simplexes ouverts de ce complexe dont l'image par la triangulation est dans Xc,. Cet ensemble semi-alg6brique Fk(,~) est l'extension ~ k ( a ) d'un sous-ensemble semi-alg6brique F de R n (on a sp6cifi6 que les sommets 6taient h coordonn6es entiSres pour en 5tre stirs). On r6cupSre donc un sous-ensemble semi-alg6brique B' de B, avec c~ E B', et un hom6omorphisme h : F x B' ~ X M p - I ( B ' ) , compatible avec les projections sur B', et qui est P-lipschitzien par rapport aux variables dans R n. La compacit6 d e / 3 permet ensuite de trouver la partition finie du th6orSme. [ ] Le rdsultat prEcEdent permet d'obtenir directement les classiques bornes m6triques uniformes pour les families, sans faire appel ~t la g6om6trie intdgrale. Pour pouvoir parler de volume, il faut ici se limiter au cas des vrais reels, on note diam (A) le diamStre d'un compact A de R'* (borne sup6rieure des distances entre deux points de A). Corollaire 5. Soil B un ensemble semi-algdbrique rdel, et soil X un sous-ensemble semialgdbrique de R n × B. On suppose que, pour tout b E B, Xb est non vide de dimension k > O. Alors il existe une constante C telle que, pour tout b C B, le volume k-dimensionnel de Xb soil majord par C × (diam (X~))k

199

Ddmonstration. On peut supposer que l'origine de R'* est dans Xb pour tout b. La fonction , p ( b ) = (diam (Xb)) -1

b,

est semi-algdbrique (pas continue a priori), et en appliquant l'homoth6tie de rapport p(b) ~, la fibre Xb on peut se ramener au cas oh Xb C [-1, 1] n pour tout b. Le tMorSme 4 donne alors le rdsultat voulu. [ ]

3. Prolongement de trivialisations Nous avons d6j~ 6voqu6 le th6or6me de triviatit6 semi-algdbrique de Hardt [HA]. Formulons en une cons6quence dans un cas tr~s particulier, qui nous occupera dans cette section. Soit X un ensemble semi-alg6brique, et f : X ~ R u n e fonction semi-algdbrique. Alors il existe un ~ > 0 et un hom6omorphisme semi-algdbrique h : f - l @ ) × ] 0 , C]

, f - l ( ] 0 , c])

tel que f o h e s t la projection sur ]0,~] et que h(., e) est l'identit6 sur f - l ( e ) . Peut-on prolonger continfiment cette trivialisation au-dessus de ]0, e] en 0 ? I1 faut ddj~i supposer que la fonction f est propre, mais ceci ne suffit pas. Prenons pour f la projection de [0, 1] x [0, 1] sur le deuxiSme facteur, et pour h la fonction de [0, 1] ×]0, 1] dans R 2 d6finie par

(x,

t), {(x,t) '

six+t>l

(tl_--Xx ,t)

s i x + t _ < 1.

Ce h ne peut passe prolonger par continuit6 ~ [0, t] x [0, 1]. Cependant, il y a toujours moyen de choisir la trivialisation pour que le prolongement soit possible. Th6or~me 6. Soit X un ensemble semi-algdbrique, et f : X --+ R u n e fonction semialgdbrique propre. Alors il existe u n ¢ > 0 et une application semi-algdbrique h : f - l @ ) × [0,~]

, /-1([0,¢])

tel que f o h est la projection sur ]0,@ que h(., e) est l'identitd sur f - 1 ( ¢ ) et que h e n restriction (i f - 1 (¢) x ]O, e] est un homdomorphisme sur f - x (] 0, ¢] ). Ddmonstration. Comme f est propre, on peut se ramener au cas off f - l ( [ 0 , 1]) est contenu

dans un pav6 [ - N , N] '~, avec N entier. On applique alors le thdorSme, qui nous donne un ¢ > 0 et un hom6omorphisme semi-alg6brique

g:K×]o,c]

, f-l(]0,c])

off K est un ensemble semi-alg6brique compact, tel que f o 9 est la projection sur ]0, 1] et que, pour tout t E]0,e], g(-, t) : K - - + f - l ( t ) est P-lipschitzien pour un entier P ind~pendant de t. Soit y E K. Puisque f est propre, alors l'intersection de {(V, 0)} x X avec l'adh6rence du graphe de 9 est non vide. Cette intersection est l~duite "a un point. En effet, soient (V, 0, x) et (y, 0, x') deux points de cette intersection. Par le lemme de s61ection des courbes, il existe deux chemins semi-alg6briques 7 , 7 ' : [0,5[ ) K

200

tels que 7(0) -- 7'(0) = y, que limt_.o(g(7(t), t)) = x et que l i m t ~ o ( g ( 7 ' ( t ) , t)) = x'. Alors la propri6t6 de lipschitzianit6 implique que x = x'. On a ainsi montr6 que g se prolonge par continuit6 ~t K x [0, e], ce qui nous donne te rtsultat. [ ] Le th6or~me prtctdent montre que f - l ( [ 0 , e ] ) se rttracte par dtformation semialg6brique sur f - l ( 0 ) , tandis que f - l ( ] 0 , e ] ) est semi-algtbriquement homtomorphe f - l ( c ) × ] 0 , e ] . On a donc ainsi des morphismes de l'homologie de f - 1 ( ¢ ) dans celle de f - l ( 0 ) (et ceci sur n'importe quel corps rtel clos de base). On peut interprtter le thtor~me prtc6dent comme une g6n6ralisation du th6or~me de structure conique locale pour les ensembles semi-algtbriques : soit Y un sous-ensemble semi-alg6brique ferm6 born6 de l'ensemble semi-algtbrique X ferm6 darts R n, et soit f une fonction semialgtbrique continue de X dans R, positive ou nulle, propre, et telle que y f - l ( 0 ) ; par exemple, on peut prendre pour f la distance ~ Y. Alors le voisinage compact f - l ( [ 0 , c]) de Y, pour ¢ suffisamment petit, est semi-alg6briquement homtomorphe au "mapping cylinder" d'une application semi-algtbrique continue de f - l ( c ) dans Y. A. Durfee [DU] a consid6r6 les voisinages semi-algtbriques d'ensembles semi-alg6briques : ce sont les f - l ( [ 0 , c ] ) comme ci-dessus. I1 obtient l'unicit6 ~ hom6omorphisme prts de ces voisinages (ils ne d6pendent essentiellement ni du choix de ¢, ce que l'on a vu, ni du choix de f). Mais les hom6omorphismes sont obtenus par utilisation du premier lemme d'isotopie de Thorn, et ils ne sont pas semi-algtbriques. I ~ thtor~me 6 s'obtient comme constquence facile du rtsultat de triangulation des fonctions semi-alg6briques (h valeurs dans R) de Shiota [SH]. Ce rtsultat dit que, si f : X --. R e s t une fonction semi-alg6brique continue bom6e, alors il existe un complexe simplicial fini K sur R, une fonction g : K ~ R affine sur chaque simplexe de K , et un hom6omorphisme semi-alg6brique h d'une rtunion U de simplexes ouverts de K sur X tels que f o h = g t u . =

4. Trivialisation de families de fonctions semi-alg6briques Dans cette section, nous exploitons le rtsultat de triangulation des fonctions semialg6briques ci-dessus, et en particulier le fait que la dtmonstration de [SH] est 616mentaire, c.-h-d, valide sur n'importe quel corps r6el clos. On peut alors lui faire subir le mSme traitement que celui qui donne le th6ortme de trivialit6 de Hardt h partir du thtor~me de triangulation des ensembles semi-alg6briques, dans [BCR] 9.3.1. Voici le rtsultat qu'on obtient : Th~or~me 7. Soient B C R m e t X C R ~ × B des ensembles semi-algdbriques. On note p la projection de X sur B. Soit f : X ~ R une fonction semi-alg~brique telle que p o u r tout b E B , la fonction fb : X B --+ R est continue. Alors il existe une partition semi-algdbrique

201

finie B = U i e i Bi, et pour chaque i E I un carrd commutatif

($1xi ,Plx~) R × B >

X i = X f l p - : ( B i)

i

hi

Fi

R

oft F i e s t un ensemble semi-algdbrique, h i, ¢i et gi sont semi-algdbriques continues et v&ifient que pour tout b G B i, h~ : Xb ---+F i e t g~ : R ~ R sont des homgomorphismes. Ddmonstration. Sans perte de g6n6ralit6, on peut supposer que f est born6e, 7t valeurs dans [-1, 1]. Soit a E B. La fibre f~ : X ~ ~ k ( a ) est une fonction semi-alg6brique continue bom6e, et on peut lui appliquer le th6or~me de triangulation des fonctions semi-alg6briques continues [SH] pour le corps r6el clos k(a) : la fonction f,~ est semi-alg6briquement 6quivalente ~ la restriction d'une fonction affine par morceaux sur un complexe simplicial fini. On insiste ici pour que le complexe simplicial et la fonction affine par morceaux sont des extensions ?: k(a) d'objets d6finis sur R. De fagon pr6cise, il y a un complexe simplicial fini K d6fini sur R, une fonction ¢ : K --+ R, affine sur chaque simplexe, une r6union de simplexes ouverts F C It', et un carr6 commutatif

fa

,

k(o,)

Go,

Ck(,,)

ofa ¢ est ~[F, et off H a et G~ sont des hom6omorphismes semi-alg6briques. On r6cup6re ainsi un sous-ensemble semi-alg6brique B' C B, tel que a C B', et un carr6 commutatif

X'=XMp-I(B

')

(,fix' ,PIX' ) ,

H

RxB' G

CxB j

F x B'

>

R x B'

202

off toutes les applications sont compatibles avec les projections sur B ' , et off H et G sont des hom6omorphismes semi-alg6briques dont les fibres en c~ sont H a et G,~ respectivement. Pour obtenir t'6nonc6 du th6or~me, il ne reste plus qu'h prendre en compte la compacit6 de pour avoir la finitude de la partition. [ ] On obtient aussi des r6sultats de finitude et de calculabilit6 pour la triangulation des fonctions. Ces r6sultats ont 6t6 montr6 dans [BS] en r6examinant le preuve de [SH]. Ici, on les obtient par un artifice logique. C'est moins fatigant, mais le d6faut est que l'on n ' a aucune pr6cision sur les bornes obtenues, ~ part le fait qu'eltes sont r6cursives. Th6or~me 8. Etant donnds n et d, il existe s fonctions semi-algdbriques continues ~i:RnDxi

~R,

i = 1,...,s

de degrd d, et un entier ~, tels que pour toute fonction q~:W~ D X

~R

de degrd d, il existe un entier i, 1 < i < s, et des hom~omorphismes semi-alg~briques h:X

)X i

et

g:R

)R

de de g r & au plus ~, et qui vdrifient ~ i o h = g o~. De plus s e t t sont bornds p a r des fonctions rdcursives de n e t d. D~monstration. On peut param6trer, par un ensemble semi-alg6brique B ( n , d), la famille de toutes les fonctions semi-alg6briques continues, ~t valeurs darts R, et dont le domaine est un sous-ensemble semi-alg6brique de R '~. On obtient ainsi une fonction

f

X

~ R

'0 h qui l'on applique le th6orSme pr6c6dent. On r6cupSre une partition semi-alg6brique finie B ( n , d) = gls=1 B i, et en choisissant bi E B i, on a pour chaque i E I un carr6 commutatif

X i - - - - - X A p - I ( B i)

(flx~,plx~)

R×B

hi X i = Xb~

R

i

203

Oh h i et g i sont semi-alg6briques continues, v6rifient que pour tout b E B i, h~ : Xb ~ X i e t g~ : R ~ R sont des hom6omorphismes et clue h~ = Idx, et g~, = Ida. En prenant pour t le maximum des degr6s des h i et des 9 i, on obtient la premi6re partie de l'6nonc6. Pour obtenir la r6cursivit6 des bomes, on remarque que, une fois que l'on a fix6 n, d, s e t t, la th~se du th6or6me 8 "I1 existe s fonctions semi-alg6briques.., qui v6rifient ¢i o h = g o ¢." s'6crit comme un 6nonc6 ~ ( n , d, s, t) du premier ordre de la th6orie des corps r6els clos, que l'on peut produire algorithmiquement ~ partir de n, d, s e t t. L'algorithme pour calculer une borne pour s e t t en fonction de n e t d consiste alors ~ appliquer un algorithme de d6cision pour la th6orie des corps r6els clos aux 6nonc6s q~(n, d, 1, 1), ~5(n, d, 2, 2 ) , . . . et en s'arr&ant quand on trouve une formule vraie. La premiere partie du th6or~me dit que l'algorithme s'arrate bien. [ ]

5. U n c o n t r e - e x e m p l e p o u r les f a m i l i e s d ' a p p l i c a t i o n s s e m i - a l g 6 b r i q u e s d6finies sur R ~

Le r6sultat de Benedetti et Shiota formul6 darts le th6or~me 8 g6n6ralise un r6sultat de Fukuda [FU] qui concerne les fonctions polynomiales. On conna~t aussi un thEor~me d'Aoki [AN] qui dit que, le nombre de types topologiques de gennes d'applications polynomiales de R 2 dans R 2 de degr6 donn6 est fini. Mais dans ce cas, on ne peut pas esp6mr une g6n6ralisation aux families d'applications semi-alg6briques continues, comme le montre le contre-exemple suivant inspir6 par Nakai [NA]. Soit a = (al,a2,aa,a4) E R 4,avec 0 < al < a2 < a3 < a4. Soit fa : R 2 ~ R 2 la fonction semi-alg6brique continue d6finie par fa : ( x , y ) '

' (Ixl,ailxl lYl)

Oh i est le num6ro du quadrant dans lequel se trouve (z, y) (les quadrants sont num6rot6s darts l'ordre direct). On va montrer que pour deux quadruplets diff6mnts a et b, les applications fa et fb ne sont pas en g6n6ral semi-algdbriquement 6quivalentes. Avec un petit peu plus de travail, on aurait le m~me r~sultat pour l'6quivalence topologique. Examinons d'abord les fibres de la fonction f~ : - f~-l(0,0) est l'axe des y, - f [ l ( u , 0), pour u > 0, comprend les deux points (u, 0) et ( - u , 0),

- f~l(u,v),

?)

pour u , v > 0, comprend les quatres points ( u , ~ ) ,

V

--Y

( - u , a--~u), ( - u , a---~u),

--V

(u,;S),

- dans les autres cas la fibre est vide.

Soit maintenant 3' : [0, 1] --+ I~2 un chemin semi-alg6brique avec 3'(0) = (0, 0) et 3"(]0, t]) c {(u,v) ; u > 0, v > 0}. Ce chemin arrive ~ l'origine avec une tangente de coefficient directeur t(7 ). Si 0 < t(7 ) < +c~, alors S(7) = fd -1 (7([0, 1])) contient les quatre points remarquables d'abscisses (dans l'ordre croissant) t ( 7 ) / ( - a 3 ), t ( 7 ) / ( - a 4 ), t(7)/a2 et t(3")/al sur l'axe des y. Ces points, que l'on d6signera par P3(7), P4(3"), P2(3") et P1(3") se remarquent dans S(3") parce que ce sont ceux dont le compl6mentaire a trois composantes. Ce pMnom~ne ne se produit pas si t(3") = 0 ou +oo. Choisissons alors un chemin semialg6brique 3' comme ci-dessus, avec 0 < t(7 ) < +e~ . On peut trouver deux suites de chemins semi-alg6briques (6,) et ( ¢ , ) , telles que 60 = ¢0 = 3", clue P3(6n+1) = P4(6,,) et

204

que P1(¢,~+1) = P2(¢n). DEfinissons s ( n ) comme le plus petit entier s tel que Cs soit en dessous de 5,, pros de l'origine. Les remarques faltes plus haut montrent que

(:,)°

et

t(¢n) = t(7 )

al

On en d6duit donc que n

lira s ( n ) = log(a3) - log(a,) ~ n log(a~) - log(~:)

On suppose maintenant que f~ et fb sont semi-alg6briquement 6quivalentes, c'est-~-dire qu'il existe des hom6omorphismes semi-alg6briques g e t h de I~2 sur lui-m~me tels que fb o g = h o f , . D'apr~s ce que l'on a vu pour les fibres, h envoie l'origine sur l'origine, le demi-axe des u > 0 sur lui-m6me, et le premier quadrant ouvert sur lui-m~me ; g envoie l'axe des y sur lui-mSme. La situation avec les suites de chemins (~5.) et (¢,~) se transporte par h en une situation semblable, avec la m6me fonction s ( n ) , et on doit donc avoir log(a3) -- log(a4) log(a1 ) - log(a2)

log(b3) - log(b4) log(b1 ) - log(b2)"

Cette derni~re 6galitE montre que l'on ne peut pas avoir un nombre fini de classes d'Equivalence semi-algEbrique dans la famille des fonctions fa quand a varie.

R6f~rences [AN]

Aoki K., Nagachi H. : On topological types of polynomial map germs of plane to plane, Memoirs of the School of Science & Engineering 44 (1980), 133-156, Waseda University

[BS]

Benedetti R., Shiota M. : Finiteness of semialgebraic types of polynomial functions, para~tre

[BCR]

Bochnak J., Coste M., Roy M-E : G6om6trie alg6brique r6elle, Springer (1987)

[DU]

Durfee A. : Neighborhoods of algebraic sets, Trans. A.M.S. 270 (1983), 517-530

[FU]

Fukuda T. : Types topologiques des polyn6mes, Publ. math. I.H.E.S. 46 (1976), 87-106

[GR]

Gromov M. : Entropy, homology and semialgebraic geometry (after Y. Yomdin), Ast6risque 145-146 (1987), 225-240

[HA]

Hardt R. : Semi-algebraic local-triviality in semi-algebraic mappings, Amer. J. Math. 102 (1980), 291-302

[NA]

Nakai I. : On topological types of polynomial mappings, Topology 23 (1984), 45-66

[SH]

Shiota M. : Piecewise linearization of subanalytic functions II. Dans : Real analytic and algebraic geometry, Lect. Notes Math. 1420, 247-307, Springer

STIEFEL ON

A REAL

ORIENTATIONS ALGEBRAIC

VARIETY

A . I . DEGTYAREV ABSTRACT. Some natural Stiefel orientations on the normal bundle of the fixed point set of an involution on a smooth manifold are constructed. This result is applied to nonsingular real algebraic varieties in order to generalize Rokhlin's construction of complex orientations on a separating real algebraic curve

INTRODUCTION 0.1. T h e main purpose of this p a p e r is to generalize Rokhlin's construction of complex orientations on a separating real algebraic curve. T h e original construction is the following (Rokhlin [6]): Denote, respectively, by CA and RA the sets of real and complex points of curve A. Since R A separates CA, the complement CA \ XA consists of two c o m p o n e n t s C A ± . T h e complex orientation of complex manifolds C A ± defines a pair of opposite orientations of their c o m m o n b o u n d a r y ]RA. A direct generalization of this construction is obvious: Let F be an m-dimensional s m o o t h submanifold of a smooth n-dimensional manifold X . Suppose t h a t the fundamental class [F] of F vanishes in H,,,(X). T h e n the linking coefficient m a p lk : H n - m - l ( X \ F) ~ Z2 restricted to the b o u n d a r y of a tubular neighborhood of F in X is an (n - m - 1)-dimensional Stiefel orientation on the normal bundle vF of F in X . (The notion of Stiefel orientation is a generalization of the ordinary orientation and Spin-structure on a vector bundle; see Definition 1.4.1 for details.) This orientation is well defined over the subgroup Ker[inclusion. : H , , - m - I ( F ) --* H n - m - l ( Z ) ] . 0 . V i r o [8] proposed a further generalization of this construction (see 4.2) for the case when pair (X, F ) is supplied with the following additional structure: X is a manifold with an involution c : X --* X , and F = F i x c is the fixed point set of c. In this p a p e r we give a n o t h e r approach to Viro's construction which shows t h a t the arising Stiefel orientations are, in fact, also the linking coefficient functionals on homology groups of some a p p r o p r i a t e spaces S s ' - l X \ s k - l F associated with pair (X, c). These orientations exist if for some k /> 1 class IF] vanishes under some "higher" inclusion m a p s k era+k_1 : H . ( F ) --+ H m + k - l ( X ) , and it is defined over the subgroup K e r e .k. . . 1 (see T h e o r e m 4.1.1 for details).

Remark. The advantage of the approach of this p a p e r is t h a t the construction is connected with the general theory of Z2-spaces (i.e. spaces with involution). In particular, if F is an M - s p a c e (i.e. d i m H , ( F ) = d i m H , ( X ) ) , this theory enables to express the Key words and phrases. Stiefel orientation, involution, fixed point set~ real algebraic variety.

206

condition necessary for existence of the orientations in terms of characteristic classes of X (Degtyarev [3]). 0.2. Applying Theorem 4.1.1 to an n-dimensional real algebraic manifold A yields a series of partial i-dimensional Stiefel orientations on the tangent bundle of RA, n - k <<. i ~< n - 1, provided that en+k_ l k [lEA] = 0. However, this condition seems to be rather difficult to be verified. If A is a projective complete intersection, it can be substantially simplified and reduced to the usual requirement that [lEA] should be either equal to zero or dual to the plane section class in H , ( C A ) . In this case some modified construction provides a series of Stiefel orientations of all dimensions on the tangent bundle of the double cover lEA of lEA corresponding to the real hyperplane section class lEh E H 1(lEA). These orientations are well defined over the whole group H,(IEA) (Theorem 4.3.4).

Remark. If [lEA] is equal to the plane section class, manifold lEA may not possess any Stiefel orientation, but the modified construction is still applicable to lEA. If A is a surface, there is another approach, also due to Viro. It prvovides a P i n - - s t r u c t u r e on RA which lifts to the 1-dimensional Stiefel orientation on lEA arising due to Theorem 4.3.4. It would be interesting to find a higher-dimensional generalization of this approach. Remark. The main difficulty in generalizing Viro's approach (see the previous remark) seems to consist in the following: In a way, P i n - - s t r u c t u r e is the structure corresponding to characteristic class w2 +w~. More precisely, let f : X ~ B O , be a characteristic map of bundle ~, and P(w) --* BO,~ be a fixed fibration killing some characteristic class w E Hk(BOn). T h e n an wstructure on ~ can be defined as a class x E H k - l ( f * P ( w ) ) which does not vanish when restricted to any fibre. (For example, Stiefel orientations are defined via the Stiefel bundles Vn,k(qn) ~ BOn, killing wn-k+l; P i n - - s t r u c t u r e is defined via the bundle B P i n - ~ BOn, which kills w2 + w~.) The problem is that there is no universal way to choose the killing fibration P(w), so in fact structures of this kind are functorially defined only modulo all (k - 1)-dimensional characteristic classes. This reduction being accepted, both Viro's P i n - - s t r u c t u r e and Nezvetaev's form (Viro [8]; in fact, this form is a w2-structure) can easily be generalized (Degtyarev [21). But in order to define the absolute (i.e. not reduced modulo characteristic classes) structures some appropriate killing spaces over BO,~ and BUn are necessary which would be functorial under some standard homomorphisms of groups On and Un. 0.3. Among the other results of the paper is a new approach to defining Stiefel orientations. Instead of the associated Stiefel bundles Vn,k(~) it involves some other spaces which can be constructed using only the total space of the associated sphere bundle S " - I ( ~ ) and the standard antipodal involution on it (Definition 3.3.1). It is this approach that, being applied to a tubular neighborhood of F in X , makes the construction of T h e o r e m 4.1.1 possible. 0.4. The paper is organized as follows: In section 1 we remind some known basic facts and introduce some notations, which are not generally accepted. Section 2 is devoted to Z2-spaces. Most of the notions and results of this section seem to be familiar to those who deal with involutions, though I could not find any systematic account of them. In 2.1 we discuss singular Smith homology groups, which

207

are more suitable for this paper. In 2.2 the spectral sequence of space 5'~ xz2 X (el., for example, Hsiang [4]) and of some natural finite dimensional approximations of this space are introduced. In 2.3 we define the stabilized spectral sequence of an involution. This sequence, first introduced by Kalinin [5], is, in fact, just a compact way to express the stabilization theorem (Hsiang [4]). Kalinin also conjectured the description of the differentials of of these spectral sequences given in 2.4. In 2.5 we define higher inclusion homomorphisms e.~ : SH.(X) --+ H.(X) and study their properties. In section 3 the new approach to constructing Stiefel orientations is introduced and its relation to the ordinary orientations is studied. Section 4 is devoted to the main construction and its applications to real algebraic varieties. 0.5. A c k n o w l e d g e m e n t s . I take this opportunity to express my deep gratitude to O.Ya.Viro for his help and interest to this work. 1 .PRELIMINARIES 1.1. C o m p l e x e s a n d h o m o l o g y g r o u p s . For a topological space X we denote by S.(X) and S*(X) its singular chain and coehain complexes with coefficients Z2. All homology and cohomology groups also have coefficients Z2. When it cannot confuse, we use the same letters to denote cycles and their homology classes. 1.2. G - b u n d l e s . Let G be a Lie group. The term "G-bundle" (over space X) means a principal G-bundle ~: G(() -+ X, i.e. a free right G-space G(~) such that a(O/a = x . For a G-bundle ~ and left G-space F we denote by F({) the associated F-bundle G({) x a F , X, where the twisted product G(() x a F is the quotient space

G(~) x F / { ( x , I ) = ( z g - l , g f )

for a n y g E G } .

1.3. S o m e s t a n d a r d s p a c e s . We use the following standard notations: IRn is the coordinate Euclidean space with the standard basis ( e l , . . . ,e~). For k ~< n we identify N k with the subspace of N" generated by ( e l , . . . ,ek); S '~ and D~ are the standard sphere and semispheres

S" = { E tiei C R~+I I E ( t i ) 2 = 1} m

= { Z t'e, C R ~÷1 t ~ ( e )

D~- = { E t i e '

~ = 1, t' > 0}

C Nn+l I E ( t ' ) 2 = 1, t 1 ~< 0};

A n is the standard simplex { E tiei C ]Rn+l ] E ti = 1, t i /> 0}; On is the group of orthogonal transformations of Nn; V,,k is the Stiefel manifold, which can be defined as either the quotient space O n / O n - k , or the space of all orthogonal k-frames in 1Rn; it is well known that ~ri(v~,k) = 0 for i < n - k, and ttn-k(Vn,k ) = 2;2 (see Steenrod, Epstein [7]); for an On-bundle ( we denote by S(() the associated Sn-l-bundle; S(() is called the sphere bundle of ~. Note that S(~) = Vn,l(~).

208

Let R = (~tl,... ,zk+l) be an orthonormal (k + 1)-frame in R n+l. Denote by Ak(R) the singular simplex A k --. S ~, ~ t~z~ , , ~ tixi/ ~ ( ti) 2. We will use the following singular chains in S":

Ek = An(_~l,...,ek+l)

k+l ~t- E ~ n ( ' ~ I ' ' ' ' ' - - C i ' ' ' ' ' " k + I ) ' i=2

k+l =

i=2 +

E~: will also be considered as singular chains in D k. If c is the antipodal involution on S n, then obviously c#E~_ = E k,_ and OE~: = E k-l+ + E k-a_. 1.4. S t i e f e l orientations. 1.4.1. D e f i n i t i o n . A (reduced) k-dimensional Stiefel orientation on an On-bundle ~ is a class xk E Hk(Vn,n-k(~)) (resp., ;¢k C Hk(Vmn-k(~)) ), such that for any fibre Vn,n-k of fibration Vn,n-k(~) ~ X the restriction of xk (resp., 5tk) to ffIk(Vn,n-k) is non-zero. The set of all (reduced) Stiefel orientations on ~ is denoted by Stk(~) (resp., Stk(~)).

Remark. A 0-dimensional Stiefel orientation is an ordinary orientation of vector bundle ~. A 1-dimensional orientation is a Pin+-structure on ~. A pair consisting of a 0- and 1-dimensional orientation is a Spin-structure. Remark. If k > 0, obviously Stk(~) = Stk(~). If k = 0, a reduced orientation is a pair of opposite ordinary orientations on ~. (So this notion is not trivial only if the underlying space X is not connected.) 1.4.2. T h e o r e m (well known). Let ~ be an On-bundle over X . Then Stk(~) and Stk(~) are not empty if and only i f w k + l ( ( ) = O. In this case Stk(~) (resp., fftk(~)) is an afl~ne space over H k ( X ) (resp., Hk(X)). The reduction map Stk(~) ~ gtk(~) is af[ine. 2. Z2-SPACES 2.1. S i n g u l a r S m i t h h o m o l o g y . Throughout this section X is a fixed Z2-space, i.e. a topological space supplied with an involution c : X ~ X. We denote by Fix c the fixed point set of c, and by X / c the quotient space. For simplicity we assume that pair (X, Fix c) allows a c-invariant cell partition. (Note, though, t h a t most results can easily be extended to general Z2-spaces using an equivariant cell approximation of (X, c).) 2.1.1. N o t a t i o n . Denote A = Z~[c]/(c 2 = 1). Since Z2 acts on S . ( X ) via the induced involution c#, this complex can be considered as a complex of A-modules. We will also consider the Z2-algebra g * ( R p k) = Z2[h]/(h k+l = 0) and the H*(Rpk) module H.(Rpk). The generator of Hi(Rp k) = Z2 is denoted by hi; both hi and h i are assumed to be zero if either i < 0 or i > k.

209

2.1.2. Definition. The Smith homology groups S H , ( X ) are the homology groups of the Smith complex S m , ( Z ) = Ker[(1 + c#) : S , ( X ) --~ S,(X)]. The inclusion homomorphism S H , ( X ) --~ H , ( X ) is denoted by smx. 2.1.3. P r o p o s i t i o n . (I) Let C , ( X ) be the cell complex corresponding to some cinvariant cell partition of pair (X, Fixc). Then S H , ( X ) = g,(Ker[(1 + c # ) : C , ( X ) ---,

c.(x)]); (2) denote by p : X ~ X / c the projection. Then the map x ~ relp#x induces isomorphism H . ( S . ( X ) / S m . ( X ) ) ~- H . ( X / c , Fix c); (3) S m . ( X ) naturally splits into S.(Fixc) @ S . ( X ) / S m . ( X ) , due to (2) this induces an isomorphism S H , ( X ) ~ H,(Fix c) ~ H , ( X / c , Fix c). Proof. The last two statements are well known for cell Smith homology groups (cf. Bredon [1]), so they follow from the first one. Proof of the first statement is absolutely anMogous to the standard proof of similar statement about ordinary homology groups H , ( X ) . We omit the details since they would require developing from the very beginning the theory of functor SH,, which is absolutely analogous to the standard singular homology theory. [] 2.1.4. C o r o l l a r y . Let (X, Fix c) is an n-dimensional cell pair. Then SH~(X) = 0 for i > n, and S H , ( X ) = Ker[(1 + c , ) : Hn(X) -~ H,(X)]. 2.1.5. C o r o l l a r y . Let X be an n-dimensional smooth manifold, and c be a smooth involution. Then there exists one and only one class [X]s E SHn(X) such that smx[X]s = [X]. We call this class the Smith fundamental class of X . ) For (smx)n : S H , ( X ) -* Hn(X) is a monomorphism, and [X] is a c.-invariant class. [] 2.2. S p e c t r a l s e q u e n c e o f an involution. 2.2.1. Definition. Denote by g k x , 0 ~ k ~ c~, the twisted product S k ×z2X supplied with the induced involution and filtration by subspaces 0 C S ° X C S1X C ... C SkX. (Here inclusions are induced by the standard inclusions 0 C S O C S 1 C -'- C Sk.) The homology and cohomology spectral sequences of filtered space S k X are denoted by kE~q(X) and kEPrq(x ) respectively and called the spectral sequences of involution c. (Here mad in other similar notations space X will often be omitted.) Note that if X is a trivial Z2-space, i.e. c = idx, then $ k X = Rp k x X, and H , ( S k X ) = H , ( R p k) ® H , ( X ) . In particular, both the spectral sequences degenerate in term E 1.

2.2.2. Proposition. (1) kE.~. = H . ( R p k) ® H . ( X ) , and kE~'* = H*(Rp k) ® H*(X); (2) d (hp ® x) = ® (1 + and ® x) = h, ® (1 + e*)x; (3) if r >1 2, both the sequences coincide with the Serre spectral sequences of ~bration S k X --, Rp k = Skpt; (4) ~E** is a bigraded H*(Rpk)-algebra, and kE:, is a bigraded kE**-module; if r >1 2, differentials d r and dr are derivatives; (5) kE~q and kE~q converge to Hp+q(SkX) and HP+q(SkX) respectively; the convergence observes both the multiplicative structures;

210

(6)

l <. k, the inclusion homomon

sm 'E;q

(resp., kE q --,

is

surjective (resp., injective) for p <~ l and one-to-one for p <~ l + 1 - r. Proof. By definition, k

1

=

= Hp+q(S p x X , D v_ x X ) = H,+q(D~_ x X , S , - 1 X X ) = Hq(X).

This proves (1). Statement (2) follows from 2.4.4(1) below. (3) is just a rewording of the construction of the Serre spectral sequence using the skeleton filtration of the base of the fibration. (4) and (5) are well known properties of the Serre spectral sequence. (6) follows from the fact that :El, (resp., IE;*) is a truncation of kEl** (resp., kE~*). [] r 2.3. S t a b i l i z e d s p e c t r a l s e q u e n c e . Denote by ~2p : o o ~ +r 1 , , --* p,* oofi'~p + 1 , * :E~ ~ _~ the mapsx~-~hClx a n d x H h o x r e s p e c t i v e l y .

oo

r :E~,, and ~

:

oo

2.3.1. D e f i n i t i o n (Kalinin [5]). Spectral sequences E ~ ( X ) = h__m(°°E;q, ~p;) and

E q ( x ) : h_m(~EV~q, ~ )

are called the stabilized spectral sequences of involution c. Natural maps E~ --~ °°E•q and °°E~q -* E] will be denoted by ¢~ and ¢ ] respectively. 2.3.2. T h e o r e m (Kalinin [5]). (1) E~ = Hq(X), and E1q = H q ( X ) ; (2) d 1 = l + c,, and dl = l + c* ; (3) E* is a graded g2-aJgebra, and Er, is a graded E*-module; d~ and d ~ are derivatives i f r >1 2; (4) provided that X is a tlnite CW-complex, Er, and E* converge to H,(Fix c) and H*(Fixc) respectively; the convergence observes the multiplicative structures, but it does not observe the grading of H,(Fix c) and H*(Fix c); (5) homology maps ¢~ are injective; cohomology maps ~b~ are surjective. 2.3.3. N o t a t i o n . The filtration on H,(Fixc) which arises due to convergence E,~ =:~ H,(Fixc) is denoted by f , ( X ) . The composed maps ~q ~ .~q/Fq+l - ~ E ~ are denoted by eq. Note that filtration 9rq can as well be defined if E,~ does not converge to H,(Fix c) (Kalinin [5]). By definition, the class of cycle x -- ~ x i , xi C Si(Fixc), belongs to 5rq if for some p the image of ~ hp+q-i ® xi E Hp+q(£ °° Fix c) in Hp+q(S°°X) comes from

Hp+q(SPX). 2.4. G e o m e t r i c d e s c r i p t i o n . Let C, (resp., C*) be a chain (cochain) complex of A-modules. Define filtered complexes s k c , and SkC* as follows: as a graduated group g k c , = H , ( R p k) ® C, (resp., S k C * = H*(Np k) @ C*); the boundary (coboundary) operator is the map hi ® x ~-+ hi ® Ox + hi-1 ® (1 + c)x (resp., h i ® z ~-~ h i ® 5x + h I+1 ® (1 + c)z);

211

the filtration is, respectively, Fp = Ira[inclusion, ®id : SvC, --* SkC,], and FV = Ker[inclusion* @id : SkC * -~ SPC*]. The spectral sequences of filtered complexes SkC, and SkC * are denoted by kE;q(C,) r and kE~q(C*) respectively. 2.4.1. T h e o r e m . There exists a natural chain map O : $ k S . ( X ) --+ S . ( $ k X ) which induces isomorphisms of (1) spectral sequences kE~.(S.(X)) ~- kEr**(X) and kE**(S*(X)) ~ kE**(X); (2) homology groups H . ( S k S , ( X ) ) ~ H . ( S k X ) and H*(SkS*(X)) ~ H*(SkX); (3) Smith homology groups H , ( R p k) ® S H , ( X ) -~ SH,($kX).

Proof. Let p be the projection S k x X --~ S k X , and ~ be the diagonal involution on S k x X . Fix some natural chain homotopy equivalence # : S, ( X ) ® S , ( Y ) ~ S, ( X x Y ) and define 0 as the map hi ® x ~-~ p##(Ez+ ® x). Since p#5# = p#, and # is natural, O0(hl ® z) - Oe(hi ® ~) = -~- p##(:~'-']~b- 1 ® C#X -~- ~iq_ ® OX) - p # # ( [~,i--1 i ®Ox)= + ® x + ~i--1 _ ® x + E+

= p##(~-~

® (1 + c#)x -- r i --- ~ ® x) = p#(1 - 5 # j~~ / E i+- 1 ® c#~) = o.

This proves that 0 is a chain map. The subquotient map

ep: sps.( x ) / s p - l s . ( x ) = hp ® S,( X)

S.($vX, Sv-IX)

can be decomposed through the map

~p : hv ® S . ( X )

~ S,(D~ x X , S v-1 x X), h v ® x ~ # ( E i+ ® x ) ,

which induces an isomorphism of terms E 1 of the corresponding spectral sequences. Hence 8 induces isomorphisms of the spectral sequences and homology groups. To prove the last statement, we need the following lemma: 2.4.2. L e m m a . Let f : C, -* D, be a chain map which induces isomorphism f , : H , ( C , ) --, H,(D,). Then ( i d ® f ) , : H , ( S k C , ) -~ H , ( S k D , ) is also an isomorphism for any k.

Proof. i d Q f induces an isomorphism of terms E 1 of spectral sequences kEr**(C,) and kE:,(D.). [~ Lemma 2.4.2 and Proposition 2.1.3(2) imply that 0 induces isomorphism

H.(,.qk(S.(X)/Sm.(X))) ~ H . ( S . ( $ k X ) / S m . ( S k X ) ) , and, due to Five Lemma, isomorphism

H (SkSm ( Z ) ) ~ SH ( S k X ) Since c# acts trivially on S m , ( X ) , the first group is equal to H,(ll~p k) ® SH,(X).

[]

212

2.4.3. C o r o l l a r y (of 2.4.1 and 2.4.2). Theorem 2.4.1 remains valid i[ S . ( X ) is replaced with the cell complex corresponding to some ¢-invariant cell partition of pair ( X , Fix c). (Note that for (1) and (2) it even is not necessary that Fix c be a cell subcomplex of

X.) The remaining statements provide a clear geometric description of spectral sequences kE.~. and E .r. (The dual description of cohomology spectral sequences is also valid.) The term "chain" ("cycle") means either singular or cell chain (resp., cycle). 2.4.4. C o r o l l a r y . Let x be a q-dimensioned cycle in X . Then (1) dl(hp ® x) = hp-i ® (1 + c#)x; (2) let r <~p. Then dr(hp ® x) vanishes in kErp--r,q+r--1 (X~ \ ) if and only if there exist some i-dimensioned chains Yi in X , q+ 1 <~ i ~ q + r , such that (1 + c# )x = Oyq+l , and (1 + c#)yi = Oyi+I t'or i <~ q + r - 1. In this case dr+X(hp ® x) = h p - r - 1 @ (1 + c#)yq+r. 2.4.5. C o r o l l a r y . Let x be a q-dimensionM cycle in X . Then (1) dIx = (1 + c#)x," (2) drx vanishes in E~+r_I(X ) if and only if there exist some i-dimensioned chains yi in X , q + 1 <. i <~ q + r, such that (1 + c#)x = 0Yq+l, and (1 + c # ) y i = Oyi+l [or i <<.q + r - 1. In this case dr+Ix = (1 ÷ C#)Yq+r. Proof. 2.4.4 is, in fact, the definition of the spectral sequence of complex S k S , ( X ) . 2.4.5 is obtained by stabilizing differentials of °~E,~,. []

2.4.6. P r o p o s i t i o n . Let in : Fix c "-+ X be the inclusion. Consider some i-dimensioned cycles xi in Fixc, and denote x = ~ xi. Then x E .Tq(X) if and only if there exist some i-dimensional chains yl in X , i <~q, such that in# xi = Oyi+ i + (1 + c# )yi for i <~ q - 1. ~rl this case eqx = in# xq + (1 + c#)yq. Proof. Replace S . ( S k X ) with t k S . ( X ) . By definition (cf. 2.3.3) x E brq if and only if i n # ( ~ hp+q_i ® x i ) = ~ + O y for somep/> 0, g: E SPSp+q(X), and y E S°°Sp+q+I(X). In this case Cpeqx is the class of ~ in °°E~. Expanding y in the sum ~ hp+q+l-i ® yl we obtain that in# xi = Oyi+l + (1 + c#)yi for i ~< q - 1, and "= h r ® (in# x, + (1 + c#)yq)

+ 0( Z

hp+q+l-i ® Yi)

(belongs to O S P S , ( X ) )

i~>¢+a

+ Z

hp+q-i ® in# xi

(belongs to S " - I s , ( x ) ) .

i~q+l

Hence Cpeqx = h r ® (in# x + (1 + e#)yq).

[]

2.5. H i g h e r inclusion maps. Define filtration S ~ ( X ) on S H , ( X ) and homomorZ;(X) as follows: The class of a cycle x = ~ xi, xi E S m i ( X ) , belongs to S~'~ if and only if

pasms e; : s a : ; ( x )

i~>q--r+l

there exist some chains Yi E S i ( X ) , q - r + 2 <. i <. q, such that X q _ r + 1 = Oyq--r+2, and xi = Oyi+l + (1 + c#)yi for q - r + 2 ~< i ~< q - 1. In this case e;x is the class of xq + (1 + c#)yq in E ; ( X ) . Denote by ~'~(X) the filtration ,9~'~ N H,(Fixc) on H,(Fixc) (cf. 2.1.3(3)). Restriction eqr p; will also be denoted by eq. r

I

213

2.5.1.

D e f i n i t i o n . Homomorphisms eq are called the higher inclusion homomor-

phisras. Remark. In this definition singular chains can as well be replaced with cell chains corresponding to some c-invariant cell partition of pair (X, Fix c). 2.5.3. P r o p o s i t i o n . Homomorphisms eq are well detJned.

Proof. We have to prove the following three statements: (1) xq + (1 + c#)yq is a cycle; (2) d~(xg + (1 + c#)yq) = 0 for s <~ r, so this cycle projects into E l ; (3) the class of xq + (1 + c#)yq in E i does not depend on the choice of yi's. (1) O(xq+(l+c#)yq) ----(1+c#)Xq-1 -[-(l+c#)2yq-1 = 0 since Xq-1 is c#-invariant, and (1 + c#) 2 = 0; (2) dS(xq + (1 + c#)yq) = 0 for any s due to the fact that this cycle is c#-invariant (see Corollary 2.4.5); (3) let y~ = yi--kzi be some other chains. Then Ozq-,-+2 = 0, and (I--kc#)zi ..= cqzi+l for q -- r + 2 ~< i ~< q - 1. Hence (1 + c#)yq = (1 + c#)yq + dr-lzq-r+2 (Corollary 2.4.5), and the image of xq + (1 + c #)yq' in Eqr coincides with that

Of Xq + ( l +c#)yq.

[]

2.5.4. T h e o r e m . (1) 85"¢ ---- (~ SHi(X), and $5"; = Ker eq_ 1r-1;

i>~q

(2) $5"; D (~) SHi(X); restriction of erq to SHi(X) is triviM for i > q and coincides

i~q (in the obvious sense) with s m x for i = q; (a) ss"; c es riction toss"; coincides with e;; (4) statements (1)-(3) remain valid if Ss"q are replaced with 5"q; (5) ~ ; = 5"q and eq = eq for any r/> q + 2 (note that in this case Eq = E ? .

r--1 = xq-1 + (1 + c#)yq-1 = Oyq Proof. If x E Sh"q, then obviously x E ~T'r--I - q - l , and eq_ix vanishes in S~q-1. Hence S~-; C K e r e q - I. Let x E Ker eq_ 1~-1. This means that Xq-1 + (1 + c#)yq-1 = dr-2zq-r+2 + Oyq for some cycle Zq-~+2 E Sq_r+2(X) and chain y'q E Sq(X). Due to Corollary 2.4.5 this is equivalent to existence of some chains zi, q - r + 3 ~< i ~< q - 1, such that Ozi = (1 + c#)zi-x, and Xq_ 1 -~- (1 -'~ e#)Yq--1 = (1 + c#)zq--a + Oy'q. Denoting y~ = Yi -k zi for

q -- r + 2 ~< i ~< q - 1, we see that x E S~-~. This completes proof of (1). Statements (2)-(4) immediately follow from the definitions, and (5) follows from comparing the definition of ,95"~ and Proposition 2.4.6. [] 2.5.5. D e f i n i t i o n . Define map ~q : $5"~ ---* Hq( s r - I x )

_";~

r %,"

2 - r--l r

as the product Hq(Sr-IX)

(here the third map is invertible due to 2.2.2(6); the last map is the edge homomorphism of spectral sequence "-lE~q).

214

2.5.6. P r o p o s i t i o n . Le~ x = ~ xl, xi E SHI(X), and r ~ O. Then i>~q s m s , x ( E hq+r-i ® xi) = 0 in Hq+r(S~X) if and only if x e Sa~q+~+a. In this case s m s . + t X ( E hq+r+l-i ® xi) = -r+2 'r+2 Proof. According to the definition the condition x C s T-q+r+l is equivalent to the equality ~ hq+r-i ® xi = 0 ( ~ hq+r+l-i ® Yi) for some yl C Si(X) (cf. Corollary 2.4.4). In this case ~ hq+~+l-i ® xi = 0 ( ~ hq+r+2-i ® yi) + ho ® (xq+~+l + (1 + c#)yq+~+l); =r+2 _ E Hq+~+I(S~+aX). [] hence this element projects to eq+~+l,

Remark. Since Ca and the edge homomorphisms ~-XE~q -+ Hq(S~-IX) are injective, Proposition 2.5.6 can as well be taken for definition of S~-~ and e~. 3. THE BASIC CONSTRUCTION 3.1 G r o u p s H n - I ( S k S ( ( ) ) . Throughout this section ~ denotes some fixed On-bundle over space X. 3.1.1 Definition. Denote by ~k the tautological linear bundle over Rp k, and by ~k ® the exterior tensor product, which is a bundle over ]l{pk x X. 3.1.2. P r o p o s i t i o n . (1) There exists a natural tJbrewise homeomorphism S(r]k ® ~) = SkS(~), S(~) being considered as a Z2-space via the standard antipoded involution; (2) if space X is connected and k <~n - 1, there exists a naturaJ exact sequence n--t

0-+

~

H i ( x ) -+ Hn_I(SkS(~))

i" Hn-l(Sn_l),

i=n--k--1

where i* is induced by the inclusion ofa/~bre S n-1 o[ l~bration S(~k ® ~) --+ IRpk x X; (3) homomo hism is non-zero if and only ..... = O; (4) if I <~ k, the following diagram commutes n--1 i* Hi(x) , Hn-l(SkS(~)) , H n - a ( S n-l) 0 i=n--k--1

i projection n--1 0

)'

Hi(x)

f restriction

• Hn-I(StS(())

~identity i*

, H"-I(sn-1),

i=n--l--1

Proof. Statement (1) immediately follows from definitions. To prove (2) and (3) consider the Gisin exact sequence of the sphere bundle S(rlk @ ~) -+ Rpk x X: eU Hn(Rp k x X) , H°(Rp k × X) H n - l ( R p k x X)~--~Hn-l(S(yk ® ~)) \i"

n--1

@

H (x)

Hn-l(Sn-l)

i=n--k-1

where e = ~ h n-i ® wi(~) is the characteristic class of the bundle. It yields that Im i* # 0 if and only if e = 0. Statement (4) follows from the naturallity of the Gisin sequence. []

215

3.2. M a p s Xk : H n - I ( S k - I S ( ~ ) )

k

, ~]~ Hn-i(Vn,i(~)). i----1

3.2.1. D e f i n i t i o n . Let f : Y --* V,,k(~) be some map, y ~-+ (x(y),el(y),...,ek(y)), where x(y) e X and ( e l ( y ) , . . . ,ek(y)) is a k-frame in the fibre at x(_y) of the associated vector bundle Rn(~). Define the coordinatewise lift of f as the map f : Y × S k-1 ---} S(~), y × ~ tiei , , (x(y), ~ tiei(y)). (Here S k-1 and S(~) are considered as subspaces of R k and R"(~) respectively.) 3.2.2. D e f i n i t i o n . Fix some natural chain homotopy equivalence # : S . ( X ) ® S . ( Y ) ----+ S . ( X × Y ) and define the following maps: (1) 8ii : Si(V,,k(~)) , SI+j(S(~)), j ~< k - 1. Let x = ~ aa be a singular chain, i.e. a sum of singular simplexes a a : A i -~ V,,k(~). Then 8ij(x) = 0 for j < 0, and O,j(x) = E ( 5 , ) # p ( i d z ~ , ®EJ+) otherwise; (2) Qk,i: Si(Vn,k(~)) , Sk-lSi+k_l(S(~)). By definition, ¢fik,i(x) = ~ h ~ _ j _ ~ ®

0~(~) 3.2.3. P r o p o s i t i o n .

(1) ~k,i is a chain map, so it induces homomorphisms

Hn-k(Yn,k(~)) vk : Hn-I(Sk-is(~))

~k :" (~k,n-k)*

:

, Hn-I(Sk-lS(~)),

and

~ H"-k(V,~,k(());

(2) i [ X : pt, homomorphism ~,k : H u - I ( S k - I S n - 1 ) (3) the following diagram commutes (c£ 3.1.2(2))

Hn-k(Vn,k ) is one-to-one;

) Hn-I(Sk-IS(~))

H"-k(x)

\ H"-k(V.,k(~)). Proof. We prove the dual statements about homology maps ~k. Denote by c the antipodal involution on S(~). Let f : Y --~ Vn,k(~) be some map, and x C S , ( Y ) be a singular chain in Y. Then obviously Oij(f#x) = f##(xQ~J+). In particular, for a singular simplex a : A i ---* Vn,k(~) this implies that 00ij(a ) = 5 # # ( 0 i d A, ® ~ + idA, ®(~j+-I + ~j_-l)) = Oi_l,j(Oa)+ (1 + c#)Oij(a), and cg¢fik,i(a) = ~k,i(Oa). This proves statement (1). To prove (2), consider the inclusion t : S n-k --* Vn,k:

E tiei ,

~, ( e l , . . .

,ek-l, E tici+k--1),

which induces isomorphism L.: H n _ k ( S "-k) --~ H,-k(Vn,k) (cf. Steenrod, Epstein [71); we suppose t h a t k < n). It is easy to see that the coordinatewise lift ~": S n-k x S k-1 --~ S n-1 is a degree 1 map, whose restriction to S n-k x S a-2 coincides with the product const x inclusion : S " - k x S k-2 ~ pt x S n-1. Hence O,_k,j(E "-k) = #(const# En-k ® inclusion# E/+) = 0 for j ~< k - 2, and 0 . - k , k - l ( E . - k ) is a fundamental cycle of S " - l ,

216

which generates H , _ ~ ( S n-~) (cf.3.1.2). If k = n, the same construction applied separately to each cycle ~ in S O gives both the generators of H n _ I ( S n - I s n-l) = Z2 ~ Z2. Prove statement (3). Let a : A i --~ V,,k(~) be a singular simplex. Denote by p the projection Vn,k(~) --+ X and by a z the simplex p o a : A i --* X, and consider the following commutative diagram 5

A i x S "-1

,

id X c°nstI

S(~)

IP s

A i x pt

ax xid

~ X x pt

Since # is natural, (ps)#Oij(a) = #(ax ® const# ~ . ) . I f j > 0, const# EJ+ = 0. Hence ~ k , , - k projects to the map a ~ hk-1 ® #(p#a ® idpt), which induces isomorphism

Hn-k(X)

) hk-1 ® Hn-k(X). k

3.2.4.

+ ~ Hn-i(Vn,i(~)) as the direct

D e f i n i t i o n . Define Xk : H " - a ( S k - I S ( ~ ) )

i=l

sum of composed maps restriction

Hn-I(Sk-lS(~))

)

Hn-a(si-IS(~))

, H rt--i (Vn,,(~)).

3.2.5. P r o p o s i t i o n . For any l <<,k <~ n the following diagram commutes k

,

(~ H n - a ( X )

inclusion*, Hn-I(SI-lS(~))

Hn-I(Sk-lS(~))

i----1

\ k

projection

(~ H"-i(Vn,i(~))

l

)

i=1

Proof. Immediately follows from 3.2.3(3).

V, (~ H n - i (n,i(~)) i=1

[]

3.3. F l a g s t r u c t u r e s . 3.3.1. D e f i n i t i o n . A k-flag structure on bundle ~ is a class x E Hn-a(Sk-lS(~)) whose restriction to every fibre S n-1 is non-zero. The set of all k-flag structures on is denoted by Flagk(~). 3.3.2. T h e o r e m .

(1) Flagk(~) is not empty if and only if wn(~) . . . . .

wn-k+l(~) =

k

O. In this case Flagk(~) is an affme space over (~ H " - i ( X ) ; i----1

(2) if I < k, the i.dusion homomo~phism H"-~(Sk-lS(~)) restricts to an agFme map Flagk(~) ~ Flagt(~);

, H"-I(St-XS(~))

217

(3) ff Flagk(() is not empty, X k restricts to an affine isomorphism Flagk(() k

1-I Stn-i((). The following d/agrarn commutes: (l <<.k)

i=1

Flagk(~)

restriction)

Flagt(()

projection [I~=I Stn-,(~)

' 1~',=1St,_,(~)

Proof. Statements (1) and (2) follow from 3.1.2; statement (3) follows from 3.2.5.

[]

3.3.3.

D e f i n i t i o n . Denote by p the projection S k - I S ( ~ ) ) Rp k-1 x X and fix a k subgroup (not necessary graded) G C (~ H ~ - i ( X ) = g n - l ( N p k-1 x X). A partial i=1 k-flag structure (defined over G) is a class g ¢ H ~ - I ( s k - I S ( ~ ) ) / A n n ( p , lG) whose restriction to every fibre S ~-1 is non-zero. The set of all partial k-flag structures on ~ is denoted by Flagk(~, G). (Remark: a k-flag structure can be thought of as a functional Hn-I(Sk-~S(()) ' Z2. A partial structure is a partial functional defined on p , I G . This explains the term.) In the case of partial structures Theorem 3.3.2 should be reworded as follows: 3.3.4. T h e o r e m . (1) Flagk(~, G) is not empty if and only ifwi(~) = 0 for i ~ n - k + 1. (Note that this condition does not depend on G.) This set is an a ~ n e .space over k (2) if l ~ k, the inclusion homomorphism induces an hi/the map Flagk(~, G) 1

Flagt (~, G fq

f~ nn-i(x)).

i=1

4. FLAG STRUCTURES ON THE FIXED POINT SET OF AN INVOLUTION 4.1. C o n s t r u c t i o n . Let X be an (n + m)-dimensional closed smooth manifold and c be a smooth involution on X. Fix some c-invariant m-dimensional closed smooth submanifold V E X and denote by F v the intersection V A Fix e, by i : F v ~ Fix c, j : F v ~ V and i v : V ~ X the inclusions, and by vv the normal bundle of V in X. 4 . 1 . 1 . T h e o r e m . Let for some k ~ 1 class (iv).[V] belongs to SJ:~+lk(X), or, equivalently, e l [ V ] . . . . . e~+k_l[V ] = O. Then there exists a natural k-flag structure xk on j * v v defined o v e r i~-l.~'nk+l(X). In particular, w p ( j * v v ) = 0 for p > n -- k. Proof. Consider an equivariant tubular neighborhood Tk-1 of S k - l V in s k - l x and denote by OTk-1 its boundary, by pr : OTk-1 --* S ~ - I V the projection, and by in : Tk-1 ~ S k - I X the inclusion. The restriction pr[ pr-~ Fv obviously coincides with the sphere bundle associated with qk-1 ® j * u v , so the orientation )
218

fundamental class of every fibre S '*-1 of pr. We let ,rk be the linking coefficient map. More precisely, consider the following commutative diagram with exact rows: (D -1 denotes the Poincaris duality isomorphism) Xk

I

)

[Sk-lV]

A

Hm+l,(Sk-l X)

rel

N

Hrn+k(,Sk-lX, Tk-1)

0

' Hm+k-i(Tk-1)

in.

.....)

~D-1 H"-l(Sk-aX)

-r-/"-a(Sk-~x \ Tk_l)

, It"(Tk_i,OTk-1)

Due to 2.5.5 the hypothesis of the theorem implies that in.[,Sk-lV] = 0. Hence there exists an element ~k E Hm+k(Sk-aX, Tk-~) such that 0~k = [,9k-av]; this element is unique up to Imrel. Consider D-X~k as an element of H o m ( H ~ _ l ( S k - x x \ Tk-1),Z2) and define ~k as the composed homomorphism H n - l ( p r -1 FV)

inclusion

' H n - l ( s k - l x \ Tk-1)

D- 1~% , Z2

restricted to Ker[inclusion, : H n - l ( p r - 1 Fv) --+ Hn-1 (,.qk-lx)]. Since (5D-l~k = D - I [ S k - I V ] is the Thom class of pr, the restriction of xk to every fibre is non-zero, and ~k is a k-flag structure. The domain of xk is the pull back of Ker[in. : H,~_l(,.qk-lFv) -+ H , _ I ( S a - I X ) ] ; due to 2.5.5 the latter group coincided with y~+a. []

4.1.3. Let (iw),[V] ~ 8,r~+~. Then due to 2.5.3 (iv),[V] ~ SOr~+~, for any l ~< k. This gives rise to series of natural partial/-flag structures xl on j*uv, 1 <. I <~k, defined over ~ + a . On the other hand, due to 2.3.4 x k can be restricted to an l-flag structure 1

x~ defined over subgroup ~-~+1 N (~ H , - I ( X ) , which contains ~"/n+1 (see 2.5.3). i----1

4.1.4. P r o p o s i t i o n . The restriction of xk to ~lnq-1 coincides with *q. Proof. Denote by in : ( S t - i X , Tl-1) ~-+ (,.qk-lX, Tk-1) the inclusion and consider the following diagraxn A~m+k(~k-lX, Tk_l)

It.,,+dSt-~X, Tt_l)

0

0

~ I-fm+k_l(Tk_l)

, II,.+~_~(r~_~)

Since i n ' [ S k - l v ] = [SI-1V], we can choose ~ = in' ~k. Then D-I~I = in*D-l~k, that shows that gl coincides with the restriction of xk. []

219

4.2. Geometric description. 4.2.1. The definition of map Xk (3.2.4), filtration ~ and homomorphisms e~, and the geometric construction of the linking coefficient map yield the following clear geometric description of flag structures xk: (For simplicity we replace ~-~+1 with its maximal graded subgroup. This allows to consider orientations of different dimensions separately.) Suppose that the total space of the sphere bundle S ( v v ) is embedded as the boundary of an equivariant tubular neighborhood of V in X. Let a : P ~ ~ , k ( j * u v ) be an ( n - k ) dimensional cycle (i.e. a singular polyhedron, a singular manifold, a cell cycle, etc.) Consider the composed map ai : P × D + i '--* P × S k-1 ---* S ( j * v v ) ~ X as a chain in X (of the same nature as a), and suppose that there exist some j-dimensionai chains yj inX, n - k + 1 ~<j ~< n, such that Oyn-k+a = a0, and Oyj = (1 + c # ) y j - 1 + #j+k-n-1 for j > n - k + 1. Suppose that y, is transversal to V. Then ~ k ( a ) = # ( Y n N F ) mod 2. Remark. It is this construction that was originally proposed by O.Viro [8].

4.3. P r o j e c t i v e c o m p l e t e i n t e r s e c t i o n s . 4.3.1. Let A be an n-dimensional non-singular real algebraic variety. Denote by CA the set of complex points of A, by RA the real part of A, and by conj the involution of complex conjugation on CA. Multiplication by v/-Z1 induces an isomorphism between the tangent bundle ~ A of RA and its normal bundle V~A in CA. Since RA = Fix conj, Theorem 4.1.1 yields: k+l 4.3.2. T h e o r e m . //' [RA] E 9v~+k(CA), there exists a natural partial k-flag structure on "~A d e a l e d ove In particular, wp(RA) = 0 t'or p > n - k.

4.3.3. Let A be a projective complete intersection. Denote by Ch (resp, Rh) the complex (resp,real) hyperplane section class in H2(CA, Z) (resp, H I ( R A , Z2)), and by RA the double cover of RA corresponding to Rh. 4.3.4. T h e o r e m . Let [RA] is either equal to z e r o o r dual to C h n/2 mod 2 in Ha(CA). Then for every k, 0 ~< k ~< n - 1, RA is naturally suppiied with a reduced k-dimensional Stiefel orientation. Remark. In particular, 4.3.4 implies that w~(RA) . . . . .

w n ( R A ) = 0. Note, how-

ever, that this holds for every projective complete intersection, no matter whether IRA] vanishes or not, since RA is a complete intersection in sphere S N = R p N. Proof. Consider the following diagram

RA

in

: Prc 1 RA

RA

inclusion

inclusion

, CA

* CA

where pr c : CA ~ CA is the S l-bundle corresponding to Ch. CA can be constructed as the pull back of CA under the standard projection S 2N+1 --* Cp N D CA. The complex conjugation on S 2N+1 C C N+I restricts to involution co~-~ on CA such that co~ covers conj, Prc 1 RA is co~-~-invariant, and RA = Fix c~onj.

220

C l a s s [pr~ 1 IRA] = pr~[I~A] vanishes in H n + I ( C A ) .

Since H * ( C A ) is g e n e r a t e d b y

p o w e r s of C h u p to d i m e n s i o n n - 1, H i ( C A ) = H ~ n + I - i ( C A ) = 0 for n + 2 <~ i ~< 2n, -{-1 a n d all m a p s eqr a r e t r i v i M for n + 2 <~ q ~< 2n. H e n c e [Pro 1 R A ] e S.T~2~+I(CA ), a n d T h e o r e m 4.1.1 a p p l i e d t o V = p r ~ 1 IRA y i e l d s a p a r t i a l k-flag s t r u c t u r e x n o n t h e b u n d l e in* ~ y = p r ~ / - ~ A ~-~ p r ~ 7~ A = T~A. T h i s s t r u c t u r e is d e f i n e d o v e r s u b g r o u p 9v~+1 0 for i ~< n. D u e t o T h e o r e m 3.3.2 +1, w h i c h coincides w i t h H . ( R A ) since Hi(C-A) XnXn is a c o l l e c t i o n of r e d u c e d k - d i m e n s i o n a l Stiefel o r i e n t a t i o n s , 0 ~ k ~< n - 1. [] REFERENCES 1. G. E. Bredon, Introduction to compact transformation groups, Academic Press, New York, London, 1972. 2. A. Degtyarev, Cohomotogy approach to structures on G-bundles (to appear). 3. _ _ , , Screwed Steenrod squares and some applications (to appear). 4. W. I. Hsiang, Cohomology theory of topological transformation groups, Springer-Verlag, Berlin, Heidelberg, New York, 1975. 5. I. Kalinin, Cohomology characleristics of real algebraic hypersurfaees, Algebra i Analis 3 (1991). (Russian) 6. V. A. Rokhlin, Complex orienialions of real algebraic curves, Funkz. Analis 8 (1974), 71-75. (Russian) 7. N. E. Steenrod, D. B. A. Epstein, Cohomology operations, Princeton University Press, Princeton, 1962. 8. O. Ya. Viro, Progress in the topology of real algebraic varieties over the last si~ years, Russian Math. Surveys 41 (1986), 55-82. A. 1. DEGTYAREV STEKLOV MATHEMATICALINSTITUTE ST PETERSBURG BRANCH qqq (LOMI) 27, FONTANKA 191011, ST PETERSBURG USSR E-mail: [email protected]

SUBANALITICITY AND THE SECOND PART OF HILBERT'S 16 th PROBLEM Zofia Denkowska

S u m m a r y : The aim of this paper is to clarify the role of subanalytic sets in the proofs concerning the second part of Hilbert's 16 th problem. After a brief introduction (§ 1), we give (§ 2) a somehow modified version of the subanalytic proofs contained, a m o n g other results, in the work of J.-P. Franqoise a n d C.C. P u g h ([FP], 1986). Finite cyclicity of elliptic points is proved in § 2 both by the methods of [FP] and as a consequence of the subanalyticity of the Poincar~ m a p (this is our m a i n modification). In § 3, which is based mostly on the r e m a r k s of R. Roussarie, we give the reasons for which singular points other t h a n elliptic do not a d m i t the same subanalytic t r e a t m e n t . The a u t h o r of this paper h a d worked with subanalytic sets for over 10 years, so n a t u r a l l y t h e p a p e r of J.-P. Franqoise a n d C.C. P u g h ([FP]), published in 1986, immediately drew her attention, as it contains, among other results, some very ingenious applications of the subanalyticity to Hilbert's 16 th problem have their limitations and why. The a u t h o r gathered together both the subanalytic results (§ 2) and the calculus t h a t explains why direct generalization was impossible a n d w h a t obstacles appear for singular points other t h a n elliptic one (§ 3). T h e a u t h o r is very grateful to P. Biler (Wroclaw, visiting Orsay) for n u m e r o u s s u g g e s t i o n s c o n c e r n i n g the proofs in § 3 as well as for his invaluable help in finding the references.

§ 1 Introduction This paper is written in such a way t h a t is could be read both by people having no knowledge of subanalytic sets and by those who know little about limit cycles, d y n a m i c a l s y s t e m s a n d Hilbert's 16 th problem in p a r t i c u l a r (although O.D.E. are more essential to the subject t h a n the subanalyticity). This is why we begin with this introduction, which gives essential facts both about Hilbert's 16 th problem and about subanalytic sets.

222

A. Hilbert's 16th problem

(see [R1], [FP] ou [D] for more details) We use the following notation equivalently: A system of analytic differential equations x' = f(x,y,X) (*)

y'

g(x,y,~.),

with (x,y,~.)~f2xU, f~ open in

~2, U open in ~l, f, g analytic in f~×U (for

simplicity sake we assume that f L U are cubes and 0eft×U) can be also regarded as an analytic deformation X~. of an analytic vector field X o, i.e. an planar analytic

analytically parametrized (by ~.) family of vector fields: X~. = f(x,y,~.)O/Ox + g(x,y,~.)O/3y

with

f

and g as above. Definition in

1.1.

A

limit

cycle

of

X~.

is

a

periodic

trajectory

isolated

the set of all periodic trajectories of X~, .

Problem

1.2.

(a

reformulation

of

Hilbert's

16th

problem

in

the

terms

of analytic functions instead of polynomials, see [R1]). Is it true that the number of limit cycles of X~. is locally uniformly bounded (with a bound independent of ~.) for every analytic deformation X~? To put it more precisely, we need Definition periodic

1.3. ([FP]) A compact subset F of ~2 is called a limit set of the analytic

deformation

XX

if there is a sequence

7n of periodic orbits of XLn such that 7n converges to F in the Hausdorff metric. Remark 1.4. ([FP], [G, p.128]) A limit periodic set of an analytic deformation

X~, is:

either a singular point, or a periodic trajectory, or curve

else in

~2

a

polycycle consisting

(a of

graphic) singular

i.e.

a

points

closed, (vertices)

piecewise analytic and

trajectories

223

for

which

these

vertices

are,

alternately,stable

and

unstable

limit

points

and possibly of lines of zeroes. We

will

not

deal

with

polycycles in

this

paper,

as

subanalyticity

cannot appear for them (cf. [D] or [I1]). Conjecture 1.5.

Finite Cyclicity Conjecture ( [ R1 ]' [ Fp ] ).

Suppose that F is a compact subset of ~2, invariant by X o, which is a limit periodic set for the analytic deformation

XX

has

e>O,

finite

cyclicity

neighbourhood V cycles of than N.

XL,

The

of

Finite XX

nonconstant

i.e.

of ~,--0 such that:

X~, contained

Remark. cycles

in in

the

Cyclicity cannot

sequence),

exist

for all X~V

E-neighbourhood

Conjecture

accumulate

because

there

the of

amounts

(i.e.

n~ ~

and

a

number of limit F

to

cannot

they cannot

of X o. Then it

is

not

saying

form

accumulate on

a sets

greater

that

limit

convergent, other

than

limit periodic sets. Definition planar

1.6. ([AP]

field

for example). Let F be a periodic orbit of the

X o and

T its

local

cross

section

i.e.

an

analytic

curve

transverse to F and intersecting F in the unique point Q. Suppose 7:(-e,E) ~ For

~2, ~0)=Q, is an analytic parametrization of y.

t~(-e,e)

and

~, in

a

neighbourhood

of

0

take

the

solution

of

the

Cauchy problem x' = f(x,y,~) (*)

y' = g(x,y,X),

(x(O),y(O)) = T(t)

and let P

= P(t,~.) be the point of the first return

the

T,

curve

whose possibly

i.e.

parameter the

the

point

x~ (-e,e)

interval

(-e,e)

of has to

intersection the a

of

smallest smaller

one

T

of this trajectory to

and

length (-v,v)

the Ix I. and

trajectory Shrinking using

the

Implicit Function Theorem we obtain an analytic map P:(-~,~)×neighbourhood of X---O ~ (t,~,) ~

x=P(t,~.) e ~,

which is called the Poincar6 map of F in X X. The notion of a Poincar6 map can be easily generalized to the case of F = Q equal to an elliptic point of X o (a singular point with purely

224

imaginary

eigenvalues),

because

for

such

a

point

a

ray

transverse

to all trajectories near Q, for ~. small enough, can be chosen and it is shown

that

all

time ([FP]).

trajectories

close

to

Q

return

to

this

ray

The only difference is that in this case P

after

a

finite

is defined only

in [0,~): P:[0,~)xneighbourhood of k=0 ~

~. For simplicity sake we will assume

here that Q=(0,0) and the

a segment of the x-axis.

We is

will

subanalytic

see

ray

is

later

that

the

the

one

for

(while

Poincar6 a

map

periodic

for

trajectory

an

elliptic

is

point

analytic,

by

the Implicit Function Theorem). Remark. Periodic trajectories of X~, correspond to the zeroes of the map 5(t,~,) = t-P(t,~). The notion of the •st

return map can be generalized to polycycles

(see [Du], [I1], [EMMR]), but then it becomes truly difficult and subtle and not subanalytic. This is why we do not even attempt to speak about polycycles in this work. B. Subanalytic sets We

will

need

only

some

basic

facts

concerning

them.

Detailed

references are to be found in [DES1], [DES2] or IBM]. Definition 1.7. (cf. above and [Ga]) Subanatytic germs in a~R n form the smallest the

family

complement

of

germs

closed

and

projection

and

under

the

containing

finite all

the

union, germs

taking ~f>0~,

~f---0~,for f a germ f an analytic function in a. A set B ~ n is said to be subanalytic in ~n if for each a~R n its germ in a is a subanalytic germ. L e m m a 1.8. ([DES1], [Ga]) Let B be a relatively compact subanalytic subset of ~k×~n and let By, y~R n, denote the section ~x~Rk; (x,y)~B}. Then there exists a constant C~ ~ such that for all y~ ~n the number of connected components of By = #co(By) is not greater than C (i.e. is uniformly bounded).

225

Proposition 1.9. ([I~__,S1]) If a subanalytic set B ~ k × ~ n is ~n-relatively compact (i.e. for each K~R k relatively compact, the intersection (Kx~n)nB is relatively compact) and g:~kx~n ----) ~k is the usual projection, then g(B) is subanalytic in R.

§2 Subanalyticity in Hiibert's 16th problem In either

this paragraph we deal with periodic orbits or algebraically

limit periodic sets which are isolated singular points of X o.

In the case where x o is a singular point of X o we suppose, with no loss of generality, that Xo=(0,0 ) and that for ~. small enough, if there is a singular point near (0,0), it is (0,0) itself.

Proposition periodic

2.1.

set F

If the

for

a

given

Poincar6

map

analytic

deformation

is well defined

X~.

in IxU,

and

its

where

limit

I=(-E,e)

if F a periodic orbit and I=[0,e) if F a singular point, and U is a small neighbourhood

of ~.---0, and

it

is

subanalytic

(its

graph

is

subanalytic

in

~×~l×~), then the limit cycles of X~. cannot accumulate on F. P

r

o

therefore

o

f.

the

subanalytic.

Observe

germ

in

Taking

its

that t=0,

~(t,~.)=t-P(t,L) X--0

of

subanalytic,

the

is set

relatively

also

subanalytic

and

A-~(t,k);

~(t,~.)---O~ is

compact

representant

and projecting it on ~ we get the subanalyticity of the set B = ~teI'; 3 ~.eU' 5(t,~.) = 0 7, where I'~I, U ' ~ U are, respectively, the interval (-e',e') or [0,e') and a neighbourhood of g----0. Now it is enough to remark that each point of B corresponds to a periodic orbit of one of the X~,, k ~ U ' , and each isolated point of B (its 0-dimensional

connected

of the XT~, ~.~U'. cannot

contain

a

component)

corresponds

to

Since #~(B) is locally finite convergent

cycles cannot accumulate on F.

sequence

of

isolated

a limit

cycle

of

one

(cf. [DLS1]) the set B points,

hence

limit

226

X~.

Proposition 2.2. Limit cycles of an analytic deformation accumulate

on

cannot

an elliptic point of X o.

P r o o f.It is well known that for such a point the Poincar6 map is defined

in

[0,e)xW

and U,

take

small

enough

(cf.

the analytic flow

[FP]).

Shrinking

~:[0,T]xW×U

~

Q,

possibly where

[0,e)

U

is

a

neighbourhood of ~.--0, W a neighbourhood of (0,0) and Q a cube in ~2 (cf. [L] for example) such that t~ is analytic in a neighbourhood of [0,T]xWxU. Let us denote t~ = ((~1,~2). We have:

graph P =~(x,~.,x')~

[0,T]xUx~; x'=P(x,k)~. Observe that x'=P(x,~.) ¢:, 3 t~ [0,T] such that ~2(0,x,0,~) = ~2(t,x',0,~.) = 0 and for all x>0, "~
C = {(t,x,x,x ,Z.)' ~ [0,T]x[0,TlxWxWxU; 0<x
and

denote

by

p

the

p(C) and therefore of the complement k,p(C) of p(C). Take now the projection g(t,x,x',X) = (x,x',~.)

and

observe

that,

since t is bounded, the projection rc(~(C)) is subanalytic. It is enough to remark that graph P = ~('~p(C)) and use Prop. 2.1. In

this

proof

taking

t

and

"~ bounded

is

essential,

otherwise

the projections might not be subanalytic. Proposition 2.3. Limit cycles of

X~

cannot accumulate on periodic

orbits. P r o o f. The same as that of Prop. 2.2, only now [-T,T] [0,T]. can

Of be

course,

obtained

the directly

result from

for the

periodic

orbits

analyticity of

P

is

will replace

classical

(Implicit

and

Function

Theorem) without ever using the subanalyticity.

Without the

hypothesis about

the

periods

being

the set A need not be relatively compact and Lemma apply.

bounded

from

above,

1.8 would not

227

Proposition

2.4. Given an analytic deformation

X~. of X o and its limit

periodic set F suppose that: (i) there exists a neighbourhood W of F in ~2 and a T>0 such that all the periodic orbits of X L, ~.eU, periods < T, (ii) the Poincar6 map P:IxU ~

contained

in

W

have

their

prime

~ can be defined for I and U small

enough (with I=[0,e) for F a singular point and I=(-e,e)

for F a

periodic orbi0. Then P is subanalytic in P,x~I. P r o o f. The same as that of Prop. 2.2 Observe that the condition (i) plays a very important role in it.

Remark. It may seem a priori that Prop. 2.1 together with Prop. 2.4 above might give more than Prop. 2.2 and 2.3, i.e. more than just the finite might

cyclicity of elliptic points apply to isolated periodic

and periodic points other

orbits; e.g. than elliptic

that they points.

We will see in §3 that this is not true. On the other hand, Prop. 2.2 and 2.3 are so simple that they have many

other proofs.

As said above,

Prop.

finite

cyclicity

elliptic

we

of

it,

also, in admitting

also

of

subanalytic,

points, given

in

2.3

is classical.

repeat

[FP].

There

below is

a

As

to

the

another

proof

third

proof

[Rz], not subanalytic, but much more useful because generalization to the hyperbolic loops, which is not ' the

case of the subanalytic methods. We

now

give,

after

[FP],

another

subanalytic

proof

of

Prop.

2.3

and 2.4, based on the following:

Theorem

2.5. ([FP]) For any analytic deformation

X X of an analytic

vector field X o and any T>0 the number of limit cycles of X~. of period < T

is

locally uniformly bounded.

228

P r o o f. The

proof is the

same

as that

of

[FP],

here because of the explanatory character of this paper. Take the flow ¢:[0,x]xWxU ---> Q, with x
such

that

~

is

in

a

neighbourhOod

detailed

a neighbourhood

of ~.---0, both small enough,

analytic

more

of

and Q a cube in

of

[0,x]xWxU

(ILl,

[G]). For each k e U , ¢(.,.,~) denotes the flow of X X in [0,'c]xW. Take the set A =3f(t,x,y,X)~[0,x]xWxU; ~(t,x,y,X) = (x,y)~. Because of the hypothesis above, this is a relatively compact, subanalytic subset of ~x~2x~/. Applying Lemma 1.8 to the sections A(t,k)4(x,y)eW;~

¢(t,x,y,~.)=(x,y)}.

such that #~(A(t,~.))<_N

we

get:

there

exists

a

constant

N

for all (t,k)a [0,x]xU.

Observe that A(t,X ) is composed of all periodic orbits of X~, for which t is a period. We assign now to each limit cycle of X7~, X~U, its prime

period

to. If to<x, we

component

of A(to,7~ ~.. in

which

can this

assign limit

to

to the

cycle

lies

unique

(in

limit cycle is in itself a connected component of to its period). We therefore have the injection limit cycle of X k --~ (to,X) ----> connected component o f A

fact

connected each

A(to,~.),

where

and

therefore # (limit cycles of X X, X~U, prime periodS.~
Since x was chosen

obtain the theorem. Remark that, as for any reasoning

arbitrarily

among

numbers

< T,

that involves subanalyticity, we

need the existence of the bound T on the periods,

otherwise Lemma

1.8

and Prop. 1.9 will not apply. Now

all

we

need

to

prove

Prop.

2.2

(Prop.

2.3

is

an

obvious

limit

periodic

consequence of Thm 2.5) is the following: Proposition

2.6.

If

(0,0)

is an

elliptic

point

of X o and

a

set of X~, then there exists a neighbourhood W of (0,0) in ~2 and T>0 such

that

all

the

prime period < T.

periodic

orbits

of

X~.

contained

in

W

have

the

229

P r o o f. (cf. [A] Chapitre 1 and also [BN], problems 1.668 and 1.558) Observe that an elliptic point of (*) i x'=f(x,y) Ly'=g(x,y) f, g analytic, is either a focus or a centre (of. [G] for instance), while it is of course always a centre for the linearization

(**)I"x'--d(°'°)f(x'Y) {.y'--dc0.0)g(x,y) and the periods of periodic orbits of (**) are constant, the angular velocity being equal to co= IIra g I, where ~t is one of the complex eigenvalues of (0,0). We will f'trst show that the angular velocities trajectories of (*) tend to co as the trajectories tend to point (0,0). To this end, take the simplest blowing up i.e. ¢:slxR 9 (x,y,r) --~ (rx,ry) • ~2,

on the

periodic elliptic

x-Z+~'Z=l,

{

which in local coordinates becomes:

(')

x--rcos~

y=rsina. The blow up of the system (*) takes the form: t r'=f(rcos~,rsina)cosa + g(rcosa,rsina)sina (#)

a'--r'l[g(rcosa,rsina)cosa - f(rcosa,rsina)sinal We calculate the angular velocity a ' using

the

Taylor

expansions

of f and g in (0,0) and substituting (o): f(rcosa,rsina) = r0f/~x(0,0)cosa + rOf/Oy(0,0)sina + r2Pl(a) + ... g(rcosa,rsina) = r~g/~x(0,0)cos~ + r0g/Oy(0,0)sina + r2Ql(a) + ... where Pi(0c), Qi(o0 are homogeneous polynomials of variables cosa,

sina,

of degree i. Up to an analytic conjugation the Jacobian matrix of if, g) has the form:

so we obtain a ' = r'l[fcosa-gsina] = co+rF(r,a). Therefore a ' tends to co as r tends to 0. All the periodic orbits of (*) remain periodic after the polar blowing up. Their periods tend to the double period of the is closed orbit slx(0)=~r--0} as the orbits approach slx(0). Since proper analytic and a two-leaf covering outside slx(0)-~1(0,0), we

230

obtain that the periods of periodic is the double period of slx(0) in (#).

orbits

of

(*)

tend

to

4x/o~,

which

It is not as much their limit that interests us, it is the fact their periods are bounded by, say, D--4rdc.o+l/2 in a small neighbourhood

that

of (0,0). Take now the system (*) with parameter: (x'=f(x,y,X) ,

[ y =g(x,y,X) i.e.

take our analytic

of the

deformation

solutions on parameters

of X o. From

we obtain

the continuous

that 0 ~

defined

dependence

as the

angular

velocity for linearized (*~) when X~. has a centre in (0,0) and as coo otherwise,

where

o~o is

the

angular

velocity

for

linearized

(*o),

tends

to coo as ~ tends to 0. This implies that periodic orbits of (*~) near (0,0) have the periods uniformly bounded (by D=4rd0)+l/2, for example). A different proof of Prop. 2.5 is to be found in [FP].

§3 The reasons for which the use of subanalyticity stops here

than

We repeat again that we do not discuss limit periodic sets other periodic orbits and singular points because (cf. ~ u ] , [EMMR]

[If], [I2]) polycycles.

(ii)

no

As

was

of

Prop.

subanalytic

said

above,

2.4

might

approach

it may be

can

seem

verified

be

a

successful

priori

for

that

singular

or

for

hypothesis points

(i)

other

and

than

elliptic ones. To show that this is not true, we make the following: Observation

3.1.

Suppose

(0,0)

is

a singular

point

X o and a limit periodic set of its analytic deformation If (0,0) is:

of

the

analytic

XX.

(a) nilpotent (two eigenvalues zero, linear part non-trivial), (b) semi-hyperbolic (one eigenvalue zero, one non-zero), (c) completely degenerate (linear part vanishing), (d) hyperbolic, then there is no uniform bound on the periods of periodic orbits of XX converging to (0,0).

field

231

P r o o f. is a

Case (a)

sequence 7.

The point (0,0) being a limit periodic set, there

of periodic orbits of X~. converging

Hausdorff metric. Let 1". denote y. take the smallest rectangle

the prime containing

to

(0,0)

in

the

period of T.. For each it, of length R. (variation

of x) and height L. (variation of y). We will consider R and L two variables tending both to zero and will estimate T as R and L tend to 0. Take the simplest normal form for nilpotent points ([AP], [GH]): x'=y+a(x,y) y'=a(x,y), where the Landau symbol a(x,y) is understood as (x,y,X) tends to 0. Suppose T is bounded for periodic orbits near (0,0) i.e. T=O(1) as R and L tend to 0. We have: By the first equation, R = variation of x < max Ix' I T=LT, hence R=O(L) as L tends to 0. By the second equation, L<(o(R)+o(L))T. Since T=O(1)

and

R=O(L),

we

obtain

thus

L_
which

is

impossible.

Case (b) The normal form for semi-hyperbolic points: x'= cx+o(x,y) y'=a(x,y), where c~0, o(x,y) are understood as above. Suppose that the minimal period T of converging orbits is bounded near (0,0). Observe that, for a periodic trajectory ((x(t,~.),y(t,~.)), there are two possibilities: either

(1) m~ ~0, r 1x(t,~.)>0 or (2) m~ r~o "T! X(t,~.)<0 .

Suppose

c>0.

Take

case

(1).

The

first

equation

implies

that,

the point (to,ko) which realizes the maximum, we have is impossible. This reasoning is valid for small X, of course.

x'(to,Xo)>0,

Case

are

(2)

is

altogether

impossible

when

neighbourhood of (0,0), with ~. small enough.

c>0

and

we

in

a

at

which small

232

If

c<0,

the

reasoning

is

analogous,

the

cases

(I)

and

(2)

being inversed: case(t) impossible near (0,0), for L small; case (2) excluded by the first equation, which would imply x'(to,~.o)<0 at the minimum point of x(t,~.). Case (c___)) Normal form is now: x'=a(x,y) y' =a(x,y) with a understood as above. On a periodic orbit of diameter r the velocity v=a(r) as r tends to 0 (as a consequence of the equations above for ~. small enough).

Therefore

their periods T satisfy: T"w~/a(r) and cannot be bounded as r tends to O. Case (d) This case is of course immediately excluded by the Grobman-Hartman theorem, but we can also reason as above, in a completely

elementary way, taking the normal form:

x'=cx+a(x,y) y'=dy+a(x,y),

c#o, d#o

(real eigenvalues)

or

x'=ax+by+a(x,y) y'=-bx+ay+a(x,y), a#o, b ~

(complex eigenvalues, real part#0).

The reasoning is analogous to that of case (b), consider only the case (1) max(x2+y2)>0 on [0,To]. (a),

It is not hard to construct (c), where arbitrarily small

course, their periods are unbounded.

except

examples illustrating periodic orbits do

that

now

Observation exist and,

we

3.1 of

233 References

[A] Arnold V.I., Chapitres suppldmentaires de la diffdrentielles ordinaires, Editions Mix, Moscou (1980)

thdorie

des

dquations

[AP] Arrowsmith D.K., Place C.M., An Introduction to Dynamical Systems, Cambridge University Press (1990) [BM]

Bierstone

E.,

Milman

P.,

Semianalytic

and

subanalytic

sets,

Publications IHES n° 67, pp. 1-42 (1988) [BN] Biler P., Nadzieja T., Differential Equations: A Collection of Exercises, Problems, Theorems, Examples and Counterexamples, Wroeraw (1990) [D] Denkowska Z., What has happened to the Hilbert 16th Problem, University of I.xxtz Publications, Workshop in Sielpia (1988) [Dt~S1] Denkowska Z., ~ojasiewicz S., Stasica J., ~l~mentaires des ensembles sous-analytiques, Bull. sdr. math. vol. XXVII, n° 7-8, (1979), pp. 529-536 [I~S2]

Denkowska

Z.,

Lojasiewicz

S.,

Stasica

J.,

Certaines propri&& Acad. Pol. Sci.

Sur

le

th6or~me

de

compl6mentaire d'un sous-analytique, ibidem, pp. 537-539 [I)LS3] Denkowska Z., Lojasiewicz S., Stasica J., Sur le nombre de composantes connexes de la section d'un sous-analytique, Bull. Acad. Pol. Sci. sdr. math. vol. XXX, n° 7-8, (1982) pp. 333-336 [Du] Dulac H., Sur les cycles limites, Bull. Soc. Math. France 51 (1923) pp. 45-88 [EMMR] Ecatle J., Martinet J.,

Moussu R.,

Ramis J.P.,

des cycles limites, Publications IRMA, Strasbourg (1987)

Non-accumulation

234

~P] Fran~oise J.P., Pugh C.C., Keeping track of limit cycles, f. Equations, 65 (1986) pp. t39-157 [G] Godbillon C., Syst6mes IRMA, Strasbourg (1979)

dynamiques

[Ga] Gabrielov A.M., Projections Appl. 2 (1968) pp. 282-291

of

sur

les

surfaces,

semi-analytic

Publications

Funct.

sets,

Diff.

Anal.

[GH] Guckenheimer J., Holmes Ph., Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer Verlag (1983)

[I1] II'yashenko Yu.S., Limit cycles of nondegenerate singular points on the real (1984) pp. 199-209

polynomial vector fields with plane, Funct. Anal. Appl. 18

[12] II'yashenko Yu.S., Finiteness theorems for limit Russian), Uspekhi Mat. Nauk t.45, vyp.2, (1990) pp. 144-200 [L] Lefschetz S., Differential Equations: Geometric Publications, New York (1977) [RI] Roussarie R., Note on the 16th Problem, preprint Dijon (1987)

Finite

[R2] Roussarie, R., Cyclicit6 finie Nonlinearity 2 (1989) pp. 73-117

des

Cyclicity

lacets

et

cycles

Theory,

Property

des

points

(in

Dover

and

H.ilbert

cuspidaux,

The decidability of real algebraic sets by the index formula J.-P.

F R A N Q O I S E AND F . t ~ O N G A

A b s t r a c t . W e propose a n a l g o r i t h m to decide if a real Mgebraic set defined by p o l y n o m i a l s with integer coefficients h a s a n o n e m p t y intersection with a given b a l l by t h e index f o r m u l a of Kronecker. We a p p r o x i m a t e t h e integral by a R i e m a n n s u m a n d we give a n e s t i m a t e of t h e t i m e of c o m p u t a t i o n which is needed. T h e m e t h o d is well a d a p t e d to the u s e of parallel time computations.

In a recent article, L. Blum, M. Shub and S. Smale [B-S-S] introduced the classes of P-problems and N P - p r o b l e m s over the real numbers• They proved that the decidability problem of real algebraic sets is N P - c o m p l e t e and they raised the question: is P = N P ? analogous to the well-known problem over the integers. This motivates the effective understanding of the geometry of real algebraic sets, to which we wish to contribute with this work. This subject has been recently considered by several authors (G. Collins [Co], J. Canny [C], M.F. Coste-Roy, Heintz, Solernb [H-R-S], J. Renegar [Re], D. Wrotman IT]). Our aim is more modest, since we do not pretend to find points on an algebraic set but merely want to decide wether it is empty or not. We use the index formula to count the geometric number of points at minimal distance from the origin in the given ball of a suitable perturbation of a polynomial equation. The difficulty is to find an explicit perturbation. Our approach is straightforward and in principle our result can be used directly to make explicit calculations, however the numbers involved may become extremely large as the dimension and the degree increase. It is well adapted to parallel computation as it needs the same computation to be done several times• G. Stengle has made several suggestions to improve our results, that we plan to develop in a joint paper. 1. T h e a l g o r i t h m . We work in a given ball B ( 0 , r ) = {x e R "-a I ~,x~ <_ r2}. Let f(x) e Z[xl,... ,x,~-l] be a polynomial of degree at least 2 with integer coefficients and let

v ( f ) = {x e B ( 0 , , 9 f

= 0}

We want to decide whether V is e m p t y or not. Let h(x) = x t + . . • + X tn-1 - 1, where g is even and g > sup{d,4}; set A =(1-c)f(x)+~h(x)

,

~eR

and set V, = V ( f , ) . Consider the m a p 6 : Ve --* R, 6(xl,. . . , xn-1) = xl2 + ' - • + X 2n-1. We build an explicit ¢ such that V(f~) is smooth, 6 is a Morse function on V ( f , ) and V(fe) Typeset by AMS-TEX

236

is e m p t y if and only if V ( f ) is empty. Let g* : R" --* R" be the algebraic m a p whose coordinates are:

f,(~l,...,~,,-l)

Of, Oxi

x,xi,

i = l...n-1

and consider the set of equations: g¢ ~ 0 D ( x ) =0

(1) where D ( z ) = det [ o _ ~

ka=iJij=i .... "

It is readily verified that if ¢ is such that (1) has no

solution, then V~ is s m o o t h and 6 is a Morse function on Ve. Consider the set of equations: rt--1

(2)

~

x 2i

_

r2 = 0

,

ge(Xl,...,Xn)

= 0

i=1

where r is fixed. T h e y express the fact t h a t the sphere of radius r centered at the origin is tangent to V~ at the point x. We can assume without loss of generality that f ( 0 ) > 0 and that r > 1. It can be verified t h a t the a s s u m p t i o n on £ and r make sure that the resultants of (1) and (2) obtained by eliminating x l , . . . , xn do not vanish identically. We can therefore apply [1%o, prop.l.1] to the set of equations (1) (with substitutions p --+ n + l , q ~ n, mx --~ d', rn2 ... mn "-~ d ' - l, m , + l --* ( d ' - 2 ) ( n - 2 ) + d ~, H ---* H ' = n l ( d ' ) " - 2 ( d ' ( d ' - - l ) H ) ") and (2) (with substitutions p --* n + 1, q --* 1, rnl --+ 2, m2 --~ d', m 3 . . . m,,+l --* d' - 1), where d' = sup {d, 4}. In fact, careful reading of the proof of [1%o, prop. 1.1] shows t h a t the estimates t h a t they provide can be improved by replacing the integer N by 1; we will make use of this. PROPOSITION. Let f ( x ) = ~ , e s f ~ , x ~, with 1 < [f~,[ < H and [a[ < d for a 6 S. Let

1 + (n + 1)o-l(H,).s! where s = (2a'+(,-2),(a'-2)+d') and let 1

C2 -~-

where t = ( ~ + d ' + ( q l ) ( a ' - , )

I + 2t-l(dH)tt!

Then i f 0 < ~ < i n / { ~ , , ~ } ,

the ~gebr=c , ~ i e t y V,, is

smooth, 6 is a Morse function on V ~ and V ( I ) is e m p t y if and only if V_, is empty.

T h e last assertion of this proposition uses the a s s u m p t i o n f ( 0 ) > 0.

237

It follows t h a t if we take for instance c = -½inf{el,e2}, then V(f) is e m p t y if and only if V, is empty. Since we have to work with polynomials with integer coefficients, we

replacefeby ( 1 - 1 )

f+h.

We want to c o m p u t e the geometric n u m b e r of solutions of the equation g~(x) = 0 with ( x l , . . . ,x,,_~) E B ( 0 , r ) . A simple calculation shows t h a t for such a solution we have:

Iz,,I <-

gH2N f(O)

where N = # {a I a,, # 0}. Therefore we can replace the last coordinates x , by (that we again denote by z . ) so that we can work in the ball

=eR"l

B=

0 x, tH-~Nr

= 2, _< : r 2 )

~ i=1 ...n

Let us introduce p = ~/x~ + ' " + X~, the vector field:

X =

zi 0 ( - 1 ) i pn Oxi

~ i=l...n

mid the volume form Q =

dxl A ... A dxn.

T h e index formula reads :

fo g* (ixa) = vol(B)(g+ - N_) B

with N + = # ( D - I ( R + ) N g - l ( 0 ) ) and N _ = # ( D - I ( H _ ) 1"1g - l ( 0 ) ) . It is not possible to decide whether V, is e m p t y or not with this formula. But, following [P] a slight change will do it. We introduce a n + 1 *h variable x , + l a n d define g~ : R n+l --* N '~+i by:

g~(Xl,...,x,,x,+l)=

gi(xl,...,x,.,),ifi < n + l, xn+lD(xl,...,x,)

, ifi = n+ 1

~i=l...n+l(-1) i pt(n+l) xi Oxt o ' with p,2 = dxl A... A dx,+l. Let B I be a ball in R " + I ; it

the vector field X ' =

x12 + "'" + x 2n+l a n d the

volume form Fg = V(f,) x {0} fl B ' is e m p t y if and only if:

is easy to show now t h a t

z = f0 g'" (ix, a') = BI

oot(B')N~. = o.

We can approximate the integral I by a Pdemann s u m E such t h a t : 1IIt is possible to decide if V 13 B is e m p t y or not on the R i e m a n n sum.

E I < vol(B')/3.

238 2. E s t i m a t e

of the number

of operations

We estimate the n u m b e r of operations necessary to c o m p u t e the integral I . TIIEOREM. Given the aJgebraic m a p p i n g g' : R "+1 ~ R n+l and a ball B'(O, r) o[ ~qxed radius r it is possible to decide ff g,-1 (0) f3 B(O, r) is e m p t y or not in a polynomial time by paraJ1el computer.

PROOF: Let us analyse more carefully the integral involved in the index formula. It splits into n + 1 integrals: n+l

_

/3'

(gl2 + . . . + g~+l)(n+,)/2 dxl A ' " A d x i A ' " A d x n + I

with Ai =

g~

Ogtl/OXl

•

:

t lg.+l

. . •

Og~/OXl

". •

Og'+x/Oxl

.

"

. .

". "

~

I

Ogtl/OXn+l

, * .

.,T,

• I

Og.+l/O,

Og.+l/Ox.+l

. . .

We can assume that n is odd, because if not we can replace gS by g" : R n+2 --* R n+2,

f ( x , , . . . , x.+2) = (g'(x,,..., =,+,), ~,+2). Now we have to repeat N times the same computation, namely evaluate the function inside the integral: Ai ¢i = ,2 ~(.+I)12 (g~2 + . . . "4- gn+l) at each point of a R i e m a n n subdivision• It is well-known t h a t the c o m p u t a t i o n of a determinant can be done in polynomial time (as function of the size of the matrix). And so we obtain t h a t this algorithm needs a parallel polynomial time. It is quite easy to get the number of terms which are necessary for the R i e m a n n sum. We assume t h a t the functions ¢i(x) axe normalized so that sup

0¢i

~

, x e B(O,r), k = l,...,n

+ l

} < C.

Let By be a Riema~n subdivision of 0 B ' ( 0 , r) with N elements centered at Pi, J = 1 , . . . , N.

l/o

Bt

^

¢,d¢, ^ . . . A d~, A . . . A a ~ . + , -

-

~, ¢,(Pj)

<

2

< i

239

v o l ( B t ) r d - l ( 1 ) ° l ( ~ B ' ) ) l[n If we want to have :

I/0B' d21dxl A " " A dxi^ A ' "

A dXn+l --

-

-

¢i(Pj)

I< v o i ( B ' ) / 3 ,

we need :

N

>

(3Cr)n.

So we have obtained that N is simply exponential in the dimensionn. S. Smale [S] computed the average complexity of several approximation methods of computation of l-dimensional integrals. It would be interesting to extend his results to higher dimensional cases for further studies of the index formula. I~EFERENCES [B-S-S] L. Blum, M. Shub, S.Smale., On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions, and universal machines., Bull. Amer. Math. Soc. 21 (July 1989), 1-46. [C] J. Canny, Some algebraic and geometric computations in PSPACE, 20th annual ACM Symposium on Theory of Computing. (1988), 460-467. [Co] G.E. Collins, Quantifier elimination for real closed fields by cylindrical algebraic decomposition, Lecture Notes in Computer Science 33,134-183. [H-S-R] J. Heintz-M.-F. Roy-P.Solern6, Sur la complexitd du principe de Tarski-Seidenberg~ Bull. Soc. Math. France t. 118 (1990), 101-126. [P] E. Pieard, Sur le hombre des racines communes ~ plusieurs dquations simultandes, Journal de math~matiques pures et appliquSes (1892), 5-24. [Re] J. Renegar, A faster PSPAGE algorithm for deciding the existencial theory of the veals, 29th Annual Symposium on Foundations of Computer Science (1988), 291-295. [1~] F. Ronga, Recherche des solutions d'indquations polyndmiales, Ast~risque 192 (1990), 11-16. [S] S. Smale, On the e~ciency of algorithms of analysis, Bull. Amer. Math. Soc. 13 (October 1985), 87-121. [T] D. Trotman, On Canny's roadmap algorithm : orienteering in semi-algebraic sets, "Proceedings of the Warwick Singularities Symposium," Springer Lecture Notes in Mathcmatics (to appear).

Keywords. Real algebraic geometry, index formula, NP-problems. 1980 Mathematics subject classifications: Primary 32C05. Secondary 12-04. J.-P. F. : Universit6 de Paris VI, UFR 920, Math~matiques, 45-46, 5~me dtage, 4 P1. Jussieu~ 75252 Paris, France. F. R. : Section de Math~matiques, Universit~ de Gen~ve, C.P. 240, CH-1211 Gen~ve 24, Switzerland. F~mail:RongaOibm.unige.ch

PROPER POLYNOMIAL MAPS: THE REAL CASE

3.M. GAMBOA(~): Fac.

Matem~ticas.

28040 Madrid.

C. UEN0: Fac.

Complutense

Spain.

Dpto Geometrga y Topolog~a Univ.

Madrid.

Spain

R be

a real

Let

Algebra

Univ.

MatemAticas.

28040.

Introduction.

Dpto.

Complutense.

closed

field.

If x = (x, ..... Xn ) E R n,

r ~ R, r>0, we denote

ItXll = %//X2+''1 "+x2n ' ' B n ( X ' r )

~ (x,r)

{y

=

~ Rn:

sn-1(x, r) = {y ~ Rn: The balls

euclidean

are

a

considered

basis endowed

= {y E Rn:

lJy-xu ~ r}

(closed ball)

IIy-xll = r}

on R n is

of

sets.

with

its

the

subsets of R n form the smallest

in w h i c h

follows

R n will

topology.

collection

ball);

p~ = (0 ..... 0, I)

topology

In what

euclidean

(open

; R + = {r ~ R: r > 0};

(n-l-sphere);

topology open

Ily-xl[ < r }

The

the be

open

always

semialgebraie

of subsets of R n containing

those of type U(f) = {x where and

e

Rn: f(x)

f E R[Xl'''''Xn ] is a polynomial, intersection,

between

semlalgebraic

its graph Gr(f) function

and

X c R n and

is a semialgebraic

f:U .......>. R,

and

complementation. subsets

where

> 0}

A

stable

under

continuous

Y c Rm

is

finite map

union

f:X

semialgebraic

) Y if

subset of R n+m. For a s e m i a l g e b r a i c

U is an open

semialgebraic

copy from the case R = ~ the notion of derivability.

(~) P a r t i a l l y s u p p o r t e d by C.I.C.Y.T PB 89/0379/C02/01 Plan ERB 4002 PL 910021 - (91100021)

and

subset

of R n, we

The ring S~(U)

Science

is

241

the

set

of

derivatives

semialgebraic of

semialgebraic

each

order

functions.

function.

A

map

coordinate

function

functions and

An

J

introduced

called

semial~ebraically

in

p:XxZ Y

and

g

and

9.4

the

in

Sm(U)

192]:

proper

closed,

identity

if

for

i.e.

semialgebraic

p maps

on

also

This

Y,

[Br.,

together

characterization

f : X ......~ Y

subsets

ma~

J

a

Nash

if

each

is a polynomial semialgebraic

map

f:X

> Y

semialgebraic

f(x) = g(y)} every

semialgebraic

map

[D-K 1] -see

Let

Nash

every

it

is map

~ Z closed

follows

X c Rn,

a

immediately closed.

[D-K I, Thm

12.5]

that

Also,

from

that

subset X of R n is proper

with

the

if and

provide

proper maps,

us

which

in this note:

semialgebraic

Y c Rm.

of Z. Taking

it follows

of semialgebraically

be

semialgebraic

subset p(C)

8.13.5]-

is the starting point of our considerations

T h e o r e m 1. -

called

If each f

proper maps are semialgebraically

if X is bounded.

the following

a

(continuous)

is

a semialgebraic

constant map on a closed semialgebraic only

is

are

partial

projection

onto a closed

semialgebraically

in

have

The notion of proper

= {(x,z) ~ XxZ:

is semialgebraically

9.1

[DKl,pg.

> Y, the canonical

Z = Y

f

is a Nash function.

map was

subset C of X x Z Y

them

> Rm

map.

which

of

f = (fl ..... fm):U f

> R

all

element

we say that f is a polynomial

g:Z

f:U

The

map

following

between

the

statements

are

equivalent: (1)

f

is

semialgebraically

proper.

(2)

f

is

semialgebraically

closed

and

its

fibers

are

closed

i n Rn a n d

bounded. Our polynomial

main

results

mappings:

are

refinements

of

this

theorem

for

Nash

or

242

T h e o r e m 2.- Let Nash map.

f:R n

> R m be a n o n - c o n s t a n t

Then f is semialgebraically

T h e o r e m 3.- Let f:R n semialgebraically

semialgebraically

proper.

> R be a non-constant

proper

if and only

polynomial

if there exists

map.

obtain

some

consequences

of

these

results.

Then f is

some M E R ÷ such

that the fibers f-l(t) are bounded for every t ~ R with We

closed

Itl > M.

For

example,

from

Thm 3 it follows: Corollary is

I.- Let

f:R n

semialgebraically

that

for

every

) R m be a n o n - c o n s t a n t

proper

if and

t ~ R + with

t>M,

only

the

if

polynomial

there

inverse

map.

exists

image

Then f

M E R ÷ such

f-l(sm-l(O,t))

is

bounded.

The r e s u l t s . proof

of

For

an easy

simple proof Proposition

the but

convenience useful

is omitted I.- Let

semialgebraic

> Y,

proper.

Proof.-

prove

g:Z---) Y

is

projection is

must

closed

a

Y c R m,

Then, that

if

Z.

map,

begin pg.

But

Xx Z Y

Our

that

says closed.

Proof of theorem 2.- Using

semialgebraic

is

the

Hence p(C)

Thm.

a

p:XxyZ

semialgebraic

= hog(z)}

semialgebraically

[D-KI,

T c Rp

Z c Rq

XxTZ = {(x,z) E XxZ:hof(x) hypothesis

in

we

with

192],

the

whose

,

such

maps

between

that

hof

f is also s e m i a l g e b r a i c a l l y

semialgebraic

and C is a closed in

reader

stated

h:Y ......)...T be

X c R n,

semialgebraically We

result

the

there:

f:X

subsets

of

subset

is

a

is

closed

projection

1 we must

prove

subset, canonical then p(C)

subset closed

p':XxTZ

is closed

only

the

of XxyZ,

and so C is also

= p'(C)

proper.

semialgebraic

....>.. Z

is

of

in XXTZ. ....). Z

is

in Z.

that

the

fibers

243

of

f

are

bounded.

b ~ R m.

Then,

Let

us

for

assume

every

Mr = f-l(B (b,I/r))\Bn(0, r) since

f is non-constant,

Prop.

8.1.13].

Thus,

that r ~ R ÷,

is open

Rn\f-l(b)

the

f-l(b)

and

is

the

unbounded

intersection

some

semialgebraic

non-empty.

is a dense

for

On

the

subset

other

of R" by

M n(Rnkf-l(b))

set hand, [B-C-R,

is non-empty

for

r

every r ~ R ÷. In other words, S = {(r,x) projects

onto R ÷ under

apply now Hardt's the

triple

that

of R+kV.

than

points

all

component

h:TxF

or,

Then,

In particular, in

V

and

arbitrary

[B-C-R,

a finite

T

set

point

the

map

H:S

be

F and

H :R÷xR n 2

> Rn

= ~2 o ~ : [ r o , ~ )

from Tarski

is

) R n we

semialgebraic

contradiction

> R÷

take the

a point

proves theorem,

are

subset the

9.3.2]

V c R ÷ such

~ R + bigger

o

semialgebraic

of ff over T means

TxF

i:T

to

that

homeomorphism

) T. The set F is and

we

pick

an

> TxF by i(r) = (r,p)

map TxF h

~-I(T ) C S c R+xR n

the

canonical

going of

theorem.

see e.g.

r

connected

surjective,

a map

) r. We

semialgebraic

a semialgebraic

is

p ~ F. Let us define

the semialgebraic

subset

over each connected

r . The triviality o

~:[ro,--> ) c T i

there

its corollary

let us

let

a semialgebraic

since

and consider

closed

> R÷:(r,x)

) ff-l(T) such that ~oh is the projection

non-empty

If

map ~:S

exists

trivial

subset

x ~ M r , f(x)~b}

better,

there

of R+kV containing

exists

r>0,

the semialgebraic

ff is semialgebraically

component

there

e RxRn:

theorem

(S,~,R÷).

the semialgebraic

to

R n,

but

The

[B-C-R,

is some point a ~ Adh(N)\N.

show

projection

that f(N)

N = ~([ro,~)) is

not

semialgebraicity

Prop.

and

2.2.7].

closed. of

N

a

This

follows

Let us assume

For some r>r O, a E Bn(0, r).

is

that

Then,

if

244

s>r,

the

point

(s,~(s))

= ¢(s) e S

and

so

In

~(s) ~ M .

particular

s

~(S)~Bn(O,s).

Thus ~(s)~Bn(O,r).

Consequently

Bn(O,r) But ~([ro, r]) 2.5.8]

and

is a closed

it does

n N c ~([ro, r])

semialgebraic

not

contain

a.

subset

Thus

of R n, by

[B-C-R,

A = B (O,r)\~([ro, r])

Thm. is

an

n

open

semialgebraic

a~Adh(N).

To

b ~ f(N),

but

since

neighborhood

prove

that

f(N)

b e Adh f ( N ) . .

(r,~(r))

e S.

of is

For

Hence b ~ f ( N )

a

which

not

closed

every

e B (b, 1 / r )

meet

shall

it

N,

check

is

hand,

i.e.,

if

that

f(~(r))vb, ¢ e

R ÷

there

(r,~(r)) ~ S, we get

¢ B (b,¢)

m

and so the intersection

we

other

exists r ¢ [r0,~) such that i/r<¢ and since

f(~(r))

not

r ¢ [ro,~)

On t h e

•

does

m

f(N)r~ (b,¢) is non-empty

as desired.

m

Remark.fibers

Thm 2 have

is also

dense

for

complement

semialgebraically instance

true

closed

in R n,

but

(non-constant)

the semialgebraic f:~

semialgebraically

a topologically In

what

.......

> ~:t

>;

0 if t if

mappings. nature

c l o s e d map b u t f - 1 ( O )

follows

However

we

is

false

maps

for

semialgebraic

whose

arbitrary maps.

For

map

t is

it

closed

shall

be

we state first

t~O t>O

i s unbounded•

mainly

concerned

an easy proposition

with

polynomial

of very general

to be used later -compare with Example 7.7 in [D-K2]- We denote

by e :R nc

> S n the

inverse

of

the stereographic

projection

from

the

n

north pole

en(X 1 ..... x n) =

Proposition

2.-

Let

z× IIx +l '"

f:R n .....) ....R m

be

•

a

.

,

zx I1×11+1

~

ilxll _l] N +lJ

semialgebraic

map.

•

Then

f

is

245

semialgebraically proper if and only if the map f:S n

> S m defined by

fOPS) = pmN and flen(R n) = emOf is semialgebraic. Proof.-

Consider

the

cartesian

square

sn

f

> Sm

Te

I°m

Ra

f

) R m

If f is a semialgebraic map it is also semialgebraically closed since S n is a closed and bounded semialgebraic subset of Rn÷1-use Thm I and [B-C-R, Thm 2.5.8]. Then, also f is semialEebraically closed. if

C is

e (C)

a

in

closed

S ~,

semialgebraic

then

f(C)

subset

= e-l(f(K))

n

of

is

Rn a n d

closed

in

K is

the

Rm. A l s o ,

In fact,

closure let

of

b be

a

m

point

in R m and let gn:Snk{P~ } N

from

p~.

Since

> Rn be the stereographic projection

f-l(b) = H~(en(f-l(b)))

and

e (f-l(b)) = (f)-l(e (b)) N

m

is closed and bounded in S n, we get that also f-I(b) is bounded and by Thm. l,

f

is

semialgebraically

proper.

For

the

converse,

note

that

Gr(f)\Gr(f) is a unique point, and so Gr(f) is semialgebraic. Hence we must

only

check

semialgebraic is open.

the

subset

continuity

of

S m.

of

f.

If pN~u,m then

Let

U

be

an

open

(f)-1(U) = en(f-I(e-1(U)))m

m

If PN~U, then e-l(sm\U)m is a closed and bounded subset of R m.

Therefore,

by

[A, Prop.

1.3], f-l(e-1(SmkU)) is a closed and bounded m

subset of R n and so (f)-1(U) = snke (f-l(e-1(SmkU))) is open in S". m

Let us denote by ~ (R) the projective space over R of dimension n

n, whose points are denoted x = (Xo:X1:...:xn)" We fix the embeddin E Un:Rn( We

can

relate

extendibility".

) ~n(R):x = (xI ..... Xn ) semialgebraic

> (l:xl:...:xn)"

properness

with

"projective

246

Proposition

3.- Let

f:R n

~ R m be

a semialgebraic

that there exists a semialgebraic

map F:~ (R)

Let

us

assume

) ~ (R) such that:

n

(i) F(H n) C H ,m

map.

m

where H n = {x = (Xo: x :...:x ) ~ ~ (R):x I

n

n

= 0}. 0

(ii) U of = Fou . m

n

Then f is s e m i a l g e b r a i c a l l y Proof.By

First

[B-C-R,

we check

that

P (R)

3.4.6],

proper.

the fibers

is

a

of f are bounded.

closed

and

bounded

Let

b ~ R m.

semialgebraic

(in

n

fact

algebraic)

subset

of

some

R s. Thus,

e (f-l(b)) = F-l(u (b)) n

closed

and

bounded

subset

of

~ (R),

is a

m

and

so

f-l(b)

is

bounded.

n

Secondly,

if

the closure

C is

a

closed

of u (C)

in P (R),

n

that

semialgebraic

subset

it follows

from

of

Rn a n d

conditions

K denotes i) and

ii)

n

f(C) = u -I(F(K)),

and

this

is a closed

subset

in R m since

F is

m

semialgebraic

and ? (R) is closed and bounded.

Now we apply Thm.

I.

n

C o r o l l a r y 2. Let f:R semialgebraically

> R m be a non constant polynomial

= ]] of i

i

. th

i

is

non

-projection.

for

f

b y P r o p . 1.

natural

Then f is

proper.

Proof.- We can assume m = I. In fact, f

map.

If

constant, f

l

Thus

is

where

for some II :R m l

semialgebraically

we c a n

n u m b e r n-~l a n d some a ° .

write . . . .

f(t) an e R,

l- R

is

proper,

the

the

same

= a t n+ a t n - l + . . . + a O

ao40.

1

Now t h e

canonical

n

is

true

for

some

semialgebraic

map F:PI(R) verifies

conditions

semialgebraically

Remarks.-

> Pa (R):x

= (xo: xa)

i)

and

ii)

n-1

> (Xo:ao xn+a I Ix I in

Prop.

~ x o +.. " +anX )

3,

and

so

f

is

Consider

fcr

proper.

(i) The converse

of p r o p o s i t i o n

3 is not

true.

247

example

the polynomial

is no extension Prop.

3.

we

C1n : ni: i)

an =

> ~2:(x,y)

) Pz(~)

F:P2(~)

Otherwise

sequences

map f:~2

take

> (x2+yZ, x(x2+y2)).

verifying

the

~ P2(~),

point

bn =

conditions

a = (0:0:I)

(!.n" 0 : I ) ,

both

i)

There

ii)

and

g P2(~)

and

convergent

in the

to

a.

Then lim u2f(l,n)

= lim Fou2(l,n)

n-)~

= lim F(an) = F(a),

n-~

lim u2f(O,n)

= lira Fou2(O,n)

n->Oo

and also

n-)Ca

= lim F(bn ) = F(a).

n->~

Consequently

n-9~

lira (l:l+n2:l+n 2) = lim (l:n2:0), n-)o0 n-~o

i.e.,

(0:I:I) = (0:I:0) a

contradiction. On the other hand, consider

g:~2

> ~2: (x,y)

enough to prove

that

H:~2(~)

(2)

mappings.

and

proper.

h = gof.

semialgebraicatly

In fact,

let us

Prop.

1

it

But i t

is

clear

By

proper.

is

map

> [P2({R): (Xo:Xl:X2)---> (x:: (x2+x2)3:x2(x2+x2)2)l 2 1 1 2

condictions

Corollary

> (x3, y 2)

h is

that the semialgebraic

fulfills

f is semialgebraically

2

i) and ii) with respect

is

not

For example,

true

for

the regular

larger

to h, and so we are done. classes

function

f:~

of

semialgebraic 1

>

> R:t

is not

l+t 2 semialgebraically

Corollary

3.-

Let

proper.

f = (fl ..... f ):Rn

> Rm be

a

polynomial

map.

For

m

every Islam, that each

let Pi be the form of highest

fl is non-constant

and

degree of fi" Let us assume

that PI ..... Pm have

common zeros in R n. Then f is semialgebraically Proof.-

Let

d I = deg

fi'

el = ~

di'

for

proper.

l~imm.

Let

e

semialgebraic 1

it

map ~:R m

suffices

to

> Rm by ~(Yl check

that

. . . . .

the

not non-trivial

Ym) = (Yl 1 . . . . . composition

us

define

a

e

Y:)'

By Prop.

g = ~of

is

248 semialgebraically

proper.

For

that

we

apply

Prop.

3.

Let

us

write

m

g = (gl .... ,gm) and d = ~ d i. Then deg gl = d >0, for every Isism, I=I

and

we write

gi = g i , d ( X ) + g i , d - l ( X ) + ' ' ' + g l , O ( X ) ' where

each

gij

is

the

homogeneous

X = (X 1 ..... Xn )

component

of

degree

/] of

gi"

e

Moreover,

gl,d = Pii' and so the set of commmon

is the origin.

Therefore,

zeros

of gl,d' " " "'gm, d

the semialgebraic map

d ) ~m(R): (Xo:X:...I :Xn ) ..... )(xd: ~- glj(x)xd-J: : 0 "" "

G: IF' (R) n

J=O

satisfies conditions

d ~. gmj(X)X: -])

J=O

i) and ii) in Prop.

3 with respect

to g, and so g

is s e m i a l g e b r a i c a l l y p r o p e r . Remark.

The converse of the last corollary

example, verify

the the

map

f: ~2

hypothesis,

(x,y) ~ R 2, it follows we get

a converse

) ~2: ( x , y )

but

since

is not true in general.

> (x4+x 2+y 2 ,x 4+x 2+y 2,)

]l(x,y)]I2~ ~f(x,y)II

that f is in fact topologically

if we restrict

ourselves

for

does every

proper.

to homogeneous

For not

point

However

polynomial

mappings:

Corollary 4. L e t

fl . . . . . fm

be non-constant

homogeneous

R[x I ..... Xn ]' let Z be the set of common zeros let

f = (fl ..... f ):Rn m

) Rm"

Then,

f

is

polynomials

in

in R n of fl ..... fm and

semialgebraically

proper

if

and only if Z = {0}. Proof.

The "if" part follows

from Cor.

3, and for the "only if" part

note that the cone Z = f-1(O) is unbounded

if Z~{O}.

Proof of theorem 3. We must prove the "if" part and we may assume that f is a polynomial

of even degree and

non-negative

on R n. In fact,

let

249

us define of

even

degree,

Isl>M 2, for g,

g = ~.f,

the

where

fiber

for

every

g-1(s) Prop.

each r ~ R +, R n k B

n

point

x ~ Rn

Hence,

that

to apply Cot.

the

set M

in R, with M

c [O,e).

if

s e R

prove

map with

the

theorem

f is s e m i a l g e b r a i c a l l y

proper.

2. Thus we assume na2 and so, for

= g(RnkB(O,r))

F

and

if we

is a s e m i a l g e b r a i c a l l y

(0, r)

of R n. Thus

> t 2. Then g is a polynomial

is bounded. I, we get

If n = I, it is enough

unbounded)

> R:t

g(x)aO

and using

subset

~:R

connected is an

Consequently

semialgebraic

interval

we have

(perhaps

a well

defined

map d:R ÷ The

map

d

has

> R:r

r ~ R +,

the

fiber

We

g-1(c)

are

has

the hypothesis.

x I g RnkBn(O'r)

square,

r

graph

going

to

choose a point c ~ R, c>b,

contradicts point

infimum of M .

semialgebraic

b = sup d(R +) ~ R u {m}. Otherwise,

>

there

exists

such

c>M.

points

Since

a point

prove

so

that

in

it

exists

fact

b = m.

We shall see that for every

with

norm

c>bmd(r)

that g(xl)
and

bigger

than

RnkBn(O'r)

and

Zl ~ Rn\Bn(O'r)' proposition

2,

as i.e.,

by

On

the other

Yl e Rn\Bn (O'r)

Bolzano's

desired.

hand,

verifying

e :R n~ n check

) S n and that

contains r ~ R÷

For

a

such

the

the

finish,

> S I :PNn---9 PN' 1

sake

e :R c 1 inverse

neighbourhood that

to

c = g(z I) it

c
image of

set

under n PN'

i.e.

g

a

Thus

connected some

point

to

apply

of g,

Rn = g,

of simplicity

) S 1 with

for

some

g being

suffices

we must prove that the e x t e n s i o n g: S n

is continuous.

theorem,

Now,

which

= inf M r , there exists

g(xl)
r

we

identify

the embeddings

theoretic

inclusions.

of

neighbourhood

for

Rn\B n (O,r) c g-I ((a,-->)),

every every which

a ~ R follows

there

We

must of

1 PN

exists

immediately

250

from what precedes.

Now we are ready for the proof of corollary Proof of corollary polynomial proper

map

~:R m

since

its

semialgebraically Thm.

3,

M ~ R+

I.

F

is

such

Let

f = (fl' " " "'fm ) ' F = f2+...+fa:Rnl m

fibers

are

that

bounded.

if and only

semialgebraically

Hence,

if f is so,

proper

if

= F -I (t)

f-l(sm-l(O,t))

> R.

The

2 2 > y1+...+ym is semialgebraically

> R: (Yl ..... Ym )

proper

l:

and

is

F = ~of

by Prop.

only

bounded

i. But,

is by

if

there

exists

for

all

t ~ R +,

t>M. Remark.

With

critical

the notations

values

of F.

there

exists

F-1(t)

and F-l(t ' ) are

is

It is a finite

~(f) e R + such

we can state Cot. "f

in the proof

that

f:R 2

Let P,Q e R[y]

bounded,

that

for

every

bounded.

If

k ~ R,

P = Q,

= z.

the

9.5.2],

Since when

set

of

and so

the

fibers

t,t'

~ R\C £

only

if f-1(sm-l(o,t))

is

ta~(f)".

be polynomials

suppose

Thm.

homeomorphic

if and

d > (xd+p(x),x +Q(y)).

we may

C £ be

way:

proper

then f is semialgebraically

W.l.o.g.

let

[B-C-R,

Cfn[~(f),~)

I in the following

semialgebraically

~ R2:(x,y)

set,

semialgebraically

bounded f o r some (and so f o r a l l ) Example.

above,

that

If

a positive

the

fiber

integer. f-lf(O)

Let is

proper.

f(O)

M = {(x,y) k

and d a l

= 0 and

it

is enough

to prove

e R2: (xd+p(y))2+(xd+Q(y))2=k 2}

is

then

f-l(O) = {(x,y) ~ R2:xd+p(y) = O} is bounded 2e and so d is even and P(y) = I 2e-j for some integer e->l and a ~ R,

y=oaJ y a >0. 0

Thus,

B = {(x,y) If P~Q,

the

e R 2: xd+p(y)

sets =-g}

the set F k = {(x,y)

J A = { (x,y) ~ R2:xd+p(y)

are E R 2,

bounded (p_Q)2(y)

for

g=[k[/vr2,

_< 2k 2, [xdl <

= t} and

and M k=AuB.

[p(y)+Q(y)[+

251

+ llip2(y)+QZ(y)-k21 2

Comment.

In

Consider

for

where

each

decide

} is bounded and M c F . k k

some

cases,

instance

f

]

condition

a polynomial

~ R[x,y]

if the fiber f-1(t)

f-l(t)

consists

is

non-trivial Otherwise,

zeros,

map

3

f:R 2

is

polynomial

and

points.

if for example

only

Of course

according

verified. d ) [ f](x,y), ]=o

of degree

j.

To

the projectivization

d ~ fj(x,y)z d-j = tz d} J=O

E ~2(R):

if

easily

; R: (x,y)

is bounded we consider

bounded

on isolated

Thm

is an homogeneous

C = {(x:y:z)

Then,

in

to our

the point

if

the

this

intersection

is the case

sufficient (0:I:0)

Cn{z=O}

if fd has not

condition

in Prop.

3.

is in C, we deshomogeneize

t o get

d

C' = { x , z )

and the question

is to decide

But this is the case of the equation

Corollary

5.

polynomial

defining

Let

such

if the origin

is an isolated

if and only if the Puiseux

that

branches

point

in C.

at the origin

C' are non-real.

f E R[x I ..... x,] the

t e R +. Then

for every

e R2: ~ f j ( x , 1)Z d-J = t z d} J=O

level

there

be

curves

exists

a

strictly

{x E Rn:f(x)

r ° e R + such

positive

definite

= t} are

bounded,

that

f(x)ar ° for all

x E Rn. Proof.

By

Thin.

semialgebraically interval. Remark. condition

Thus, The

3,

the

non-constant

proper

and

so

its lower bound r

boundness

of

the

in the last corollary:

o

polynomial

its

image

does

the work.

level

map

f:R n

f(R n) c R +

curves

the polynomial

is

not

f(x,y)

is

a

> R a

is

closed

superfluous

= (xy-l)e+y 2 is

252 strictly

positive

and

it

attains

all

positive

real

values

since

f(n,I/n) = I/n 2.

Corollary 6. Let f = (fl '''''fm ) : R n .... > Rm be a non-constant map. Let A be the polynomial by

~

the

sums

of

semialgebraically g ~ A and SI,S 2 ~ [

an

Let

Cor.

1.

true

for

if and

only

if

in

there

A.

exist

Then,

f

is

C ~ R +, p e ~,

1=1

1

> R.

m

If

such

F - l ( t 2) c B (O,V/C[I+t2) p) n

Conversely,

if

f

is

an

for

identity

every

semialgebraically

F a n d s o F(R n) = [ a , e )

a non-empty,

polynomials

j

F = f2+...+f2:Rn

inclusion

of

such that

[J~l

Proof.

ring A = R[x I..... Xn, T] and let us denote

squares

proper

polynomial

for

bounded and closed

t • R,

proper,

some p o i n t

algebraic

exists,

the

a • R.

subset

and

we g e t we

apply

same h o l d s

Hence F - l ( t )

o f Rn f o r

every

is tea,

and the well defined map

has

C

1

semialgebraic

e

R÷

and p ~ ~

preimage

r: [a,->): t

> max ~11×11 FOx) = t>

graph.

[B-C-R,

such

F-l([o,b])

F-l([O,t]) c Bn(O,C 2) get

By

Prop.

that

r(t)~C .tp for

is

a

for

1

bounded

n

for

every

there

every

tab.

subset

of

C 2 ~ R +" Then,

some

F-l(t 2) c B (O,C(l+t2) p)

2.6.1]

if

t ~ R,

exists

By Thm. R n,

C = max and

bza,

1 the i.e.,

{CI,C z}

we

the

set

so

n

{(x,t) ~ Rn+l: ~ x~-C(I+T2) p z O, i=l

F(x)-T 2 = O}

is

empty.

Applying

[B-C-R, 4.4.2] we get the desired identity. Let follows

i = ~ we

denote

and C = R(i)

be the a l g e b r a i c

x = (X 1..... Xn )'

Y =

closure

(Yl ..... Yn )'

o f R.

Z =

In what

(Zl ..... Zn )--

253

x+iy, can

and we get

identify

some

C n with R 2n via

insight

on

f = (fl ....

f ):Cn * n

> Cm'

polynomials

gj,hj

E R[x,y]

identify

f

F=(g I . . . . .

with

gm, hl . . . . .

) R 2", z = x+iy proper

~ C[z I ..... z ]. m

J

such

) (x,y).

....

semialgebraically f

the

C n

that

fl(z)

polynomial

map

polynomial

There

F: R 2n

maps

exists

= gj(x,y)+ihj(x,y)

We

real

and

) R 2m,

we

given

by

f = (fl ..... fm ): C A .......>. Cm

is

hm). S i n c e m

we g e t f r o m Thin.

Corollary

= Z If J cz, l 2

J

j=l

]=1

3:

7.

The

semialgebraically

polynomial

proper

for every r ~ R, r>M,

map

if and only

if there

exists

M E

such

R÷

that

the subset {Z E cn:

~ Ifj(z)I 2 = r} J=l

is bounded. In the one variable Corollary

8.

Every

semialgebraically

case we have an analogous non-constant

polynomial

As in the proof of Cot. 2, we d = Z aj(z) d-j with aj ~ C, a 0 40 J=0

P" = (g,h):R 2

> R 2 is given

whose homogeneous

by

components gd(x,y)

Thus,

real

F is semialgebraically

between

the

f:C

2:

) Cm

is

other

may

assume

and

polynomials

the g

and

m = I.

Then

induced

map

h of

degree

d

gd and h d of degree d verify

+ i hd(X,y)

gd and h 4 have not non-trivial

On

map

to Cor.

proper.

Proof. f(z)

result

= a0(x+iy) d.

common

zeros and by Cot. 3 the map

proper.

hand,

for

nz2,

the real and the complex

there

cases:

is

a

significant

difference

254

Proposition map f:C n i.e.,

4.

The

maps

open

(semialgebraic) For

part

it is not

Each a l g e b r a i c

compact.

subset

by s e p a r a t i n g

of real

order

from"

lemmal]

onto

that b o u n d e d

algebraic

subset of C n w h i c h that W k Z ~ o and Z is

and so Z is not closed

the result

V I ..... V s .

in the general [B-C-R,

of C n and

such

fact,

number

following

formula

in

Z(R):

R = ~ and

so

particular

bounded algebraic

is also

of R 2n

For every

algebraic

subsets

expressible

R 2n

of

to

C = C,

Thm.

such

some

as

V

]

that and

"V ~ B for

every d

~

and

the

can

card(V)

arbitrary

real

closed

a

then exists

V

Ba

in

is

as M(n,d) components

integer

subsets of C n are finite.

first

exists

take

~ M(n,d)"

set

there

connected

positive

the

of

~2d.

V ~ Ba,

every

we

in

there

9.9.4],

semialgebraically for

true

Moreover,

if

[B-C-R, of

homeomorphic

in R.

that,

by

V I, . . . , V s

the

5.2.4].

subset

as a formula

in

case

in R[xl,...,Xn, Y 1 ..... ym ] of degree

Consequently,

it

of bounded

parameters

M(n,d) In

semialgebraically of

with

open

the field of complex

of C n can be seen as an a l g e b r a i c

subsets

open,

f is surjective.

algebraic

"V ~ B " is expressible d

subsets

the m a x i m u m

§2.1,

Cn

for real closed fields -IT] or

algebraic

integer

algebraic

of

closure W verifies

be the collection

Z(R)

card(V)sM(n,d).

of

d

fact

language

positive

polynomial

the real and imaginary parts of its equations. let B

the

to see

To obtain

zeros of a polynomial

Clearly,

subsets

is true for C = C,

[S, Ch.7,

we use Tarski P r i n c i p l e

R 2n "coming

This

proper

then f is s e m i a l g e b r a i c a l l y

it is enough

its p r o j e c t i v e

a dense subset of W,

integer d>0,

If m = n,

if Z c C n is an irreducible

is not a point,

W, hence

semialgebraically

subsets of C n and in p a r t i c u l a r

the first

since

a

semialgebraic

subsets of C n are finite. numbers

of

> C m are finite.

f

Proof.

fibers

is

d,

the

true

for

field

R.

In

255

Assume now n = m. Once

the openess

an open and closed s e m i a l g e b r a i c denote and ~d

F = (F 1 ..... F2n): R 2n

by

let deg F = max be

the

polynomial

map

Cn

of

" For the

every

set

formula F ~ ~

) R 2n the

F(B2n(a,r))

maps

R 2n

degree

in ~(R)

is

that an

map

every

f

induced

integer

) R 2n

-
in case C = C -see

such

d

For

then f(C n) is

f(C n) = C n. Let us

polynomial

F2m}.

polynomial

> C n with

then F is an open map the f o l l o w i n g

subset of C n, i.e.

{deg F I ..... deg

collection

of f is proved,

d>O,

"coming

has

F

let

from"

finite

iN, Prop 4, pp

by

a

fibers,

132]-

and so

is true for R = ~ and every d>O: card

open

F-1(b)

subset

s M(n,d)

of

for

R 2n for

every

every

b e R 2n,

a ~ R 2" and

every r E R, r>O". Therefore closed

field R and since

the e u c l i d e a n Remark.

map

is not

of

map f:C n f:C 2

implies

f

but

is

polynomial

the

point

map

f:R n

) Rm

and

of open

of

induces

maps

maps C n

with

finite

f

sets

condition

finite for

another

in

for

a

For example, fibers every

one

but

it

natural

f :C n c

semialgebraically

c

proper,

proper

is

a

closed

converse

is

not

true:

> x2+y 2, which has not finite

fibers

but

fibers

since

) Cm

Rn

C n = R 2n,

) ~: (x,y)

real

open map.

proper.

but

every

is in f(C2).

Clearly,

semialgebraical ly subset

has

(l,O)~f(C 2)

by Thm 3, s e m i a l g e b r a i c a l l y

semialgebraic

d>O

a sufficient

) (xy, x-y(xy-1))

(l,I/n) = f(I/n,n)

the map f:~2

it is,

every

form a basis

is not

the same polynomials.

semialgebraie consider

the fibers

In fact

polynomial

by

for

) C m to be s e m i a l g e b r a i c a l l y

number n, the point

defined

true

the open balls

) C2: (x,y)

proper.

Each

is also

topology of R 2n, F is a s e m i a l g e b r a i c a l l y

Finiteness

polynomial the

the formula

the

proper. -e.g.

To f i n i s h we show semialgebraically

) C m - have some nice properties:

that

proper

256

Proposition fibers.

5.

Let

Rnkf-l(F)

Let F < ~m

From

9.3.1]

it follows

the

be

a

the

that

be

a

semialgebraic

semialgebraic

finiteness

13.2 of

Acknowledgement:

shortening

> Rm

is semialgebraically

Proof.

Theorem

f:R n

set

map

with

with

finite

dim F~n-2.

Then

connected.

of

the

fibers

dim f-l(F) ~ n-2

and

and

Hardt's

now

theorem

it suffices

[B-C-R

to apply

[D-K] with M = R n and N = f-l(F).

We

wish

proofs

to

of

thank

Thm.3,

the

Cot.4

referee

and

for

Prop.5

his

and

contrlbutlon

some

other

In

useful

comments.

References

[A]

Alonso,

M.E.

Real proper morphisms.

pp. 237-243 [B-C-R]

Bochnak,

Vol 43,

(1984).

J.;

Rdelle.

ArchLy der Math.

Coste,

Ergebnisse

M.; der

Roy,

Math.

Gdomdtrie

M.F. 3

Folge,

Band

Algdbrique

12.

Springer.

(1987).

[Br]

Brumfiel,

Partially

G.W.

geometry. Cambridge: [D-K ] 1

Dells,

H.;

Zeit. [D-K 2 ]

1173.

[N]

175-213

H.; Knebusch,

Springer

Narasimhan,

spaces. L.N,M. Shafarevich,

[T]

Tarski,

A.

geometry. Berkeley

Univ.

and

Press

Semialgebraic

semialgebraic

(1979).

Topology

over

a Real

Spaces.

Math.

(1981).

M. Locally semialgebraic spaces. L.N.M.

(1985). Introduction

R.

[s]

M.

rings

If: Basic Theory of Semialgebraic

178, pp.

Delfs,

Cambridge

Knebusch,

Closed Field

ordered

25, Springer I.R.

to

the

theory

Basic algebraic geometry. Springer

(1951).

for

analytic

(1966).

A decision method for elemmentary Prepared

of

publication

by

J.C.C.

(1974).

algebra and Mac

Kinsey,

S U R LES O R D R E S DE NIVEAU

2n

ET S U R

U N E EXTENSION DU 176me PROBLEME DE HILBERT

DANIELLE GONDARD-GOZETTE UNIVERSITE PARIS VI, 4 p l a c e J u s s t e u , 75252 PARIS c e d e x 05, FRANCE.

ABSTRACT. L e t K be a c h a i n a b l e f i e l d , w h i c h m e a n s a f i e l d a d m i t t i n g o r d e r i n g s o f e x a c t level

2n

field there exists

f o r any ~ e K

n B 1 , f r o m [G1] a n d [G2] w e k n o w t h a t in s u c h a such that

z

~ E K¢ ; w e t h e n d e f i n e t h e n o t i o n

o f ~ - c h a i n o f o r d e r i n g s and u s e it t o o b t a i n e x p l i c i t f o r m u l a s f o r o r d e r i n g s of e x a c t level

2 n in s o m e s p e c i a l f i e l d s . We a l s o o b t a i n f o r s o m e

c h a i n - c l o s e d f i e l d s a n e w e x t e n s i o n o f H i l b e r t ' s 17th p r o b l e m .

INTRODUCTION. Les c o r p s o r d o n n a b l e s s o n t d e d e u x s o r t e s : les c o r p s n o n c h a ~ n a b l e s (i.e. il n ' e x i s t e p a s de c h a i n e d ' o r d r e s d e n i v e a u s u p E r i e u r de niveaux exacts

2 n ) off t o u t e s o m m e de c a r r E s e s t une s o m m e de p u i s s a n c e s

quatriEmes (par exemple

~ , © et ses extensions algEbriques ordonnables), et

les c o r p s c h a i n a b l e s off il e x i s t e un ElEment ~ t e l que

z

ne s o i t p a s s o m m e e s t un c o r p s

de p u i s s a n c e s q u a t r i E m e s ; d a n s ce c a s n o u s d i r o n s que

K

o~-cha~nable ( p a r e x e m p l e ~(X) e t ~((X)) s o n t d e s c o r p s

X-chainables) .

L a n E c e s s i t 6 de f a i r e i n t e r v e n i r une c o n s t a n t e

cc

a 6t6 m i s e en Evidence

par les a x i o m a t i s a t i o n s des t h e o r i e s des corps cha~nables et des corps c h a i n e - c l o s que n o u s a v o n s donnEes d a n s [G1] ( p r e u v e s darts [G2]). Ce p o i n t de vue n o u s a dEj~ p e r m i s de d E m o n t r e r d e s r E s u l t a t s du t y p e 17~me p r o b l E m e de H i l b e r t d a n s [D-G] e t d e c r E e r un a n a l o g u e ~ l ' a t g E b r e r E e l l e d a n s [B-G] , e t n o u s le c o n s e r v o n s ici p o u r o b t e n i r une e x p r e s s i o n d e s o r d r e s de n i v e a u d e c e r t a i n s c o r p s (~(X)

2n

p a r e x e m p l e ) e t une n o u v e l l e f o r m e de g E n E r a l i s a t i o n

du 17~me p r o b l E m e de H i l b e r t ; l ' i n t r o d u c t i o n de la p a r t i e III e x p l i q u e r a c o m m e n t la g E n E r a l i s a t i o n p r E c E d e m m e n t 6tudiEe d a n s [D-G] e t [B-B-D-G] e s t d i f f E r e n t e de c e l l e p r E s e n t E e d a n s c e t a r t i c l e .

258

I-NOYAUX DES CHAINES D'UN CORPS CHAINABLE.

D6finition 616ment

I-1. Un c o r p s ~

t e l que

d ' 6 1 6 m e n t s de

D6finitions

K

2

sera dit ~-cha}nable s'il existe dans

K

un

ne soit pas somme de puissances quatri6mes

K .

I - 2 . Un p r 6 o r d r e

T

de n i v e a u

2n

(i.e.

T + T ~ T

, T

. T c

T

,

n-I

n

K2

_c T ) sera dit un

a-pr6ordre si

2

~ T

(i] est alors de niveau exact

n~l

2n

car

~ K2

n ' e s t pas contenu dans

T ). De m~me un ordre

P

de niveau

n-l

exact

2n

sera dit un

~-ordre si

Definition

I - 3 . Nous a p p e l e r o n s

sup6rieur

(PI)I~

t e l l e que

au moins une

~ p .

~-cha}ne une chaine d'ordres

2

I - 4 . Un c o r p s

Proposition

2

de n i v e a u

t~ P2"

K

est

a - c h a } n a b l e si e t s e u l e m e n t s ' i l e x i s t e

~-chaine.

Ceci r 6 s u l t e d e s d 6 f i n i t i o n s I-1 e t I - 3 e t du r 6 s u l t a t qui d o n n e l ' e x p r e s s i o n

s u i v a n t e s d e s s o m m e s de p u i s s a n c e s :

K# = ( ~ Pl2 ) a ~ K 2 q u e l c o n q u e de

de B e c k e r - H a r m a n

o~ a

d 6 s i g n e un o r d r e de n i v e a u e x a c t

4

W e s t p a s u n e s o m m e de p u i s s a n c e s q u a t r i ~ m e s 2 j tel que ~ ~ P . D ' a p r ~ s le c o r o l l a i r e J2 I - 4 de [H] il e x i s t e a u m o i n s une c h a i n e d ' o r d r e s de n i v e a u s u p ~ r i e u r p a s s a n t

darts

K . Donc si

P12 2

K , il e x i s t e a u m o i n s u n

par ce

P

j2

Proposition I-S. Dans un corps exact

2n , n -~ 2 , d'une

n-I 2

En e f f e t si P

6tant de

~

ct-cha}ne

nl_1

a2 2p

a

l

~ Pl ~ P

que si alors

~ P

1

et

a2

on a

a

x ~ K°

K

~-cha~ne

~-eha~nable tout ordre (PI)I~N

n

de niveau

n-I 2

, alors ~ P

P

est un ~-ordre.

P est bien un ~-ordre. Si ¢~ e P n n , on en d6duit qu'i! existe i -~ 2 tel que

2

~ Pl+l ( p u i s q u e d ' a p r ~ s ta r e l a t i o n de c h a ~ n e d~s que 2 p+q

et si

x ~ P u -P

e P xz

pour tout

q -~ 1 ) ; o r il e s t c o n n u (cf. [Bel])

l+q appartient & un ordre

: en effet si

x ~ P

alors

P

de niveau exact

2n

P + Px est un pr6ordre propre

259

contenant

strictement

P

et donc

P + Px = K , d'oO

- 1 -- a + b x a v e c a e t b |

dans

P

et

entra~ne

x = (-1-a)/b

que

~

2

~ - P . On d6duit de ce rdsultat

que

2

a

e P

1-1

e P

u - P

, puis par

I+i

la relation

1+1

de cha~ne

I+I I-1

P

u

1+1

- P

contraire

=

1+1

( P

P

) u

t

- ( P

~ P

0

)

l

que

c¢2

e P

ce qui est

1

& i'hypoth~se.

Lemme I-6. Soit n

T

~

0

K

un corps

2 n- 1

= ~ K 2 - c~

~-chainable.

Pour tout

n e ~* ,

n

~ K2

est un

~-prdordre propre (i.e.

) de niveau

-i ~ T

n

n

exact

~n.

C ' e s t t o u t & f a i t c l a i r pour

0c2 ~ ~ K 4

~-chalnable

(Pl)l~

; par

I-5 tout

la relation

de chaine

qUne 1Pn_ 1

contient

P P

de

n

~-cha~ne

u - P

c~

e P

u - P n

2

- c~

d'dl6ments

de niveau

n'appartenant

pas & & T

de

0

P

qui est

n

k-1

o n a T k : ~. K 2 - 0c2

clair

Proposition

I-8.

2n

et

n

0

donc

T

que

T

n

exactement

K

~-chalnable

~ Pk ' p o u r t o u t

la preuve

&

clairement

est

n

n-1

car

T

; de m~me

n

a

2

2n

- 1

pour

toute

a-chaine

(Pi)le~

k -> 2 .

du lemme I-6.

Dans un corps

K

~-chainable, n

exact

~ P

au fait

k

~ KZ

d'aprds

n-I

donc propre.

Proposition I-7. Dans un corps k

2

c o n t e n a n t t o u t e s les

ne peut donc pas appartenir

n

6tant

n

contient

2 n , e t le n i v e a u e s t

~

n P ) jointe

montre

; P

K

~-chaine

donc

n-i

2n-l-~mes e P

K

une

~-ordre

n

2n-6mes

appartenir

un

n P ) u - (P n-I

n -> 3

I - 4 il e x i s t e

est

l e s puissanceSn_l

e t donc que

un prdordre

C'est

n = 2 ; pour

n

puissances

peut

= (P n

toutes

et

la proposition

n

2

n = I

et par

n >- 2 , q u i c o n t i e n t

T

tout

P

n-i

= ~ K 2 - ccz

, ordre

de niveau

nn

~ K2

est un

P

n

de n

a-chaine.

n-I

En effet

seulement

l ' u n d e s d e u x dldments

z

n-I

ou

- c~z

appartient

n-1

P

n

.Donc

une ~-chaine

(z

2

~ P

puisque

et

n

si

on

2

en

~ p

ddduit 2

que

toute

, en utilisant

chaine

passant

la condition

par

de chalne

P

It

(qui

est

ne

260

s'exprime

au niveau 3 par

:

P3 u - P 3 = ( P z r~ Po ) u - ( P 2 ~ P o ))

on obtiendrait n-I

2

e P

u - P

3

ce qui est

3

d'o~

~

impossible.

4

~

P

Toute

; on d6duirait

3

chaine

(PI)I~N

en it6rant passant

que

par

~

Pn

2

e P

n-I

est donc une

~x-chaine .

n

I-9. D a n s un corps

Th4or6me

a-cha[nable

T

2n- 1

= ~ Kz - ~

n

~ K2

est dgal

n

l'intersection de t o u s l e s a-ordres de niveau exact

au moins

une a-cha£ne).

D'apr~s est

le t h 6 o r ~ m e

6gal & l'intersection

q u i le c o n t i e n n e n t

1 de Becker de tousles

; un ordre

[Bell, un pr6ordre ordres

propre

de m6me niveau,

qui contient

T

n-1 2

- c~

2 n (qut apparttennent

est

un

de niveau

exact

~-ordre

2n

ou non,

car

n-1n

~ T

~ P n

ce qui entra~ne

que

~

2

~ P

n

; enfin

P

n

est bien de n

n-I

niveau

exact

2n

puisque

~ K2

n'est

pas contenu

dans

P n

n

C o r o l l a i r e I - I 0 . Dans un c o r p s

K

a-cha~nable , K

2n - 1

= ~. K2 - a

n

~ K2

n

est 4gal ~ l'intersectton

des

P

, ordres de ntveau exact

des

2n

n

~-cha£nes.

Cela r6sulte

imm6diatement

de I-7,

I-8 et I-9.

n

Remarque. toujours niveau

Dans un corps

¢~-chainable

T" = ~ K 2

2 n- i

+ ~

n

~ K2

est

n

un pr6ordre exact

D4finition

K

propre

de niveau

2n

mais

il n ' e s t

pas forc6ment

de

2n .

1-12. Appelons

, pour

n

-> 2

,

n o y a u a u n i v e a u _n d e s a - c h a i n e s .

T n

I I - O N O R D E R I N G S OF SOME C H A I N A B L E F I E L D S .

II-1. St un corps

Th4or~me (PI)I~

, ~i 4 c h a n g e

de

P0

K

~-cha£nable n'admet

et

PI

qu'une

est pour tout

n -> 2 ,

a-cha~ne

p r e s , a l o r s l ' o r d r e d e nLveau e x a c t n

de celle-ci

seule

P

= T n

= ~ K2 - ~ n

2 n- 1

n

~ K2

.

2n

261

En e f f e t p a r l a p r o p o s i t i o n I - 7 t o u t e Qn ~ T seule

pour tout

n

a-chaine

Exemple

le c o r o l l a i r e 1-10 d o n n e a l o r s

:Dans

trois ordres,

de n i v e a u e x a c t 2 n intersection

(Ql)le~

n

= T

avec

archim6dien, et admettant

n -> 2 donc p o u r t o u t

A(P o) = A(P I)

de Pasch) que d'apr~s

2n

pour chaque P

616ment d e

et

sdparant

II-2. Soft

le ddbut

les o r d r e s

= ~. ( P

d'une

n P 0

P

K un corps

n >- 2

)2

- a2

~. ( P a-ordres

2n

de

et sont

PI

sont

K

n'admet qu'un

; d ' a p r 6 s le t h d o r ~ m e I I - I n-I

n

~ K2

off

a est un

1

et

P

; pour

n P

, P

0

deux

1

ordres

qui

n >- 2 , s o f t

)2

, alors

C

I

P

n

est

~ g a t ,~

n

le contenant

(de

tels

P

n

sont

afors

a-chatne).

e s t a n a l o g u e ~ c e l l e de 1-10 d6s que l ' o n a m o n t r d q u e

e s t u n p r ~ o r d r e p r o p r e de n i v e a u e x a c t

n

n >-- 2 . De

n-I 0

La d6monstration

P

(P{)IE~

de tousfes

exact

o

n

n-i

I

l'tntersectton

avec et

= ~ K2 _ 2

a-cha~nable

a-chatne

n-I

C

K

e t Po = P1 " On e n d d d u i t (ce c o r p s d t a n t

cet ordre est explicitement donn~ par : K

2n P0

n

de niveau

des ordres

n : il s ' a g i t d ' u n c o r p s

le t h 6 o r 6 m e IV (if) d e [Bell c e c o r p s

s e u l o r d r e de n i v e a u e x a c t

n

qu'une

d ' u n e c l o t O r e r d e l l e de O(X) p o u r u n o r d r e a r c h i m 6 d i e n a v e e u n e

co-chainables

C

n'admet

n

[G4] il d 6 c o u l e q u e l e s d e u x o r d r e s (non a r c h i m d d i e n s )

sotent

K

[Bel] p a g e 61 e s t d o n n ~ u n e x e m p l e de c o r p s a y a n t

dont l'un Pest

pour un

P

e l 6 t u r e r 6 e l l e g ~ n d r a l i s S e p o u r u n o r d r e de n i v e a u e x a c t

Thdor~me

est telle que

n ~- 2 , d ' a p r ~ s l ' h y p o t h ~ s e f a i t e q u e

d'application

exactement

a-cha/ne

2 n . Si

(Pi)leN

est une

rl-1

a-chaine n = 2

commen~ant par

Po r~ P1

d'apr~s la relation

alors

(Po a P1) 2

c Pn : c ' e s t v r a i p o u r

P2 u - P2 = (Po a P1) u - (Po a P1) , e t p a r

n-1

rdcurrence

si

(Po n P1 )2

-¢ P n

a l o r s p a r l a r e l a t i o n de c h a i n e , a u n i v e a u n

n

+

1 ,

Pn+l u - Pn+l = (Po n Pn) u - ( P o n

Pn)

on d d d u i t q u e

(Po n P1)2

n-I

est contenu d a n s

d o n c on a

P

n+l

P

n

6 t a n t de

a-chaine

~

2

n-I

~ P

n

et

- a

2

~ P

C

n

a Pn . On e n d d d u i t qUne IC n , qui e s t c l a i r e m e n t u n p r ~ o r d r e d e n n n-I n i v e a u a u p l u s 2 n (Vx x 2 = (X2) 2 E (P n P )2 , est un pr~ordre 0

1

p r o p r e d e n i v e a u e x a c t 2 n . I1 e s t d o n e 6 g a l ~ r i n t e r s e c t i o n ordres

P

de tousles

qui le c o n t i e n n e n t , e e u x - c i s o n t d e n i v e a u a u p l u s

2n

c a r si le

n

n i v e a u d t a i t s u p d r i e u r on a u r a i t

~ K2

non contenu dans

P

ce qui e s t

262

n-i

impossible puisque

(P

r~ P )2 0

K2n

_~

. Le niveau e s t 6 v i d e m m e n t au

1 ri-1

moins

2n

puisque

~ K2

n'est pas contenu dans

P

qui c o n t e n a n t

n-I

peut contenir ces ordres

2

P

C

ne

ri

n-I

. E t a n t de n i v e a u e x a c t

contenant

(2

ri

2n

e t t e l s que

_ 2

s o n t d e s o r d r e s de n i v e a u e x a c t

2n

~ p

,

ri

de

e-chatnes.

Exemple

d'application

: la p r o p o s i t i o n I I - 2 p e r m e t d ' o b t e n i r une e x p r e s s i o n

d e s o r d r e s de n i v e a u s u p E r i e u r de En e f f e t

~(X) :

~(X) e s t un c o r p s de P a s c h e t , c o m m e dEj& u t i l i s 6 p l u s h a u t , il

dEcoule de [Bel] e t de [G4] q u ' u n c o u p l e d ' o r d r e s c o - c h a i n a b l e s ne l ' e s t que p a r une s e u l e c h a i n e . E n f i n il e s t m o n t r 6 d a n s [Di] que les c h a i n e s de ce corps sont c o n s t r u i t e s sur des paires d ' o r d r e s "symEtriques" qui p l a c e n t

X

i n f i n i m e n t prEs de a ~ ~ a v e c

r e s p e c t i v e m e n t , ou s u r

P

CO+

et

P

130-

X-a

qui r e n d e n t

P

et

P

,

p o s i t i f ou n 6 g a t i f X

infiniment grand positif

ou n 6 g a t i f r e s p e c t i v e m e n t . II y a d o n c d a n s d'ordres

P

0~(X) d e u x

et

&+

de n i v e a u e x a c t

P

&-

2n

( X - a ) - c h a i n e s b&ties s u r les d e u x c o u p l e s

d'une part et

= ~. (P na

r~ P a+

)2

riCO

= ~ (P

CO+

CO-

n P a+

)2

P

o0-

d ' a u t r e p a r t . Les o r d r e s

n-1

~ (P

a-

r~ P

et

n-I

- (X-a) 2

)2

- (X-a) z

et par

a-

n-I

n-1

P

o0+

de c e s d e u x c h a i n e s s o n t a l o r s d o r m , s p a r n-I

P

P

n-1

~ (P

CO+

r~ P

o0-

)2

respectivement •

c e c i r 6 s u l t e 6 v i d e m m e n t du f a i t que c e s d e u x e x p r e s s i o n s s o n t d e s ( X - a ) - p r 6 o r d r e s p r o p r e s de n i v e a u e x a c t

2n

distincts .

III-UNE AUTRE GENERALISATION DU 17~me PROBLEME DE HILBERT AU NIVEAU 2 n

A l ' o r i g i n e le 17~me p r o b l ~ m e de H i l b e r t s e p r 6 s e n t a i t s o u s la f o r m e :

solt

f ~ ~[X] t e l l e q u e V ~c ~ o~P f >- 0 , a - t - o n

f ~ ~. ~ ( ~ ) 2 ? A r t i n le

r 6 s o l u t en 1927 (cf. JR]). n ' a y a n t q u ' u n s e u l o r d r e d o n t les 616ments p o s i t i f s s o n t

p = R2 = Z ~2

une

premiere extension possible est :

soit

K

un c o r p s o r d o n n a b l e ,

soit

f ~ K [ X ] t e l l e q u e V ~ ~ K p f ~ ~ K z,

a - t - o n f ~ ~. K ( X ) 2 ? Ceci a 6t6 Etudi6 d a n s [G-R], [McK], [P2] e t [Z-G]. Une a u t r e e x t e n s i o n e s t le p a s s a g e a u x s o m m e s de p u i s s a n c e s

2n-~mes :

263

soit

K

un c o r p s o r d o n n a b l e e t s o r t II

f

f

~ K[X]

telle que

V ~c • K p

n

• ~ Kz

a-t-on

f

• ~. K ( X ) 2

? C e t t e e x t e n s i o n a 6t6 r 6 s o l u e p o u r une

v a r i a b l e e t c e r t a i n c o r p s c h a i n e - c l o s d a n s [D-G]. P u i s clans [B-B-D-G] ce p r o b l ~ m e e s t r 6 s o l u d a n s un c a d r e p l u s g 6 n 6 r a l , p o u r les p u i s s a n c e s

2n-Omes

et plusieurs variables. Nous p r o p o s o n s ici une e x t e n s i o n d i f f 6 r e n t e du l?-&me p r o b l ~ m e de H i l b e r t a u x o r d r e s de niveau sup6rieur sous la f o r m e : Soit P

K

~. K 2 ~ ~ K 4 ), s o i t

un c o r p s chatnable (i.e. ordonnable et tel que

un o r d r e d e n i v e a u e x a c t

2n ( n a 2 ) d e

K , sort

~ K[)(I t e l l e que

f

n

V ~¢ • K p

f(~C) ~ P

, peut-on

caractdrlser

f

?

n

Nous a p p o r t o n s une r 6 p o n s e d a n s le c a s de

K

chalne-clos (<2>-r6el-clos dans

l a t e r m i n o l o g i e de [ B - B - D - G ] ) n ' a d m e t t a n t q u ' u n e s e u l e v a l u a t i o n h e n s e l i e n n e c o r p s d e s t e s t e s r 6 e l - c l o s . I n s i s t o n s s u r le f a i r que darts [D-G] ou [ B - B - D - G ] l ' o b t e n t i o n d ' u n r 6 s u l t a t n 6 c e s s i t e de c o n s i d 6 r e r l e s off

L

f(x)

e s t une e x t e n s i o n a l g 6 b r i q u e o r d o n n a b l e de

consid~re f(x)

que p o u r

Th6orSme l I I - 1 .

Soit

K

n >- 2

( l ) f ~ E K(X)2

- ~ f

~ corps des restes rdel-clos,

et sort

f • K(X)

:

E K(X)2 f(x)

est ddf~nie

2n

on ne

n

~ P

, oa

P

n

de n i v e a u e x a c t

,

un corps chatne-clos ~-chatnable n'admettant qu'une

2n-I

( H ) Yx ~ K p 00

K , alors qu'ici

les propridtds su$vantes sont dqulvalentes

n

x e Lp

x ~ Kp .

seule valuation hensdlienne alors pour

pour

de

K

dds~gne

l'unique ordre

n

; n

n-I

( l i t ) Yx • K p

o0

f

est ddflnte

f ( x ) • E K2

a2

(iv) Yx • Kp

oO

f

est d d f t n t e

f ( x ) e K Z U - a,

-

n

2n-I

n

E K2

K2n

La p r e u v e u t i l i s e le c o r o l l a i r e 1-10 du p r 6 s e n t a r t i c l e e t le l e m m e s u i v a n t o b t e n u a v e c Delon d a n s [D-G] :

L e m m e I I I - 2 [D-G]. S o i e n t et

K

K

et

L

d e u x c o r p s c h a I n e - c t o s t e l s que

n ' a d m e t q u ' u n e s e u l e v a l u a t i o n hens61ienne & c o r p s d e s t e s t e s r 6 e l - c l o s

alors sont 6quivalents : (i) (ii) (iii)

K _c L

K t~ L z = K z

;

K est relativement alg6briquement clos dans K ~ L

( off

"~"

L .

e s t une i n c l u s i o n 6 1 6 m e n t a i r e ).

;

264

De l ' e x p r e s s i o n

d ' u n c o r p s chaYne c l o s

K = K 2 u - Kz u a K z u - a Kz qu'une

a-chainable

sous la forme

et du lemme pr6c6dent

seule valuation hens61ienne & corps des restes

dans un autre

corps chaine-clos

a-chainable

it r 6 s u l t e

r6el-clos

L , alors

on a

que si

K

n'a

et est contenu K ~ L

Preuve de III-1. (ii) ~ (iii) o (iY) ~-chainable

K

est imm6diat

: En e f f e t

l'unique ordre

dans un corps chaine-clos

de n i v e a u e x a c t

2n

est donn6 par

les

n 2n-I n n 2n-i K2n = ~ K2 - a ~ K2 = Kz u - ~

expressions suivantes : P

; la

n

premi&re f o r m e r6sulte du fait qu'un corps chaine-clos uniquement

a-chainable est

a-chainable et du th6or&me II-2 , et la seconde vient de

]'expression des ordres de niveau sup~rieur d'un corps n'admettant que deux ordres usuels donn6e par Becker dans [BI], II suffit de montrer le th~or&me pour n

f ~ K[X]

n

f = gh 2 -1/

f = g /

car si

h

alors

n

h2

et l'on sait que d'une part

n 2n-I n T CKCX)) = ~ K(X) 2 - ~ ~ K(X) 2

~ K2

e s t un

_c p

et que d'autre

n

pr6ordre

part

(voir § I } .

n

(i) ~ (iii) : il s u f f i t n

~. K(X)2

- a

n ~, K 2

de v 6 r i f i e r

2n-I

~ K(X)2

2n-I - a

f , appartenant

&

, est ddfinie en

x , alors

f(x)

appartient

n ~ K2

nous autoriser

q u e si

n

. C e t t e p r e u v e e s t d u e ~ B e c k e t e t n o u s le r e m e r c i o n s

~ la reproduire n

Notons

f ~ ~ K(X)2

o4 les

r I , s3 ~ K(X)

ici :

2n-1

- ~¢

n

~ K(X)2

; soit

de

zn

sous la forme

x = (Xl, ..., xp) ~ Kp

f = ~ rI

2n-1

- ~¢

2n

~. sj

et soit

Ox= K[X](x~x,i..,x_x ) l'anneau local en x ; soit A : K(X) - - > K u {00} p une place telle que A(Xl) ~--> x i et soit VA l'anneau correspondant ; alors si

f

est d6finie en

x ,

f ~ O

x

a VA . II s u f f i t de montrer que

r i , sj ~ VA d'o~ l'on d6duit =~

Mf) = f(x I . . . . . x ) = p ~(rl)2n - ¢~2n-i ~ A(SJ)2n ~ P n

Soit p a r exemple

rI

tel que

alors on a

f = r2 I

v(r I) = min { v(r i) , v(sj) } n-I

n

[ I + ~ ( r l / r l ) 2n - ~2

f 6 VA et le crochet , not6

, si r I 6 VX ,

n

~. ( s j / r l ) 2

]

• on s a i t que

z dans la suite , d a n s l'expression ci-dessus

est une unit6 ce qui donne une contradiction ; en e f f e t si

z

n ' 6 t a i t pas une

265

unit& a l o r s d a n s le c o r p s r ~ s i d u e l r/

r1

et

s/r

1 n

;~(z) = I + E yl z

VA / m A = K

appartiennent 2 n-I

- ~

&

on aurait,

p u i s q u e les

VA :

,2 n

~. yj

= 0

ce qui est impossible puisque - 1 ~ Pn

L'autre cas o~ l'on a par exemple

s

d4fini par : I

v(s I) = m i n ~ v ( r I) , v(sj) } s e t r a i t e

de mani~re analogue.

(iv) ~ (i) : On c o n s i d ~ r e la t h 6 o r i e d e s c o r p s c h a i n e - c l o s

a-chainables

u t i l i s e le l a n g a g e d e s a n n e a u x a u g m e n t 6 d ' u n s y m b o l e d e c o n s t a n t e corps par

K

chalne-clos

P / Q

avec

a-chainable

l ' h y p o t h ~ s e (ii) se t r a d u i t ,

= 0

v

f(x) = y

2n- 1

v

qui est ~-ehainable on fixe une ct-chaine cl6tuve chaine de

K(X)

l'hypoth~se faite sur

f(x) = - ~ (PI)I~I

K

on a

K ~ L ,donc

= 0

a u s s i que a-chaine

de niveau

II

f de

appartient

v

f(x) = y x = X

2n &

de

2

dans

L

f(x) = - ct

dans

L

P

n

2n

des

de

Ce r a i s o n n e m e n t f

n

c o r o l l a i r e 1-10 q u ' e l l e e s t 4 g a l e & y. K(X) z

formule

z

2

) " f

L ; cet ordre prolongeant

K(X) , n o u s a v o n s

& l'intersection

K(X)

de t o u s l e s

d o n t on s a i l p a r le

zn-1

- a

est

e s t f a i s a b l e p o u r r o u t e s les

appartient

~-chaines de

une

n

et on obtient que 2n

L

K(X)

III-2 et

la m ~ m e 2

v

; dans

et on consid6re

I' ~ - c h a i n e c h o i s i e s u r

K(X) , p a r c o n s 4 q u e n t

ordres de niveau exact

) "

n-i

appartient ~ l'unique ordre de niveau exact

P

f

2n

pour cette chaine . D'apr~s le l e m m e

satisfaite ; on ehoisit alors

l'ordre

en notant

z

n

" V x 3y 3z ( Q ( x )

ct . D a n s le

P , Q ~ K[X] , p a r l a f o r m u l e s u i v a n t e : 2n

" V X 3y 3Z ( Q ( x )

e t on

n

~ K(X) z

.

REFERENCES

[Bel] E. B e c k e r :"Hereditarily p y t h a g o r e a n f i e l d s and o r d e r t n g s of higher" t y p e s " , I.M.P.A., L e c t u r e s N o t e s # 29 (1978), Rio de J a n e i r o . [Be2] E. B e c k e r : "The real h o l o m o r p h y ring and s u m s o f 2n-th p o w e r s " , in G ~ o m 4 t r i e A l g 4 b r i q u e r 4 e l l e , L e c t u r e N o t e s in Math. # 959, p.138-181, S p r i n g e r - V e r l a g 1982. [ B - B - D - G ] E. B e c k e r , R. B e r t , F. Delon e t D. G o n d a r d : " H i l b e r t ' s 17th p r o b l e m and s u m s o f 2n p o w e r s " , p r e p r i n t . [B-G] E. B e c k e r e t D. G o n d a r d : "On r i n g s a d m i t t i n g o r d e r i n g s and 2 - p r i m a r y

c h a i n s of o r d e r t n g s of h i g h e r level", M a n u s c r i p t a M a t h e m a t i e a , 69, 2 6 7 - 2 7 4 , 1990. [D-G] F. Delon e t D. G o n d a r d :"17~me p r o b l d m e de H i l b e r t au ntveau n dans l e s c o r p s c h a f n e - c l o s " , The J o u r n a l o f S y m b o l i c Logic, vol. 56, # 3 S e p t . 1991, p . 8 5 3 - 8 6 1 .

266 [Di] M. Dickmann :"Couples d ' o r d r e s chatnables", Expos4 au S6minaire " S t r u c t u r e s alg6briques ordonn6es" (D.D.G.), Univ. p a r i s VII 1987-88. [GI] D. Gondard :"Thdorie du premier ordre des corps chatnables et des corps cha~ne-clos", C. R. Acad. Sc. Paris, T o m e 304, if 16, 1987. [G2] D. Gondard : "Chainable fields and real algabratc geometry", in Proceedings "Real Algebraic and Analytic Geometry" (Trento Oct. 88) , Lectures Notes in Math. 1420, Springer-Verlag, 1990. [G3] D. Gondard :"Kernels of chains and chatnable fields", in Abstracts A.M.S., vol i0 #5 Oct. 1989. [G4] D. Gondard :"Sur l'espace des places rdelles d'un corps cha~nable", preprint. [G-R] D. Gondard et P. Ribenboim :"Sur le 17dine p r o b l d m e de Httbert", C. R. Acad. Sc. Paris, tome 277, 20-08-1973, p. 303-304. [H] J. Harman :"Chains o f h i g h e r l e v e l ordertngs", Contemporary Mathematics, vol. 8, 1982, pp. 141-174, A.M.S.. iLl] T. Y. Lam :"The t h e o r y o f ordered f i e l d s " , Proceedings of Alg. Conference, pp 1-152, M. Dekker (1980). [L2] T. Y. Lam :"Ordertngs, Valuations and Quadratic Forms", C.B.M.S. regional conference, # 52, 1983, A.M.S.. [McK] K. McKenna : "New f a c t s about H i l b e r t ' s 17th problem", L e c t u r e notes in Math. 498, S p r i n g e r - V e r l a g , pp. 220-230. [PI] A. P r e s t e l :"Lectures on f o r m a l l y real f i e l d s " , I.M.P.A., Monografias de Matematica, if 22, 1975, Rio de Janeiro. [P2] A. P r e s t e l : "Sums o f s q u a r e s in f i e l d s " , Bol. Soc. Brasil., Rio de Janeiro, 1975. [R] P. Ribenboim : "Arithmdtique d e s corps", Herman, P a r i s 1972. [Z-G] Zeng Guangxin : "A c h a r a c t e r i z a t i o n o f p r e o r d e r e d f i e l d s w i t h the w e a k H t l b e r t p r o p e r t y " , proc. of the A.M.S. 104 (1988), 335-342.

CURVES OF DEGREE & WITH ONE NON-DEGENERATE DOUBLE POINT AND GROUPS OF MONODROMY OF NON-SINGULAR CURVES

I.V.ITENBERG

Abstract. The paper is devoted to the plane

projectiv~

real

rigid

algebraic

isotopy

curves

of

classification degree 6

of

~ith

a

non-degenerate double point and to ~he calc~lation of the groups

of

monodromy of non-singular curves of degree 6.

Key words and phrases : curve

of degree 6 with a non-degenerate

double

point,

rigid

isotopy c l a s s i f i c a t i o n , group of monodromy,

Address • Oep, E~f Math. and Mechanics Leningrad St~.ite Unive,~sity Bib!iotechnay~, pl. 2, Leningrad~

Petredvoretz,

~98904 ~ UoSR e-ma~!

address

: degt@!omi,spb.su

Introduction

There are two main problems for every class of plane projective real algebraic curves :

the classification up to real

isotopy

and

the c l a s s i f i c a t i o n up to r i g i d isotopy. Two curves-A and B are r e a l l y isotopic i f f

sets ~A and

~B

of

268

their

real

points

are i s o t o p i c

in ~pz

are rigidly isotopic iff there

Two c u r v e s of

exists

a

real

a

fi>:ed

isotopy

class

connecting

these curves and consisting of the sets of real points of the curves of th~ same c l a s s ,

Real isotopy and r i g i d isotopy c l a s s i f i c a t i o n s of

non-singular

curves of degree ~4 ~nd of the curves with a non-degenerate point

of

degree

~4

were known

in

the

last

century.

c l a s s i f i c a t i o n up to real isotop7 of curves of

degree 5

known.

the

In

1969 D,A.Gudkov

classification

of

[I~

obtained

non-singular

curves

double

of

The

was

real

also

isotopy

degree

6,

The

c l a s s i f i c a t i o n of non-singular curves of degree 5 and of degree 6 up to r i g i d isotopy was completed in 1978-198! in works of [2]~ V,V.Nikulin [~] and V,M,Kharlamov [ 4 ] ,

V.A.Rokhlin

The results of Kharlamov

[4] also allow to obtain the r~gid isotopy c l a s s i f i c a t i o n of

curves

of degree 5 ~ith a non-deg~ner~ite double point, in 1991 the author ( [ 5 ] ,

[6]) obtained the

classifications

up

to reai isotopy and up to r i g i d isotopy of the curves

of

with one non-degenerate double point using

scheme ( [ 3 ] )

and the results of E,B.Vinberg

Nikulin's

degree

6

([7]~ [8]) on the groups generated by

reflections, in the present paper we consider the scheme of obtaining

these

c l a s s i f i c a t i o n s , Besides, we discuss the c l a s s i f i c a t i o n up to

rigid

isotopy in the cases of the most interest,

Kharlamov noticed that the scheme developed gives a method for

non-singular

caiculation

of

the

by

groups

([3])

Nikuiin

of

monodromy of

curves of degree 6 ( if the beginning and the end of

rigid isotopy coincide with non-singular isotopy defines some permutation real points of A; we w i l l

curve A,

then

of t h e components

call the group of

all

the group of monodromy of curw~ A ). Kharlamov

this

of

the

a

rigid set

of

s u c h permutations has

calculated

groups of monodromy of non-singular M-curves of degree 6 is a curve with ~a~imal possible number of components

of

(

the

M-curve

the

real

point set for' given degree; this number is equal to 11 for degree 6) in

the

present

paper

we

consider

the

general

scheme

of

269

c a l c u l a t i o n of th~ group of

monodromy

of

curves

non-singular

of

discussed

degree 6. Only the cases of ~-curves and (M-l)-curves are in details,

l.Statements of results

We ~il! consider the curves D~ degree 6 with double point

as

the

results

of

the

a

simplest

non-degene:ate

degenerations

of

non-singular curves, Rigid isotopy

classification

of

these

~urves

is

given

by

Theor;~m 2. t and Propositions 3.4, 3.5 of the present paper. Calculation of the g~'oups of monodromy of

non-singular

curves

oF degree 6 c~n be done using Theorem 3.6. Zn the present part of the p~p~r ~e w i l l

discuss the

cases

of

M-:urves and (M-l)-curves only. For the sc~emem of d i s p o s i t i o n of ovals of non-singular (an oval is ~ connected component of the set of

tea!

curves

points

of

curve which is homeomorphic to c i r c l ~ and embedded two-sidely ~pZ) we w i l l

a

into

use the system of notations suggested by Viro [9] . The

curve consisting of a single oval w i l i

be denoted by symbol <1>, the

empty curve - by symbol <0>. I f

symbol stands for

ovals then the

set

obtained

surrounding a l l

old ovals ~ L i l be denoted by symbol >.

of

ovals

by

some set

addition

of

an If

curve is the union of two non-Jntersectino sets of ovals denoted

of oval the by

and r e s p e c t i v e l y with no ova~ of one set surrounding an oval of the other set, then t h i s curve w i l l Besides, i f

A i s the notation

for

be denoted by symbol

some set

of

ovals

A~...UA of mnother notation where A repeats n times w i l l by nxA; parts nxl of notation w i l l

.

then

part

be denoted

be denoted by n.

Let us notice that the c l a s s i f i c a t i o n s up to real up to r i g i d isotopy coincide in the c:ases of

isotopy

non-singular

and

M-curves

and (M-i)-curves of degree 6 ([3]). Thus ~n these cases it is enough to point out the scheme of d i s p o s i t i o n of ovals in order to f i x r i g i d isotopy type of a curve.

the

270

No~ we w i l l present the schemes of

possible

conjunctions

c o n t r a c t i o n s of ovals of non-singuiar if-curves and degre~ 6. Not~ that no

oval

of

these

curves

and

(M-ll-curves

can

conjunct

of

with

itsmlf. An oval of a non-singuiar curve i~ c a l l e d even (odd) i f

it

lies

i n s i d e of ~ven (odd) number of ovals of t h i s curve. For convenience we f i r s t

fortnulate a general statement :

each empty ov~i of a non--singular c~rve contracted and th~r~ exi~to on!y one

rigid

of

degree

isotopy

6

eiass

can of

be the

r e s u l t s of such degeneration.

M-curves There arm three r i g i d isotopy types of non-singular M-curves of degree ~ :

, U 9),

Schemn_ U g",.

,'i,"5>.. , U ~x,~" xl<9~, . U I>,

Tho

~c hm.~,e of

possibl~

conjunctions

c o n t r a c t i o n s of ovais of a curve of t h i s isotopy type Fig.

I.

kl

i

\

.C

c.

Fig.l

is

shown

and in

271

it is stated that the even empt7 ovals of the to this rigid isotopy typ~

curve

belonging

can be numerated by numbers from I to

9

in such a way that the ¢ i r s t oval cmn conjunct with the second one, the second oval can

conjunct with the t h i r d one and so on with

last nineth oval conjuncting

with

first,

seventh

the

~ourth

and

the

the

first

the

o n e . Moreover,

owls

can

conjunct

the

with

a

non-empty one while the other conjunctions of ovals are impossible. The l e t t e r s standing near two dotted

lines

coincide

iff

the

r e s u l t s of the conjunctions corresponding to these l i n e s are r i g i d l y isotopic.

The l e t t e r s placed inside the empty ovals coincide i f f

the

r e s u l t s of contractions of these ovals ~re r i g i d l y i s o t o p i c . ~e w i l l

nam~ such schem~s the marked schemes of

conjunctions

and contractions of ovals. It

is clear that the even empty ovals of a non-singular M-curve

of degree 6 with the scheme < I < ! ) U g> can conjunct only in a c y c l i c order. There is natural cycli~ order for these ovals ( t h i s order defined up to i n v e r s i o n ) . Let us take a point inside the

odd

is

empty

oval and a l i n e containing t h i s point; we can r o t a t e the chosen l i n e around the chosen point

and

mark the

i n t e r s e c t s the even ~mpt7 mva!s. I t

order

is

the

in

w h i c h the

corollary

line

~rom Bezout

theorem that the c y c l i c order obtained cannot be changed by a isotopy. Thus non-neighbouring (in sense of the given c y c l i c

rigid orderl

ovals cannot conjunct.

Rigid isotopy cl~ssification of the

results

of

the

simplest

degenerations of curves with scheme U 9> can be formulated

as

follows : the numbers of r i g i d isotopy classes contmined

in

i~otopy

classes represented in Fig. 2 are: 2a) - i~ 2b) - I , 2c) - 2, 2d)

But in t h i s statement

some information is l o s t .

The group of monodromy of a curve of under

discussion

is

- 2.

equal

permutations of 3 elements.

to

$3,

the

where S

rigid is

isotopy the

group

type Of

272

b)

o)

c)

d)

Fig.2

Schem~

l•'~,~-" t~ U 5>.

The s~heme of conjunctions

and contractions

ovals is given in Fig.3. This scheme is not marked because case it is convinient

to say that the results of

the

in

this

contractions

of non-coincident ovals are not r i g i d l y isotopic and the r e s u l t s conjunctions of

non-coincidEnt

isotopic as well. We w i l l

pairs

of

ovals

are

of

not

of

rigidly

always present non-marked schemes

in

the

cases like this, It

is necessary to note that there are l i n e a r

the even empty and on the odd ovals of a curve of

orders degree

both 6

on

having

scheme U 5>,

The group of monodromy under discussion is trivial,

of a curve of

the

rigid

isotopy

type

273

Q

Fig.3 Scheme U I>.

The

~arked scheme

of

conjunctions

contrmctions of oval5 is represented in Fig.4,

tE

~c

®°

Q

Fig.4

and

274 Note that there is

a

natural

inversion) on the odd ovals of a U I>. So i t

linear

curve

order

of

(defined

degree

6

with

up

to

scheme

i~ clear that the odd ovals can conjunct only

in

chain. The group of monodromy of a curve of under discussion is equal to

the

rigid

isotopy

type

~/2Z.

(M-1)-curves

There are 6 rigid isotopy types of non-singular degree 6 :


(! U 8>,

(I<4> U 5>,

(M-1)-curves of

(i<5> U 47, (I<8> U I>,

>. Only those degeneration~ are

discussed

whose r e s u l t s

cannot

be obtained as the r e s u l t s of a degeneration of an M-curve.

Scheme . The marked scheme of conjunctions and contractions o~ ovals is given in Fig.5.

/

/

!

,A

i /

f

4' %

/

' I

;}_

-

' %

I /

I

' ^" ~ < - ~'~,

x %

I I

,A

A~ x

I

/

X

!

%

% %

Q

'6

6"

/

A Fig.5

1 I

275 The group of monodromy of a curve of under discussion is equal to A , where

A

the is

rigid

isotopy group

the

of

type even

permutations of 5 elements..

Scheme U B>.

The

marked scheme

of

conjunctions

and

contractions of ovals i s given in Fig,G,

k,

Fig.6 The group of monodromy of a curve of

rigid

isotopy

of

symmetries

of

and c o n t r a c t i o n s

of

The two dotted lines between the non-empty oval and one of

the

discussed is equal to De, where D e is the

the group

type

the regulmr 8-gon,

Scheme (L<4} U 5>. The scheme of c o n j u n c t i o n s ovals i s given in Fig,7,

even empty ovals denote the existence of two rigid

isotopy

of the results of conjunctions of the corresponding ovals,

classes Thus

it

276

is riot enough to point out a pair of conjunct ovals for pointing out the rigid isotopy cl~ss of the res,a!ts of conjunctions.

© df /

"'0-

Fig.7

The group of monodromy of a c~rve of this rigid isotopy type is trivial.

Scheme (I<5> U 4>. The scheme of conjunctions and contractions

ovals is given in Fig,8,

/

0

"'Q)' !

Fig.8

of

277

The group of monodromy of a curve of this rigid isotopy type is trivial.

Scheme

,

The

of

marked scheme

conjunctions

and

contractions of ovals i~ given in Fig,9,

#

F F

,

F

A

Fig.9 The group of manodromy of a curve of t h i s r i g i d isotopy type is

equal to Z / ~ ,

Scheme 7.

The

marked

scheme

of

conjunctions

and

contractions of ovals is given in F i g . t 0 .

Thus one can introduce a natural c y c l i c inversion on the odd ovals of a curve

of

order

degree

defined 6

having

up

to

scheme

>. The group of monodromy of a curve of discussed i s equal to S ,

the

rigid

isatopy

type

278

B!

'B

Fig.lO

The ~econd part oF the present paper is the rigid i~otopy classification

of

non-degenerate double point and to

curves

Let A be a curve of

of

calculation

monodromy of non-singular curves of d~gree

2.The r e d u c t i o n

d~voted

to

obtaining

degree 6 of

the

~ith

a

groups of

6.

to a problem on K3-surfaces

degree

6

with

a

non-degenerate

double

point, C A be the set of complex points of this curve. Set ~ A of real

points of curve A divides real projective plane ~pZ in two sets with common boundary ~A, I f one o÷ them

is

non-orientable

(both

sets

cannot be non-orientable) then l e t us denote i t by ~P= and the other

279

by ~P~. If both sets are orientable then let us denote an one

by ~ p _ Z the other by ~P~. Consider

the

arbitrary

t~o-sheeted

branched

covering space of complex projective plaf~e ~pz with branch locus Variety

X obtained as a r e s u l t of resolution of the singular

~A.

point

of the branched c o v e r i n g 5pace i s a K 3 - s u r f a c e . The i n v o l u t i o n of ~omplex conjugation acting to X by two

different

wa~s. L~t

us

denote the

i n v o l u t i o n on X with s~t of the fixed points

lifted

antiho!omorphic

lying

conj . The other ~ntiholomorphic i n v o l u t i o n on X

is

CP 2

on

above ~p2

(its

set

of

by the

fixed points l i e s above ~P~) w i l l be denoted by conj÷. Choose one of these i n v o l u t i o n s and denote i t

by

conj.

We can

identify

group

Hz(X;Z) together with ~he i n t e r s e c t i o n form and involumtion conj~ on i t with l a t t i c e L having the rank equal to 22, the

signature

equal

to -16 and i n v o l u t i o n ~. Denote by L+ and L_ the eigenspaces of

the

i n v o l u t i o n ~ corresponding to eigenvalues I and - l respectively. Let B be the b i l i n e a r form on L. The

diagonal

r e s t r i c t i o n s of B on L+ and L_ have one

f o r m s over

and

two

~

of

positive

the

squares

respectively . In H (~;~) we can point out two elements: class h of z hyperplane section and class ~ of exceptional d i v i s o r , Denote the

images in L of classes h and ~ by h'and ~ ' r e s p e c t i v e l y . The following statements hold true: B(h',h')=2, ~(h')=-h', The

B(~',S')=-2,

B(h',~')=O,

~(~'}=-~',

isomorphism

of

lattice

h ' ~ ~'(mod 2L) f:

L+

L

I

~e

2

will

call

the

isomorphism of quadruples, (Li'~i'hl '~i ') and (L2,~z,h2 'Sz '} if f o ~ =~ of I

2

f(h '

')=h ' , ~

f(~

2

')=~ ~

". 2

Let us fix quadruple (I)

(L,~,h',~') s a t i s f y i n g a l l conditions above. We w i l l

call

K3-surface

X with

fixed isomorphism of the quadruples :

the marked

(H2(X;Z),conj.,h,~}- - - ~

(L,~,h',~')

K3-surface.

Let subspace H2'°(X) of complex vector

space Hz(X,C)

is

the

2Be

image of the de Rham isomorphism

of

Hz'°(~) consisting

space

coho~otogous classes oF holomorphic 2-forms, The dimension of H~'°(Y) is equal to I. Denote by H the real one-dimensional of

HZ'°(Z)

consisting

of

x

vector

satisfying

~ondition: the image of vector x under the induced by the i n v o l u t i o n of coincides with c ~ j

complex: conjugation

(x). We w i l l

denote by

H'

space

subspace

the

involution

of

foilowing of

Hz(X,C)

of

coefficients

the

subspace of

H (~,~) generated by a ~i~ss dual (in the Poincare sense) to a class generating space H. Define (as was done in C 3 l , [ I 0 ] ) the space c

c

B (~,h')=O, B (~,B'l=O, ~(~)=~ } I R * , c

where B i s the extension of b i l i n e a r form B on L to b i l i n e a r

form

c

on L ® ~ , ~* acts by : i u l t i p ! i c a t i o n . The

period

mapping bringing

z~,.n~:ona[ s~bspace ~(H ] of L ~ ~ i~ corr'espondance with marked KS- s~rface m~,,~_ space ~ the ~pace of K~-sur~aces with

a

non-degenerate

double

period~

point

of

~ith

the

marked

involution

equivalent to @ and with linear s~stem Jhl.

~e will ~a!l

equival~nt

transformed one into

periods

'the periods

which

can

be

another by automorphism of quadruple ( I ) .

us describe the im,~ge of the period mapping up to

the

Let

equivalence.

Let element ~ ~ L ~ ~ generate class [ ~ ]~ ~, Let ~ = ~)+÷i(~_ where ~+ ~ ~

~,

~_ ~ L_~ ~. Denote the orthogona] complement in L_ of

l a t t i c e generated by vectors h' and ~' by L_h,~. L_h,~® ~ ,

~

= ~

of B on L+ and

N o t e that

~o

>0. The diagonal forms over ~ of the r e s t r i c t i o n s

L_h,~hav~ one

positive

square.

T h u s the

consists of two connected components. Each of these

space

0

components can

be transformed to one another by i n v o l u t i o n - ~ . We are interested in periods only up to ~quivalenc~ and thus we can consider the quotient space 0 / ( i , - ~ ) spaces

of

spaces.L+~

as the space of periods. Let ~+ and ~-h,~

Lobachevsky obtained

and L _ A , ~ ® ~

by

respectively,

the standard

be

the

procedure

from

It is easy to verify that

281 Period~ ([~+1,[~_]) b~longing to ~ +

x ~-h.~ are the periods of

the marked k~-s~rfaces " "~ with ~ non-degenerate double point is no vector v ~ h

if

that B (~,v)=O, B(v~h')=O, B(v,v)=-2 in

lattice

c

L. Consider a reflection with

respect

to

the

there

hyperplane

in

~+

orthogonal to vector v~L+ with square -2 (this reflection transforms x to x + B ( x , v ) . v ) .

I t can be transposed with ~ leaving vectors h' and

~ ' i n their places. Thus the reflection described gives us equivalent periods. I t also

h o l d s true

for

a

reflection

with

respect

to

hyperplane in ~-h,~ orthogonal to vector v ' ~ L_h,~ having square -2.

Let O+ and O_h,~ be the fundamental domains of groups generated reflections with respect to hyperplanes orthogonal to

by

vectors

squares -2 in ~+ and ~_ respectively. We have shown that the

with image

of the period mapping belongs to ~+x /~-h,~ up to equivalence. Consider element ~ ~ L ~ ~ such that there is vector v~L

with

the properties: B (~,v}=O, B(v,h')=O, B(v,v)=-2. c Let ~=v++v where v+e L÷ ® ©, v_e L_ ® @. We w i l l denote the orthogonal complement in 2~, E L_h. Note that i f

L_ to vector

h"

by

L_h

. Then we have

[~]~+XD_h,G , then v_# 0. Now we consider

t~o case~:

a) v+= 0. i f vector v

does not belong to space L_h.~ ~ ~

projection of vector v

, then the

on L h,~ ® ~ w i l l be denoted by v'

It

is

easy to notice that 2vj~ L_h,~ , B(2~'L,2v~)

0 or B(2v',2v~) = -6,

2~'_~ ~'

(mod 2L).

Consider hyperplanes in L_h,~ orthogonal to vectors v'_~ L_h,B where

B(2v',2v'_}=-6 and 2v_~ ~'

(mod 2L). Let

t h e s e hyperplanes divide

open polytope (~-h,~ int~ .polytopes ~t_h,B ( i = I , 2 , . . . ) (the number

of

these poiytopes may be infinite).

b) v+~m0. I t should be noted that B(V+-v ~v+-v _ ) = B(v++v _~v++v _) _

I f vectors v+-v_,v++v_, ~'

(*) except 6"

=

-2

are linear independent, t h e n class

[c~]

282

belongs to the sertain subset of codimention 2 This

subset is

vectors

the

and 2v÷_= ~ '

~ + x ~-h,G

intersection of two hyperplanes orthogonai

v++v_ and v÷-~'_. Linear

v -v_, ~ '

of space

is possible

only if

(mod 2L}.

Consider

dependence

v = ~'/2.

of

But

vectors

in ~ .

v e c t o r s 2v+~_~+ where B(2v+,~°v+)=-6 and 2v+= ~"

orthogonal

(mod 2L)

the previous case. Let these hyperplanes divide

v++v_

,

B(2v÷,2v+} = - 6

then

the h y p e r p l a n e s

to

open

similar

polytope

J (j=i,2 ...) (the number of these polytopes may into polytopes £~+

to to

~+ be

i n f i n i t e as well}. W.p have proved that the imago equivalence can be obtained

by

of

mapping

removing the

of

periods

closed

up

subset

to with

condimP.nsion at least 2 ( t h i s subset w i l l be denoted by 7.) from the j union of components £)xO . ~ (i=I ;j=l ) I t should be noted +.3 -tl,~ ~... 1. . . . 1, that some components ~+xD h,~ can be transformed to other ones by the automorphisms of quadruple ( I ) . Similar to [33 i t [co]~(u~J~xC~_h,8)\7._

is

possible

defines

only

to

show that

one

class

every

of

the

class

of

projective

equivalence of real K3-s¢Irfaces being t,~o-sheeted covering spaces of

C P z branched over a curve of degree 6 with a non-degenerate

double

point.

It is easy to prove that one of lattices L+ and

L h.~

has

no

vectors v such that v _----~' (mod 2L). Thus one of collections ~+J and O_h,~. consists of the single poiytope. Let O, stand £~+ i f

there exists vector v in L such that v = ~'

for

collection

(mod 2L} and

for

collection ¢)s -t~,~ in the other case. We have proved the theorem. Theorem 2 . 1 .

Consider

tile

curves

of

non-degenerate double point having quadruple

degree

6

with

a

(H2(X,Z),conj,5 h,

~)

isomorphic to quadruple (1). The rigid isotopy types of these curves are in one-to-one correspondence with the elements of collection

~,

considered up to the automorphisms of ( l ) .

We can choose for each curve of degree 6 with a

non-degenerate

283

double point involution conj in such a way that l a t t i c e L+ would not have such vectors v that ~ ~ ~'

(mud 2L), i . e .

collection ~ ,

coincide with collection 0 -h.~" t We suggest that involution

would

conj

is

chosen in this way.

3.Corollaries of theorem 2,1.

Let ~-h be the space of Lobachev.sky obtained in a standard way from space L_h. The group generated by the reflections with

respect

to the hyperplanes orthogor,al to the vectors in L_A having square -2 acts in ~

.. Let ~

stand for ~ polytope b~ing a fundamental

domain

- t%

of this group and having a fa~e orthogonal to ~'. The Coxeter scheme ( see ~or example appears to contain all

information

[7])

of

polytope

required about collection [~, (let

us remind that the involution conj was chosen in

such a

way

that

collection ~ , coincides with collection O_tL~).. p

Let C be the Coxeter scheme of polytope 0_, e be the vertex scheme C corresponding to ~ ' .

Remove all thick

and

dotted

from scheme C (two vertices of a scheme are connected by

of

edges

thick

(or

dottedl edge i f bilinear form B has the vaiue equal (or ~ore than] 2 on corresponding vectors). Let C' stand for

the

scheme obtained.

Consider the symmetries of scheme C' generated by the of (L,~o,h').

The group of these symmetries acts on C'. Let

the corresponding quotient-scheme and e' for vertex e. C''.

it

automorphisms

the

class

C''

be

containing

Let K stand for the connected component of vertex

e'

in

is easy to prove the following statement.

Proposition

3. I.

The number of the polytopes

of

collection

~,

which cannot be transformed one into another by the automorphisms of (1) is equal to the number of vertices of scheme K.

Corollary 3,2. The number of the r i g i d isotopy types of the curves of degree 6 with a

(H2(X,Z},conj,,h,

non-degenerate doubl~ point

having quadruple

~} isomorphic to (I] is equal to the

number

o{

284

of s c h e m e K.

vertices

The construction of s c h e m e s

for

K

all

possible

classes

of

isomorphism of quadruples (H ( ~ Z ) , c o ~ j . , h, 3) cor?esponding to the 2

curves of degree 4 ~ith a

non-degenerate

double

point

allows

to

obtain r i g i d isotopy c l a s s i f i c a t i o n of the curves mentioned.

It

necessary to note that

a!ways

scheme K (unlike c o l l e c t i o n 0 , )

is

is

finite. ~e ~ i l l

reformulate now the above r i g i d isotopy

in terms of conjunctions and contractions of ovals

classification of

non-singular

curves of degree 6. But !~t

~s f i r s t

ma;~ some remarks.

Let A be m curve of degree 6 mith a non-degenerate double o point. There ~re two types of ~moothings of singular point of th'is curve. [.~t At

(t~O) be a smoothing of one

of

these

types

and

At

(t~O) be a smoothing of the other type. Let the number of ovals

of

non-singu!ar ~,~L.... ,~ A

,:~' <0) be no ~;ess "~'~,~n the number of ovals

of

(~. >0)o Then the f~mily At (to._
be

t

o

o

non-~;,,'~',~.,,~.l,.,~ curve At

c~lled the non-increasing simplest degeneration of curve A

L 0

family

At

(O
~ill

be

ca!led

the

non-decreasing

and the simplest

degeneration of curve A t

We w i l l double

point

consider the curves of degree 6 with as

the

results

of

the

a

non-degenerate

non-increasing

simplest

degenerations of non-s~ngular curves of degree 6. There are G ca!~es of th~

non-increasing

non-~ingular curve of degree 6 ( i f

its

degeneration

scheme of

of

disposition

ovals does not coincide ~ith (I~I< ~ • ' ' ~,x,., ' . I)

conjunction of an even empty oval ~ith the non-empty one;

I')

conjunction of an odd oval with the non-empty one;

~)

conjunction of two even empty ovals;

2') conjunction of two odd ovmls; ~)

contraction

of an even empty oval;

3') contraction of an odd oval;

the of

285 4)

conjunction of the non-empty oval with i t s e l f

that the component obtained

in

such

a

of the set of real points of the

way

curve

is embedded in ~p2 l i k e the union of two l i n e s . 5)

conjunction of an even empty oval with i t s e l f

that the component obtained

in

such

a

of the set of real points of the

way

curve

is embedded in ~p2 l i k e the union of two l i n e s .

Cases 4) and 5) are impossible for M-curves and (M-l)-curves of

degree 6.

Proposition 3,3.

Two curves being the results of degeneration

of

types i - 3 of the same non-singular curve of degree 6 have isomorphic quadruples (Hz (Y,Z~,co~js, ., ,

toni_.

h, ~)

and i n v o l u t i o n

S i m i l a r l y two curves being the r e s u l t s

types I ' - 3 ' isomorphic

of

the

same non-singul~r

quadruples (Hz(X,Z),coD]~ , h,

cenj

of

c~rve ~)

coincides

with

degenerations of

and

degree

6

of

have

involution

co~i

D~fine two graphs P and P" for an a r b i t r a r y non-singular

curve

coincides with toni÷.

of degree 6. The v e r t i c e s of graph P (P' r e s p e c t i v e l y ) correspond to the r i g i d isotopy ciasses of the r e s u l t s of degenerations I-3 ( I ' - 3 '

respectivelyl

connected by edge

if

the

of

the

given

curve.

c l m s s corresponding

Two to

v e r t i c e s can be r e a l i z e d as conjunction of ovals A class

corresponding

to

the

other

vertex

can

of

vertices one

and be

types

of B

are

these

and

the

realized

as

contraction of ova! A.

Proposition 3,4,

The graph P (P" r e s p e c t i v e l y ) of m non-singular

curve of degree 6 is isomorphi~ to scheme K constructed for a result of an a r b i t r a r y degeneration of types I-3

(I"-5 °

respectively)

of

the given curve.

Proposition 3,5. type 4 (type 5

Two curves being the r e s u l t s of degenerations of

r e s p e c t i v e l y ) of

the

same non-singular

curve

of

286

degree 6 arm r i g i d l y isotopic. Proposition 3.5 is the corollary of Theorem 2.1

and

the

fact

that in the cases of such degenerations both l a t t i c e s L_h,~ and

L+

do not have vectors which are congruent to ~' modulo 2L.

Construction of graphs P and P' for a l l r i g i d

isotopy

of the non-singular curves of degree 6 together with of Proposition 3.5 gives the r i g i d

isotopy

the

classes statement

classification

curves of degree 6 with a non-degenerate double point

in

of

the

terms

of

conjunctions and contractions of ovals of the non-singular cl~rves of

degree 6. The schemes K

for

.~ii

possible

classes

of

isomorphism

of

quadruples (H2 (X,Z),conj., h, ~) corresponding to the curves degree 6 with a non-degenerate double point (or, as is just

the

same, graphs

the

P

and

P'

for

all

rigid

isotopy

ciasses

non-singular curves of degree 6) have been calculated by the of the present paper. The r e s u l t s

of

Vinberg

([7],

[8])

of

of

auther on

the

groups generated by r e f l e c t i o n s are used. if ~

has i n f i n i t e number of faces i t

i t s Coxeter scheme. In t h i s case i t

is d i f f i c u l t

to calculate

is possible to calculate

scheme

K directly.

Now we w i l l

discuss the question

on

groups

of

monodromy of

non-singular curves of degree 6. Consider a non-singular curve of

degree

6

and

polytope

constructed for the result of an a r b i t r a r y degeneration of types I-3 (types i ' - 3 '

respectively). Consider vertices

Coxeter scheme C of polytope ~ contractions of even empty ovals

corresponding (odd ovals

a2, . . . ,

as, to

the

an

classes

respectively)

of

of of the

given curve Let

S

be

the

group

automorphisms of (L, ~, h ' ) ,

of

symmetries

of

C

generated

by

S' be the subgroup of S consisting

of

the symmetries leaving vertices al, a z , , . . , a n on t h e i r places.

287 Theorem 3 . 6 , monodromy

Quotient

of even ovals

~jrol.~p S/S"

is isomorphic

(odd ovals

the

to

respc-ct~ve]y)

of

group

of

non-singular

CUrV~,

It ~s easy to calculate

the group o f

non-sinuular curve using Theorem 3 . 6 .

monodromy

of an

arbitrary

288

References l.Gudkov, D . A ~

Utkin, G.A.: Topology of the curves of degree 6 and

of the surfaces of d~gree 4 (in R~ssian). Uch. Zap. Gor'kovskogo Universiteta, v. 87, (1969) 2.Rokhlin, V.A.: CoMplex topological characteristics

algebraic curves (in Russian).

of

the

real

Usp. Mat. Nauk, v. 33, N 5,

77-89

(1978) 3.Nikulin, V.V.:

Integer

quadratic

applications (in Russian). N I, I l t - t 7 7

f o r m s and

some geometrical

izv. Akad. Nauk SSSR Set. Mat. v.

43,

(1979)

4.Kharlamov, V.~.: Rigid isotopy c l e s s i f i c a t i o n of the

real

curves of degree 5 (in Russian). Funk. Analis i ego Rrii

plane v. 15,

N I , 88-89 (198!) 5. iteT~berg~ I . V . :

Rigid

isotopy

classification

of

degenerations oF M-curves of degree 6 (in R~ssian).

the

simplest

Vestn.

Len.

Unlversiteta. Set. Mat. v. 3, 32-37 (1991) ~nb~,

~

~ R~gid iso~opy c o ~ i f . c a t i o n

6 with a non-degenerate double point

(in

of curves of

Russian).

degree

Z a p . Nauch.

Sem. LOMI AN SSSR. v. 193, 72-89 (i~91) 7.Vinberg, E~B.: On the groups of units of some quadratic forms (in Russian)~ Mat. Zb. v. 87~ ~ i ,

18-~6 (1972)

8.Vinberg, E.B.: The two ~ost algebraic

K3

surfaces.

M a t h . Ann.

v. 265, 1-21 (1983) 9.Viro, O.Ya.: The ~urves of degree 7, the curves of degree

hypothesis of Ragsdale v.

254,

lO.Nikuiin,

(in Russian).

Dokl.

Akad.

B

Nauk

and

SSSR

N 6, 1305 -~310 (J980) V ~ V . : Involutions

of

integer

quadratic

forms

applications to real ~igebrmic geometry (in Russian). Izv. Nauk SSSR Set. Mat. v. 47,

~ I~ 109-188 (19831

and Akad.

The 17-th Hilbert problem for noncompact real analytic manifolds Piotr Jaworski, University of Warsaw, Institute of Mathematics, u l . B a n a c h a 2, 00-913 W a r s z a w a e , P o l a n d

Abstract In this paper we deal with the ring of real analytic functions on a real analytic manifold. We investigate under what assumptions this ring and its field of quotients posses the Artin-Lang property and give the positive answer to Hilbert's 17 problem.

1

Introduction

Let Q be a real analytic connected paracompact manifold. By A we shall denote the ring of (global) real analytic functions on Q, and by M the field of meromorphic functions on Q, i.e. the field of quotients of A. The investigation of these objects gives rise to certain basic problems; under what assumptions the following properties are valid: P r o p e r t y 1 ( H i l b e r t ' s 17 p r o b l e m ) Every positive-definite analytic function on Q is a sum of squares of meromorphic functions i.e. k

(VxeQ f ( x ) >_O) ~

3h,,...,hk • M f = ~h~. i-----1

The field of meromorphic functions is formally real hence it may be ordered (see [9] §XI.2). P r o p e r t y 2 ( A r t i n - L a n g p r o p e r t y ) Let f l , . . - , fk be analytic functions on Q. If in every point of Q at least one fi is not negative then in every ordering a on M at least one f~ is not negative; (Vx • Q 3i f~(z) >_ 0) =-~ Va Si fi _> 0. We remark that the first property is a consequence of the second one. It is known that the answer is "yes" if: i) Q is compact ([10],[8] both properties); ii) Q is lowdimensional: property 1 is valid for dim Q < 2 ([3], [2], [7]); property 2 is valid for dim Q = 1 ([1]), iii) there are additional conditions on functions: property 1 is valid if f - l ( 0 ) is discrete ([2]).

290

The aim of this paper is to show that it is enough to assume that certain preimages are compact. Namely the property 2 (resp. 1)is valid if N~k=lf ( l ( ( - c a , 0]) (resp.f-l(0)) is compact. Moreover we shall show that the property 1 is valid if f - l ( 0 ) is a union of a discrete set and a compact set. We record here that independently the similar results were recently obtained by Ana Castilla from the University of Madrid.

2

Orderings and f i l t e r b a s e s

Let a be any ordering on the field M. Let mo (resp. W~) be the set of all infinitely small (resp. finite) elements of M: m~={f:VnEN Wo={f:3neg

Ifl
We remark that W~ is a valuation ring and mo is its maximal ideal (see [9] §XI.1). Moreover all bounded analytic functions belong to Wo (between the others the constant ones) - see [1]. Hence the residue field is isomorphic to the field of the real numbers: w o / m o = R.

Every equivalence class If] mod mo contains a constant (a constant function). Hence we put If] = r where r E R and f - r e mo. Let ~ : M ; R U ca be the place associated to a;

~(f)=

0 [/] oo

for for for

fEmo f c Wo fEM\W,

Obviously ~ is compatible with the ordering a: f > O : :. w(f) >0 or ~ ( f ) = ca. We identify R U ca = RP1. We shall consider both the place ~p and the meromorphic function f as a mapping to RP1. We shall describe the orderings on M with the help of filter bases of closed semianalytic sets (compare [1]). We associate to the ordering a the family of sets: Go = { f - r ( [ - 1 , 1 ] ) : f E mo V1A } P r o p o s i t i o n 1 The family Ga has the following properties: a) The empty set does not belong to Go

291

b) The intersection of any two elements of G~ contains the third one A, B EG~ ==~ 3C EG~

C cAMB;

c) If the closed subsets of Q - V~, ½ intersect every element of G~ then their intersection is not empty ( V V E G ~ VMVI~t 0 and V M V 2 ~ O )

~- V1AV2 ~t O;

d) If the compact subset of Q - Vo intersects every element of Ga then the intersection of all elements of G~ is not empty and consists of one point which belongs to Vo

(vv co

vnvo#o)

--->

l-'l v = {p,,}CVo; VEG~

e) The intersection of the closures (in RPI) of the images of elements of Ga under the analytic function is not empty and consists of one point equal to the value of the associated place ~ at this function Vf E A

N

Cl f ( Y ) = {~,(f)};

VEGo

f) If the analytic function maps a set from Go to an interval then the value of the associated place ~ at this function is contained in the same interval VfEA(3VEGa

VxeV a
~

a<_~(f)<_b;

g) If the analytic function is positive on some set from G~ then it is positive in the ordering 0"

VfEA(3VEG~

VxEV

0
----->, 0 < ~ f .

Remark. Conditions a and b imply that the family G~ is a filter base. Condition c implies that there is an unique ultrafilter G~ of closed subsets of Q which contains G~; namely

= {w

Q:vveco vnw#

}.

Hence the filter base Ga determines a point in the (~ech-Stone compactification of Q. Moreover ~ ( f ) equals to the limit of f at this point (cond. "e'). More details about filters and their connection with Cech-Stone compactification the reader may find in [5] §§1.6 and 3.6. Proof. a. If the set f - 1 ( [ - 1 , 1]) is empty then either f is positive on the whole domain either - f . In the first case Vx f(x) > 1 hence the function f - 1 is a square in A. Thus f is greater then 1 in the ordering a and does not belong to m~. Analogicaly in the second case. Therefore the family G~ does not

292

contain the empty set. b. If A = f - l ( [ - 1 , 1]) and B = g - a ( [ - 1 , 1 ] ) then we put C = (f2 + g2)-1([_1, 1]). c. If the sets 171 and V2 were disjoint then there would exist a bounded analytic function f which would separate them.

fly, > 2 fly2 <

-2

Let r = q o ( f ) t h e n the set V = ( f - r ) - 1 ( [ - 1 , 1]) would belong to Go and intersect at most one of Vi's; the contradiction. d. The family of closet sets V A V0 has the finite intersection property and is contained in the compact set V0 hence the intersection is nonempty ([5] Th.3.1.1). Moreover it follows from point c that the intersection of all elements of the family G~ may contain at most one point. Hence this point must be contained in V0. e. The manifold RP1 is compact, the family of closures of the images of the sets V has the finite intersection property hence the intersection is nonempty ([5] Th.3.1.1). Moreover. Let ~o(f) = r ~¢ oo (resp. = oz) then the functions n ( f - r),n • N (resp. (1 + n2)/(1 + i f ) ) belong to rn,~. Therefore the sets V~ = f - l ( [ r - 1/n,r + l / n ] ) (resp. V,~ = f - l ( ( - o o , - n ] U [n, oc)) ), n E N, belong to the family G~. Obviously n C l f(V, 0 = {r}. f. Point f is a direct corollary of e. Indeed if one image is contained in the interval then the intersection of the closures of all of them is also. g. Let W ---- f - l ( ( - - ( x ) , 0 ] ) .

The function f is positive on the set V hence the sets V and W are disjoint. Let X : Q - - * [o, 1] be a C °° function which equals 1 on W and 0 on V. We put ~b=X+(1-x)fObviously ¢ is positive on Q and equals to f on V. We approximate ¢ by an analytic function g such that 1

Vx E q

I g(x) - ¢(x) I<

(see [6], also [2] ). The analytic function g is positive on Q hence it is invertible and moreover it is a square of an analytic function, so it is positive in every ordering on M. On the other hand "1

Vx E V

[g(x) - f(x) 1< ~f(x) z,

hence

Vx E V l f(x) < 9(x) < 3 f(x) i.e. WeV

2

_<

f(x) < 2. _

293

T h e fraction S/g is an analytic function, thus from the point "f" we obtain that 52 - < ~ ( g f) - < 2 . Therefore the analytic function f / g is positive in a (compatibility of the place and the ordering), g is a square hence f is positive in a.

3

The

Artin-Lang

property

with

a compact

assump-

tion

Theorem

1 Let f l , . . . , fk be analytic functions on Q. If k

N f~-~((-oz,0))

a)

= O,

4=1 k

b) N f/-l((--°°,0])

is compact

i=1

then in every ordering on the field of meromorphic functions M at least one fi is positive. Proof. Let us assume that the thesis of the theorem is not valid; i.e. there exists an ordering a on M such that all fl are negative. Case 1. The filter base G~ is free i.e.

N

y=o.

VEG~

T h e set

k

N f/-l((--o0'O])

w =

i=1

is compact hence there exists a set V in Go which is disjoint with W (property "d"). k

0= w

n

v =

N (v

N

fi-l((--oo, 0])).

i=l

The sets

v n L-'((-~,0]) are closed therefore for some i there exists a set V~ E Go disjoint with V N f ~ - l ( ( - c ~ , 0]). Indeed, it follows by induction from "c" that the intersection of sets V n f~-~((-oo, 0]) would be not empty if each of them intersected all elements of G~. fi is positive on V n V1 hence the properties "g" and "b" assure us that fi is positive in a - a contradiction. Thus in any "free" ordering at least one f~ is positive.

294

Case 2. The filter base G , is fixed i.e.

N

v=

{po}.

VEGa,

In this case the value of the associated place T at an analytic function f equals to the value of f at the point p, (property "e "). Hence the valuation ring W~ contains the localization of the ring A at the point po. Next one may either apply the method of desingularisation - see [8] Th:C, or the method of excellent rings - see [10] Pr.2.2. Both lead to a contradiction. This finishes the proof of the theorem. We remark that the hypothesis "a" is used only in the second case, while the hypothesis "b" only in the first one.

C o r o l l a r y 1 Let f be a positive-definite analytic function on Q. If f has compact zero set then f is a sum of squares of meromorphic functions. Proof. From theorem we have that f is positive in every ordering on M. Thus f is a sum of squares (see [91 §XI.2).

4

17th Hilbert p r o b l e m with c o m p a c t + d i s c r e t e assumption

T h e o r e m 2 Let f be a positive-definite analytic function on Q such that f-l(O) is a union of a discrete set and a compact set then f is a sum of squares of meromorphic functions. Proof. The theorem follows directly from the following lemma: L e m m a 1 Let f be a positive-definite analytic function on Q such that f-l(O) is a union of a discrete set and a compact set then f is a product of two positive-definite analytic functions on Q f = gh

where g-l(O) is discrete and h-'(O) is compact. Indeed, both g and h are sums of meromorphic functions (see [2] for g and corollary 1 for h ), hence f is also a sum of squares. Proof of the lemma. Let D be a compact component of the zero set of f . We consider a subsheaf ,7 of the structure sheaf O defined by: ,Tv=fp'Op Jp = Op

if p C D if p ¢ D

295

,.7 is coherent hence there is a finite number of global sections hi,..., each stalk tip, p E Q (see [4] ). We put

ht which generate

h=h~+...+h~+f. The analytic funtion h is positive-definite and h -1(0) equals to D. Moreover at each point

pED hv= L + qvf2p where qv is a germ of an analytic function. Thus the fraction g = f is an analytic function which does not vanish on D.

References [1] C.Andradas,E.Becker, A note on the real spectrum of analytic functions on an analytic manifold of dimension one. in: Real Analytic and Algebraic Geometry. L.N. in Math. 1420. Springer V. 1990. [2] J.Bochnak,W.Kucharz, M.Shiota, On equivalence of ideals of real global analytic functions and the 17th Hilbert problem. Inventiones Math.63 (1981) 403-421. [3] J.Bochnak,J.J.Risler, Sur le theoreme des zeros pour les varietes analytiques reelles de dimension 2. Ann.Sci.Ec.Norm. Super.8 (1975) 353-364. [4] S.Coen, Sul rango dei fasci coerenti. Boll.Un.Mat.Ital.22 (1967) 373-383. [5] R.Engelking, General Topology. PWN Polish Sc. Publ. 1977. [6] M.Hirsch, Differential Topology. Springer V. 1976. [7] P.Jaworski, Positive Definite Analytic Functions and Vector Bundles. Bull. Acad.Pol.Sc. serie math. 30:11-12 (1982) 501-506. 329-339. [8] P.Jaworski, Extensions of Orderings on Fields of Quotients of Rings of Real Analytic Functions. Math.Nachr.125 (1986) 329-339. [9] S.Lang, Algebra, Reading Mass. 1965. [10] J.M.Ruiz, On Hilbert's 17th problem and real Nullstellensatz for global Analytic Functions, Math.Z. 190 (1985) 447-459.

CONSTRUCTION 0F NEW M-CURVES 0F 9-th DEGREE.

A. B. Korchagin Nizhny Novgorod University, USSR This paper continues investigations in the l-st part of Hilbert 16-th problem. I construct 6 new M-curves of 9-th degree ( Theorem 5 ), give all the information (Th.6 and 7) about M-curves of the 9-th degree that I know and formulate five conjectures about ovals arrangement of odd degree nonsingulsr algebraic curves. I use Viro techniques of gluing of charts of polynomials [1,2]. Theorem I. For any two numbers k,k'E ~ (0
X1 (X[]X2_kI1 2 ) (XoX2_kl Xl 2 ) (XOX2_k2X12 ) (XoX2_k3Xl 2 ) =0

(1)

can be smoothed by g l u i n g the charts shown on f i g . 1.

E

~

~o

4

6

%

o

o

#

#

7

I

3

Fi 6. I Proof. I ) There exists the 4-th degree curve Z=0 placed in projective system of coordinates as it shown on Zig.2.

C--O

°

r~6.2

z~.3

,

z~6.4

297

Let us consider conics C=O and line I~O placed in projective system of coordinates as you see on fig.3. The 4-th degree CUl~e CLx O + tx14 = 0 for sufficiently small ItJ and suitable sign of t is located such as shown on fig.4. Obvously the arc pq of this curve has a flex point r. Now we choose the new system of coorginates. Let the new axes xo=O and x1=O be the tangents at the double point of this curve ( one of the branches crossing through double point has a flex point). And let the axis x2=O be a taugent to this curve at flex point r. So we obtain the 4-th degree curve F=O shown on fig.2. 2) Let us consider the 6-th degree curve I g(xo:x I :xa) -

(Foqu-loqu)(xO:x 1:x~2) = 0

Xo4X12x24 where qu(xo:x1:x2) = (XoX2:XlX2:Xo 2) is quadratic projective involution and l(xo:x ~ :x2)=(xO:x2:x I ) is linear projective involution, qu-1=qu, 1-'=l. The curve G=O has double point in (0:0:1), cusp in (I :O:O) and degenerate 4-fold point in (0:1:0). It's shown on fig.5.

~(.--0 y % ~ ' = 0 0

////>z .5

~ig.6

3) Keeping double point and degenerate 4-fold point we can obtain the 6-th degree curve G 1=0 shown on fig.6 as a result of sequence of small perturbations shown on fig.7.

Zig.7

298 4) Let us consider the 5-th degree curve

(c t , , ~ ) ( x o : x 1 :x,z)

H I (x0:x I :xz)-

-- o

Xo4Xl 3 where hy(x0:x I :x2) = (x0x I :x12:xOx2) is birational quadratic transformaZion and by -I (xO:x I :x2)=(xO2:xoXl :XlX2). It is shown on fig.8. This curve has 3-fold point in (I "0:O), and 3 points

x~-~°/ k J\ /

~ig.8

Fig.g

of intersection with axis x0=0 one of which is flex point. Tangent to the curve H I=0 in this flex point is axis x0=O. I~t (0:0:1), (0:1 :~" ), (0:1:~) be coordinates of points of intersection ( ~ < ~" ). There exist two numbers a,b ~ ~,b~0 and linear projective transformation li(x0:x I :x2)=(xO:x1:(ax1+bx2)) such that li(0:0:1) = (0:0:1), li(O:1:~') a (0:1:k'), li(0:1:~) = (0:1:k) for any k < k'. So for any 0 < k < k' there exist 5-th curve H(xo:x I :x2)-(H I°li -I )(x0:x I :x2)=0, located in s shown on fig.9 and H(0:x I :x2)= cx I (x2_k, Xl )(x2_kXl ) 3 where c~0. It is not difficult to see ~here exists neig_j~,bm~hood U ~- ~ of number k" such that for any different k I ,k2,k3~U there exists the 5-th degree curve H(x0:x I :x2)=H(x0:x I :x2)-H(0:x I :x2)+

+cx I (x2-kx I )(xz-k Ix I ) (xz-kzx I )(x2-ksx I )=o shown with its chart on fig. 10. 5) Under transformation by -I the curve ~(x0:x I :x2)=0 maps onto 9-th degree curve S(x0:x I :x2)=~ohy)(x0:x I :x2)/cx1=0 that

299

Fig. I0 has a chart shown on fig. 1 I. This curve has N28 germ defined by equasion (I) in point (0:0:1) and has Z15 germ defined by equaZ ion x I (x_ox,2-mlXl 2 ) (xOx,2-m2xl 2 ) (xox2-m3xl 2 ) =0 in the point (I :0:0), where

(21

O<m 1<m2<m3.

FJ~E.11 6") In [3] it was proved that for any three numbers O<m1<m2<m3 any Z15 germ of plane curve defined by equation (2) can be smoothed by gluing the charts shown on fig. 12.

~

(o

2.

0

o

,~

oo

o

o

z

G

Fig. 12 7) iet us glue by Viro method the charts shown on Zig.1 t with ~he charZs s h o e on f i g . 1 2 we obtain the charts shown on

fig.1.

300 Theorem 2. For any two numbers m,m'~ ~ (0<m°<m) there exists neighbourhood U~- ~ of number m" (m~U) such that for any different m I ,m2,m3EU any plane curve NZ8 germ that defined by equation

Xl (xOx2_ml Xl 2 > <xOx2_m2xl 2 ) (xOx2_msxl 2 ) <xOx2_mx 12 ) =0 can be smoothed by gluing the charts shown on fig.13.

o

o

#

#

"7

'1

3

"1

Proof. T et US transform the curves provided with theorem I to transformation (x0:x I :x2)~-,(xO2:X0Xl : (x0x2-tx 12)), where t>k' and then to symmetrization (XQ:XI:XZ)~-~(Xo:XI:(-X2)) we obtain the charts we need. The next two theorems were proved in [4]. Theorem 3. For any two numbers k,k'~ ~ ( O
Xl (XOX2_k](1 2 ) (X(]X1 _k 1Xl 2 ) (xOx2_k2Xl 2 ) (X(]x2_k3x 12 )=0 can be smoothed by 61uin~ the cl3B~ts shown on f i 6. 14. o( #

4o

o

o

o

o

Fig.14 Theorem 4. For any two numbers m , m ° ~ ( O<m'<m ) there exists neighbourhood Ue~. of number m" ( m~U ) such that for any different m I ,m2,m3~U any plane curve N28 germ defined by

301 equation

Xl (xOx2_ml Xl 2 ) (xOx2_m2xl 2 ) (XoX2_m3xl 2 ) (xOx2_mx12)=0 can be smooChed b y g l u i n g the charts shown on f i g . 15.

z

6 I

o

0

4

Fi6.15 Theorem 5. The schemes contained i n the l a s t colllmu o f Table 1 can be r e a l i z a b l e by M-curves o f 9 - t h degree. Table 1 Cluing Scheme o f M-curve

N

1

2 3 4 5 8

14 14 14 15 15 !5

4,7,0,0 10,1,0,0 10,1,0,0 6,0,4,1 2,5,4,01 2,5,4,0i

fig.

.~,~,'0

1 1 1 13 13 13

4,0,7 4,0,7 10,0,1 10,0,1 10,0,1 4,0,7

<2 <5>117<7 >UJ> 111<5>UI <7>UI UJ> <2117UJ> <8UI UI UJ> <6U1 <1 >U1 <5>U1<13>UJ> <5UI <5>U2UJ>

Proof. As it iollowed irom theorems 1-4 the germs of curve XI (XoX2_]~ 12 ) (XoX2_kl XI 2 ) (XoX_F~2Xl 2 ) (XOX2_]~XI 2 )=0 in points (I :0:0) and (0:0:1) can be smoothed by gluing the charts shown on fig.1,13,14,15. I~t us do that according to Table I and we obtain M-curves that we need. There are 1723 admissible (satisfying to Bezout theorem) schemes for M-curves of 9-th degree [12], All the information that I know about 9-th degree M-curves is contained in the following two theorems.

302 Theorem 6 ([4-9] + Th.5). There exist M-curves o f degree which have the following 396 admissible schemes 1 ) <JU28>

2) <JU~UI<#>>, a=27-#, t<-P~--16, #=18,19,22,23 3) <JUo,U1 <#>U1<~>>, 0,=26-#-~.-, where #=1, 1-.<-z-<19, z'=22 #=2, 2~y-<10, z.=14 #-..=3, 3-U1U1<6>>, a=25-#-z---&, where (13,z')=(1,1), 1~6-<7, 6=9,10,11,13,14,18 (P,z')=(1,2), 6---.3,5,7,9,11,13,1"( (#,z')=(1,3), 3<-6---12 (#,z')=(1,4), 6=5,7,9,11,15 (#,z')=(1,5), 6=5,5,7,9,10,11,13,14 (#,~')=(1,5), 6=7,9,11 (~,~')=(1,7), 7<-6-<12 (n,~)=(1,8), 6=1 1 (#,;,,')=0 ,g), 6=9,10 (5',~.)=(1,10), 6=11 (#,z,)=(2,3), 6=3,5,7,11 (f~,~)=(2,5), 6=5,7,9 (#,~)=(2,7), 6=7,11 (#,z.)=(3,3), 3<-6-<7, 6=9,10 (#,z.)=(3,4), ,5=5,7,9 (#,v)=(3,5), ~6<-8, 6=10 ( # , z ) = ( 3 , 6 ) , 6=7 (#,z,)=(3,7), 6=7,9,10 (/?,~')=(4,5), 6='7 (#.;.')= ( 4 , 7 ) , .5=9 (#,~)=(5,5), ,~=5,6,7 (#,z,)=(5,6), a=7 ( # , ~ - ) = ( 5 , 7 ) , 6=7 ( # , r ) = ( 5 , 8 ) , ,~---8,10 ( # , ~ . ) = ( 6 , 7 ) , 6=7

9-th

303

5) <Jl.J~U1U1<~>111<6>U1 <~>>, ~=24-/9-z-&-~, where (/9,;*,6)=(1 ,! ,! ), ~=1,5,9,13,1T,21 (/9,;*,6)=(1,1,3), ~'--3,7,11,13 (#,;*,6)=(1,1,5), ~=5,9,13 (/~,;*,6)=(1,1,7), ~=7,11,15 ( # , ~ , 6 ) = ( 1 , 1 , 9 ) , ~=9 (f~,~',6)=(1,1,11), ~=11 (/9,;*,6)=(1,3,3), ~=5,9 (#,;~,6)=(1,3,5), ~=7,11 (/9,;*,6)=(1,3,7), ~=9 (/9,;*,6)=(1,5,5), ~=5 (/9,;*,6)=(1,5,7), ~=7,11 (#,;*,6)=(1,7,7), ==9 (#,;*,6)=(3,3,3), ~--~3,7 (#,;*,6)=(3,3,5), ~=5 (#,;*,6)=(3,3,7), ~=7 (#,;*,6)=(3,5,5), ~=7 (#,;*,6)=(5,5,7), ~=7 6) <Jt~U1 <#U1<;*>>>, a=26-/9-r, where ;*=1, /9=0, 2-
304

~=22, , -~, Tiuese 896 M-curves conzaLn 22 M-curves conszruczed by G.M. Theorem 7 d e g r e e w. . . . . . .

([19-15]), .i n e .

, ) 2 schemes <JUI<2T>>,

There

non

.

exis:

M-curves

of

9-zh

e ~,,eme~.

cm ......

<JUIUI .25.....

3) 52 schemes <JH",<~>UI ,.'.r~>H1<~>>, where a+n+~=25, I~-~-pr~_-~ 4) 12 schemes <JU;H~ Ul <~->>, where a+n+~=24, ~<-~<#~ ~nd

~,n,;~-

-

even

0

......e~ <JUaUI <~>ll1H1 .~.U,/~-,~ where •- ~ + F ~ + ~ . + 6 + ~ = 2 4 , I _U1 U1 <~>>, where

a+/~+Z+&=2,

even. 7) 7 schemes <JUt >>, <JU1UI <1 <25>>>, <JU1 <1U1<25>>>, i~f'~.~,

and

a,r~,~.,~

-

<JUt UI >>, <JUI <2U1 <24>>>, <JU1 <3U1 <23>>>, <JUt <25UI
- ~

aO~nizsible schemes c o n t a i n .e~..~c.e~ b y T.Yledler ~" ~.,.~.~.

So

zhe

.~.~ .,~.~ ~ .

9-zh

~.~.~nea.

~e have

Lez

remLnd

us

52 schemes w h i c h were

4-98

degree

M-curves

classification

ir2zers~iga~ed abou~ o$

~he

7-zh

51~ admissible

degree

is

non

schemes.

nonsJmgular

curves

~ ~. £,8,~ 1 . , . ~ C ~ - , ~ . 0 n .

Theorem

8

(

Viro

~16,17]).

There

exis~

7-~h

degree

nonsir4£alar plane c ~ v e s which have zhe ~~"o ~ o w z "n g schemes: I ) < J ~ : > , where 0~o~15 2) <JL~UI<~>>, where a+P~-14, 0 ~ < 1 3 , 3 ) <JUt .'.>. There

no%

exis~

~-c'J~ves

These ~heorems 6 , 7 , 8 o$ 11 and 1 3 - ~ h d e g r e e s conjec~es.

which

have

~he

1_<~_13 scheme <JU1 >.

and my c h e c k i r ~6 o f n o n s ~ u g u l a r c u r v e s a l l o w e d ~o f o r m u l a t e ~he f o l l o w i n g

~,.u~,'.,'az real algebraic c,'~-ve o$ odd degree and i~s oval S is sulTounded by o~her r ovals of Zhe curve, i$ r . . . . .

..~o o ~ (even) -~...enzne oval S is called odd (even) oval. Lez n(p) be zhe number of odd (even) ovals oZ Zhe curve F=0.

305

Conjecture I. If F=O is nonsingular curve of odd degree d then

(3d2~5d+3)/8

if d~--1 rood 4

(3d2-6d-1)/8

if ~-- mod 4

n-p <- i

L

Conjecture 2. If F=O is nonsingular curve of odd degree d then

P~

i

(3d2-6d+3)/8

if @~I rood 4

(3e2-6 -1)/8

if cL=~ -° rood 4.

Conjecture I is analoguos to Petrovsky theorem about even degree nonsingular curves [14]. Conjecture 2. is analogous to famous Rag#dale conjecture [15]. By analisis of theorems 6 and 7 I have also formulated the following conjectures. Conjecture 3. If <JI/~UI

UIUI<6>>, ~+~+z+6=25, P->I, z->1, 6_>I is the scheme of M-curve of 9-th degree then I ) p,z,6 are odd or 2) two of P,z,6 are odd and one of them is even.

Con,jecture 4. I f <JU~UI

U1<~,>Ul<6>U1<=>>, ~+P+z+6+~=24, P->l, z->1, 6->I, ~->1 is the scheme of M-curve of 9-th degree then ~ 0 mod 4 and p,~,6,~ are odd. Conjecture 5. If <Jk~UI>>, e+#+z+=26, the scheme of M-curve of 9-th degree then ~,0.

I.

~zl

is

Viro O.Ya. Gluing of plane real algebraic curves and constructions of curves of degrees 6 and 7. -Lect. Notes in Math., 1984, N 1080, pp. 187-200. 2. Viro O.Ya. Gluing of algebraic hypersurfaces, smoothing o f s i n g u l a r i t i e s and constructions of curves. - Proceedings o f I~ningrad I n t . Topological Conf., 1983, pp. 149-197 (in Russian).

306

3. Korchs~in A.B. Isotopy classification of plane seventh degree curves with the only singular point Z15. - Iect.Notes in Math., 1988, v. 1346, pp. 407-426. 4. Korchs~in A.B. M-curves of the 9-th degree: realizability of 167 types. - P a p e r placed in VINITI 10.11.87, N7884-B87 (in Russian). 5. Polotovskii G.M. To the problem of nonsingular algebraic cureve ovals arrangement topology classification in projective plane. - Methods of quA]itative theory of diff. equations, Gorky State Univ., 1975, pp. 101-128 (in Rassian). 6. Korchsgin A.B. New capacity of Bruzotti method for construction of M-curves of degree >- 8. - Methods of qns]itative theory of diff. equations, Gorky State Univ., 1978, pp. 149-159,(1n Russian). 7. Korchagin A.B. M-curves of the g-th degree: constructions of !41 curves. - Paper placed in VINITI 29.10.86, NY45g-B86 (in Russian). 8. Korchsgin A.B. M-curves of the 9-th degree: realizability of 32 types. - Paper placed in VINITI 14.04.87, NZ566-BS7 (in Russian). 9. Korclqagin A.B. M-curves of the 9-th degree: realizability of 24 types. - Paper placed in VINITI 29.11.87, N3049-B87 (in Russian). 10. Fiedler T. Shears of lines and real algebraic curves topology. -Izvestia Acad. of Sc. USSR, math., v.46, N 4, 1982, pp.853-863 (in Rassian). 1 I. Korchagin A.B. M-curves of the 9-th degree: nonrealizability of 12 types. - Methods of q11~]itative theory of diff. equations, Gorky State Univ., 1985, pp.72-76, (in Russian). 13. Korctmgin A.B. M-curves of the 9-th degree: the new prohibitions. -Math. Zametki, v.Sg, N2, 1986, pp.277-283, (in Russian). IB. Korchsgin A.B. M-curves of the 9-th degree: nonrealizability o5 !2 types with two nests. - Paper placed in VINITI 04.03.88, N1&32-B88 (in Russian). 14. Petrovsky I.G. On the topology of real plane algebraic curves.- Ann. of Math., v.39, N I, 1938, pp. 187-209.

307 1 5. Ragsd~!e V. On the arrsngment of the real branches of plane algebraic

curves.

-

Amer.

Jour.

Math., v. 28,

1906,

pp. 377-404. 18. Viro O.Ya. Curves of degree 7, curves o$ degree 8 and RagsdAle conjecture. Soviet D o k l . , v.254, N6, 1980, pp. t306-1310 ( i n Russian). 17. Viro O.Ya. Curves of degrees 7 and 8: new r e s t r i c t i o n s . I z v e s t i a Acad.o$ Sc. USSR, math. ,V.47, NS, 1983, pp.1135-1150 ( i n Russian).

ON LINEAR DIFFERENTIAL JACOBIAN CONJECTURE

OPERATORS

RELATED

TO

THE

n-DIMENSIONAL

Tadeusz KRASI~SKI Stanislaw SPODZIEJA I n s t i t u t e of Mathematics, U n i v e r s i t y of Lod±, 90-238 L6dz, P o l a n d

ul.S. Banacha 22,

Introduction Let

En

be

the

ring

standard

topology

of

C n,

let

F =

(fl'''"

into en. In operators

the

and

of

uniform

paper

entire

convergence

fn ) ~ En be u n we consider

AT1 : En E f ~-~ J a c ( f l , . . . , f i _ l i

=

l,...,n,

where

functions

Jac(.)

on

Cn

compact

a fixed the

on

subsets

mapping

linear

such

Let

that

F =

JacF

(i) F is

the

a polynomial

(ii)

A~(E ) lF n (iii) AI(En)

is d e n s e

means

the

is d e n s e

following

automorphism in

E

in

n En

of

from

Cn

' f ' f i + l ' ' ' ' ' f n ) ~ En' usual

(Pl .... 'Pn ) : cn --~ Cn

= I. T h e n

the

differential

jacobian.

operators AT w e r e s t u d i e d m a n y t i m e s in c o n n e c t i o n 1 Jacobian Conjecture ([i],[3],[9],[i0],[ii],[12],[13]). In t h e p a p e r we p r o v e the f o l l o w i n g t h e o r e m Theorem.

with

for and

with

be a p o l y n o m i a l

conditions of

The the

mapping

are e q u i v a l e n t :

C n,

i = i,

.. n-l, " ' kerA[ = { f(P2,...,Pn)

:

f ~ En_l}. For n = 2 the e q u i v a l e n c e (i) ~ (ii) w a s p r o v e d by Y . S t e i n in [I0]. He c a l l e d the i m p l i c a t i o n J a c F = 1 ~ (ii) the Analytic Jacobian Conjecture. A s c o r o l l a r i e s we o b t a i n s i m i l a r t h e o r e m s if we r e p l a c e E n by t h e r i n g s of p o l y n o m i a l s C[X] := C [ X l , . . . , X n ] or R[X] := R[Xl,...,Xn] i. O p e r a t o r s Let fi ~ En'

F =

(in the

latter

case

for a r e a l

polynomial

mapping

F).

AT l (fl .... 'fn ) : cn

i = l,...,n.

The

basic

~ Cn

be an e n t i r e

properties

of

a~l

mapping,

i.e.

(defined

in

309

Introduction)

are g i v e n The

in the f o l l o w i n g p r o p o s i t i o n s

Proposition

I.I.

functions

differential

o p e r a t o r s on

A F1

En, c o n t i n u o u s

T h e y are a l s o d e r i v a t i o n s

of the r i n g

Proposition

for

constant, such that

1.2

(cf.[3]

n=2,

k ..

for any

ik(f ) =

1.3

(cf.[l],

,n, c o m m u t e .

.

i e.

2.Operators Let

i

if

JacF

A[ I

are

F =

°

in the t o p o l o g y of

[9]).

and

If

JacF

f ~ En

is

a

E n-

nonzero

U

of

x

o F

o

[3], [9]). The o p e r a t o r s

3

3

for any

i,j

~FI ' i = = 1 ..... n,

i

is c o n s t a n t ,

(PI,...,Pn)

: Pj(x)

and its d e c o m p o s i t i o n

o

: en

~ en

be

a polynomial

i ~ {l,...,n} and C o n s i d e r the a l g e b r a i c set

= yj,

mapping numbers

j = 1 ..... i - l , i + l ..... n}

into irreducible

S = S1 u ... u S k

components

•

J a c F = I, t h e r e f o r e and d i s j o i n t

SI,...,S k

algebraic

Proposition

2.1.

If

Ai(En)

superscript

F

in

A[)

curves

are

biregular

Sj

linear

a n d level sets of p o l y n o m i a l m a p p i n g s

S := { x e ~n

PiISj,

n,

E n. o

a F = AF o AF

with JacF = i. Fix yl,...,Yi_l,Yi+l,...,y n e e

mensional surfaces.

t

aYil--.aYik

p.297,

AF

.

if and o n l y

Since

i,...

ak(f°(FIu)-l)

(JacF)

i l , . . . , i k ~ {l,...,n}

Proposition ' .

=

then, for any x e C n and a n e i g h b o u r h o o d FIU is a b i h o l o m o r p h i s m , we h a v e in U

Ii

1

i

t

sets.

(if is to

are n o n s i n g u l a r ,

F

is

They

are

f i x e d we

dense

in

En,

C

for

j =

also

open Riemann

shall

then

one-di-

omit

the

the

algebraic

1,...,k.

Moreover,

j = 1 ..... k, are b i r e g u l a r .

Proof. (cf.[lO] for n = 2). S i n c e the p r o o f is a n a l o g o u s for e a c h of the i r r e d u c i b l e c o m p o n e n t Sj of S, we m a y a s s u m e t h a t S is irreducible. Let us d e n o t e f u n c t i o n s on S w i t h the c o m p a c t s u b s e t s of S, and

by O(S) the ring of h o l o m o r p h i c t o p o l o g y of u n i f o r m c o n v e r g e n c e on by I(S) the ideal in En of all

entire

on

functions

vanishing

S.

By

Cartan's

theorem

O(S)

310

En/I(S ).

Since

Ai(I(S))

c I(S),

Ai

induces,

in

the

canonical

way,

a unique continuous derivation D of the ring O(S). The p r o o f w i l l be d i v i d e d in several steps: i. D(O(S)) is dense in O(S). It follows from the a s s u m p t i o n that Ai(En) is d e n s e in E n2. For any (PilS)

where the

is

(PilS)

other

=

f ~ O(S)

we have

a holomorphic

chart

is invertible)

hand,

we have

I.

This

gives

=

exist

g n , h n e O(S)

Fix

zo ~ S. We may a s s u m e

any c o n t i n u o u s z ~ S we have

Hence,

on

S. Locally,

locally

(i.e.

S

On

= D(fo(PilS)-Io(PilS))

equality

on

because

locally

is exact.

and

hence

Since

(PiIS)

S, therefore, for a g i v e n ~ = gd(PilS ). By step 1 there

gn

by step

have

(fo(Pils)-l)'o(Pi~S)

e

such that

gn = D(hn)"

D(f)

desired

form

we

(fo(Pils)-l)'o(PilS)d(PilS).

is l o c a l l y a h o l o m o r p h i c chart on there exists g ~ O(S) such that

and

So,

locally

the

globally. 3. Each h o l o m o r p h i c

So

df =

also

(fo(Pils)-l)'o(PilS)D(PiiS)

D~PilS ) =

df = D ( f ) d ( P i l S )

on

) g

in the t o p o l o g y

of

0(S)

2, dh = D ( h n ) d ( P i l S ) = gnd(PilS).

that

hn(Zo)

piecewise-differentiable

= 0

for

path

n ~ ~. Then,

~

joining

z o

for with

hn(Z ) = Idh n = Ignd(PilS). Hence,

there

exists

the

limit

h(z)

:= l'mn~ hn(Z)

= /gd(PilS )

and

it does not d e p e n d on the choice of ~ Of course, h is h o l o m o r p h i c on S and dh = ~ . 4.S is b i r e g u l a r to C. From the above step and the R i e m a n n theorem S is b i h o l o m o r p h i c to the unit disc or to e. Since S is an a l g e b r a i c set, then by the L i o u v i l l e p r o p e r t y of a l g e b r a i c sets, S is b i h o l o m o r p h i c to C. Let ~ :C ~ S be such a biholomorphism. Again, since S is an a l g e b r a i c set, ~ is a b i r e g u l a r m a p p i n g (see [7], Th.4). 5. (PilS) is a b i r e g u l a r mapping. Obviously, (PilS) is regular, mapping and

locally invertible ~ :C ) S we have

locally

invertible.

So,

mapping. Hence, for any b i r e g u l a r that (PilS)o~ : C ) C is r e g u l a r this

az + b, a,b ~ C, a ~ O. Hence 3.

composition

(PilS)

has the

= Lo #-I

form

L(z)

is biregular,

o

The m a i n t h e o r e m Let

such

that

F =

(PI,...,Pn)

JacF

F(C n) = C n . well-defined

=

I.

: Cn

Then

F

will

a

So, the d e g r e e degF ([6], §3A). Put d := degF

B F := {y ~ C n : #F-l(y) which

) Cn is

be c a l l e d

the

set

of

be

a

polynomial

dominating

mapping

mapping,

i.e.

of the m a p p i n g F and d e f i n e the set

is

~ d} bifurcation

points

of

F.

It

is

=

311

known that On

the

BF

is a c o n s t r u c t i b l e

other

hand,

from

set

tile

([5],

fact

Lemma

in

that

1.8.4).

F

is

a

local b i h o l o m o r p h i s m we have from P r o p o s i t i o n 3.17 in [6] that C n \ B F is open and non-empty. Hence BF is an a l g e b r a i c set d i f f e r e n t from C n. For any i ~ {l,...,n}, e n-I

the

we

canonical

shall

denote

projection

by

~i

: ~n

>

~i(Yl,...,yn):=

(Yl,---,Yi_I,Yi+I,---,Yn)Lemma

3.1.

Fix

there e x i s t s

i 6 {I .... ,n}.

an a l g e b r a i c

If

set

Ai(En)

is d e n s e

B i c C n-I

En,

then

has

the

such t h a t

-i BF = gi (Bi)"

(3.2)

Proof. We shall first show that f o l l o w i n g property:

the

(3.3)

nil(=i(y))

if

Indeed,

y ~ e n \ BF, then

let

2.l,the Yi-l'

us

take

set

S

=

y

=

{x

~

complement

(y~,...,yn) e

:

function

components

Pi[Sj

BF,

therefore

the

fact

z E C,

is a b i r e g u l a r

k = d.

that

Hence,

Sl,...,S d

and,

mapping

from

are

if

(3.3)

it

easily

y ~ B F , then

we

obtain

the

an a l g e b r a i c is o b v i o u s . o

set in

follows

(i) F

is a p o l y n o m i a l

(ii) Ai(En)

= En,

of

yl,...,Pi_l(X)

=

into d i s j o i n t

any

Sj

such

Sj,

the

onto e. since

we

obtain

= d.

This

BF

that Bi

the

of

Pi[Sj

that, gives

for

y and any

property

has t h e p r o p e r t y :

automorphism

set

B i := ~i(BF)

so defined,

assumptions

on

of

F,

equality

the

is (3.2)

following

C n,

i = 1 ..... n-l,

(iii) Ai(En)

is d e n s e

in

En,

(iv)

is dense

in

En

AI(En)

=

Proposition

c BF .

For the

T h e o r e m 3.4. U n d e r the above c o n d i t i o n s are equivalent:

By

bijectivity

that

difficulty

C n-l.

BF

for

disjoint,

~il(~i(y))

without

g

decomposes

#F-l(yl,...,Yi_l,z,Yi+l,...,yn)

(3.3) . From

Hence

Sl,...,S k

~\B

c en \ BF .

Pl(X)

Pi+l (x) = Y i + l ' ' ' ' ' P n (x) = Yn }

irreducible

E

in

i = 1 .... ,n-l, and

kerA 1 =

{f(P2,...,Pn)

n : f ~ En_l}.

Proof.

I.

function Propositon

(i) such 1.2,

,

(ii).

that

f

Take

og/0Y i

=

c E . Let g be an e n t i r e _I n . Put h := goF. Then, by

foF

312

A i (h) = a(h°F-l) a¥. 1 2. 3. be

oF = ~ . o F 1

= f.

(ii) ~ (iii). Obvious. (iii) ~ (i). Let d

the

bifurcation

set

algebraic

set

B* c C

therefore algebraic

B ~ C. Take set

of

the

F.

degree

By

Lemma

of

F

3.1,

and

there

let

BF

exists

B F = cn-lx B * . Since

such that

,

an

B F ~ C n,

,

V := Pnl(z) Since

be

z ~ B

and c o n s i d e r

the n o n s i n g u l a r

= F-l(cn-lx{z}).

(cn-lx{z})

n B F = ~, the m a p p i n g

has the same number, Moreover, FIV is a

equal local

FIV : V

) en-lx{z}

to d, of e l e m e n t s in each fibre. b i h o l o m o r p h i s m . H e n c e we e a s i l y get

that FIV is a covering. Since en-lx{z} is simply connected, then F is a b i h o l o m o r p h i s m on each t o p o l o g i c a l c o m p o n e n t of V. So, the n u m b e r of t h e m is equal to d. Let Vl,...,V d be these components.

Since

gradP n

:=

(aPn/aXl,...,aPn/aXn)

vanishes

nowhere, therefore, from the c o n n e c t i v i t y of the i r r e d u c i b l e components of a l g e b r a i c sets ([6], Coro4.16) we o b t a i n that Vl,...,V d are also the irreducible components of V. The mappings Hence

FlVi,

and

FIV i

from

i

=

is a b i r e g u l a r Let

Qi(x)

=

Since

be

regular

of

Vi

([6],

onto

polynomials

gradP n

Pn - z = ~QI...Qd,

that

nowhere,

and

biholomorphic. we get that

en-lx{z},

such

vanishes ~ ~ e,

and

Th.3°20)

i = l,...,d.

Vi =

{x

therefore,

QilVj

= const,

~ ~n: by

the

for any

Qi = TijQj + ~ij w h e r e Tij ~ C[X], ~ij ~ e. T h e n we degTij = O. In consequence, there e x i s t s a p o l y n o m i a l

i,j. H e n c e e a s i l y get H ~ e[Z]

are

Main T h e o r e m

mapping

QI,...,Qd 0}.

above,

l,...,d,

Zariski's

such

that

because, otherwise, g r a d P n vanishes.

degH = d there

and

would

Pn - z = HoQ I. H e n c e

exist

points

in

en

d = 1

at w h i c h

Since d = i, t h e r e f o r e F is injective. H e n c e F is a p o l y n o m i a l a u t o m o r p h i s m (see [8], Th.l.4, [i], Th.2°l). 4. (i) ~ (iv). By the i m p l i c a t i o n (i) ~ (ii) it s u f f i c i e s to show only the e q u a l i t y ker~ 1 = { f(P2,...,Pn) ~ E n : f ~ En_l}. D e n o t e the set on the right hand side of this e q u a l i t y Obviously, sition

we have

Enc

ker~ I. Let

depends

E n-

By Propo-

i.2, a(g°F-l) 8Y 1

Then

g ~ kerA I.

by

B(goF-l)/aY 1

=

O,

and

only on the v a r i a b l e s

oF = 0

hence

the

Y2,...,Yn.

entire

function

Denoting

goF -I

goF -I by

f,

313

we have 5.

g = f(P2,...,Pn). (iv) ~ (i). Put

dominating

mapping.

So,

ring

C[YI,...,Yn_I]

ring

e[X]

of

So, g ~ En" F~= (P2 ..... Pn ) it

of

regular

induces

regular

C[X]

then

P

~

functions

functions

assumption kerA 1 = E ~, n F (C[YI,...,Yn_I]) =C[P2,...,Pn] Indeed, if an element P ~ C[X] kerA 1

: C n --~ e n-l.

a monomorphism on

on

C n.

C n-I

We

implies

It

F

claim

that

is

a

of

the

into

the

that

the

the

ring

is integrally closed in e[X]. is integral over e[p2,...,pn] c

(because

P

is

constant

on

each

irreducible component of any fibre of F). Hence P = f(P2,...,Pn) for some f ~ En_ I. Since P'P2' .... Pn are polynomials, therefore, f is also a polynomial. This means that P ~ ~[P2,...,Pn]. From the fact that e[P2, .... Pn] is integrally c l o s e d in C[X]

it

easily

follows

that

the

field

e(P2,...,Pn)

is

a l g e b r a i c a l l y closed in C(XI,...,Xn). Hence F is primitive, i.e. there exists an a l g e b r a i c set V c ~n-l, V ~ C n-l, such that,

for

any

y

~

en-l\v,

the

algebraic

set

irreducible ([4]). So, for any y e en-l\v, c o n n e c t e d ([6],Cor.4.16). On the other hand, the a s s u m p t i o n that

the

En

each

implies

mapping

(by

Proposition

PlIF-l(y)

F-l(y).Hence,

for

2.1)

is b i r e g u l a r each

that

for

on each

y E Cn-l\v,

the

topological mapping

automorphism

Remark. 3.5. = En in that

is

~-l(y)

AI(En)

injective. This implies that in the open set mapping F is injective. Also, degF =i. This polynomial

~-l(y)

set

is

is dense

in

~ c n-l,

the

y

component

of

Pl[F-l(y)

is

F-I(cx(cn-I\v)) the gives that F is a

([8],Th.I.4,[I],Th.2.1).D

In the above theorem,

(iv) can be omitted

for

because

n = 2

it follows

the c o n d i t i o n

kerA 1

from the a s s u m p t i o n

J a c F = i. G

4. O p e r a t o r s Let

A[

k = ~

polynomial

mapping

operators

A~l

in the polynomial

rings

or

(PI,...,Pn)

C.

such

Let that

to the ring

F = JacF

k[X 1 t ' "

= °

i.

,Xn]

The

: k n -~ k n restriction

we shall

be of

also denote

a

the by

1 T h e o r e m 4.1. Under the above assumptions, the f o l l o w i n g are equivalent: (i) F is a polynomial a u t o m o r p h i s m of k n,

conditions

(ii) A[(k[X 1 ..... Xn] ) = k[X 1 .... ,Xn], i = 1 ..... n-l, (iii) ~ & ~~( k [ X 1 ..... Xn]) = k[XI,...,X n ] and kerA~TM

=

314

k[P2,...,Pn]. Proof. i. (i) ~ (ii). A n a l o g o u s l y as the i m p l i c a t i o n in t h e o r e m 3.4. 2. (i) ~ (iii). A n a l o g o u s l y as the i m p l i c a t i o n in t h e o r e m 3.4. 3. (ii) ~ (i). a) k = C. From (ii) it follows that is d e n s e

in

En

for

i = l,...,n-l.

a polynomial automorphism. b) k = ~. F has a :C n

--~ C n.

We

shall

Hence,

canonical

show

that

by t h e o r e m

extension ~[(C[X])

to

=

(i) ~

(ii)

(i) ~

(iv)

A~l(e[X]) 3.4.,

the

C[X]

F

is

mapping for

i

=

t

l,...,n-l. T a k e i ~ {l,...,n-l} and P ~ C[X]. Let P = P' + iP'', P',P'' ~ ~[X]. By the a s s u m p t i o n there exist Q',Q'' ~ ~[X] such t h a t =

P.

From

automorphism the t h e o r e m

A[(Q') the of in

= P'

and

case

a)

we

en [2]

Hence F is

~[(Q'') obtain F = also

= P''. that

Hence F

~[(Q"

is

a

+ iQ")

polynomial

Fi~ n is an injection. a surjection. So, F

bijection. Since F -I = F -ll~n, t h e r e f o r e F -I p o l y n o m i a l mapping. 4. (iii) ~ (i). a) k = e. A n a l o g o u s l y as the (iv) ~ (i) in t h e o r e m 3.4.

is

a

By is a real

implication

b) k = ~. As above, let F : Cn > Cn be a c a n o n i c a l e x t e n s i o n of F. From the a s s u m p t i o n s it f o l l o w s easily that ~l(e[X])

= ~[X]

and

kerA~

= C[P2,...,Pn].

From

the

case

a)

we

have that F is a p o l y n o m i a l a u t o m o r p h i s m . P r o c e e d i n g a n a l o g o u s l y as in 3b) we o b t a i n that F is a real p o l y n o m i a l automorphism, a Acknowledgement. We thank Mr K. R u s e k for our a t t e n t i o n to the p a p e r by Y. Stein [i0].

his

having

called

References [I]

H.Bass, E . H . C o n n e l and D.Wright, The J a c o b i a n C o n j e c t u r e : Reduction degree and formal expansion of the inverse, Bull.Amer. Math. Soc.7(2) (1982) 287-330. [2] A . B i a l y n i c k i - B i r u l a and M. R o s e n l i c h t , I n j e c t i v e m o r p h i s m s of real a l g e b r a i c varietties, P r o c . A m e r . M a t h . S o c . 1 3 ( 1 9 6 2 ) , 200-203. [3] Z . C h a r z y ~ s k i , J . C h ~ d z y n s k i and P . S k i b i n s k i , A c o n t r i b u t i o n to K e l l e r ' s J a c o b i a n C o n j e c t u r e III, Bull. Soc. S c i . L e t t r e s L6d~ 39, N O . 4 ( 1 9 8 9 ) , I - 8 . [4] T . K r a s i n s k i and S.Spodzieja, On the i r r e d u c i b i l i t y of fibres of c o m p l e x p o l y n o m i a l m a p p i n g s (to appear). [5] S . L o j a s i e w i c z , I n t r o d u c t i o n to C o m p l e x A n a l y t i c G e o m e t r y (PWN, Warszawa, 1988) (in Polish). [6] D.Mumford, A l g e b r a i c G e o m e t r y I, C o m p l e x P r o j e c t i v e V a r i e t i e s (Springer, B e r l i n - H e i d e l b e r g - N e w York, 1976). [7] K . R u s e k and T.winiarski, Criteria for regularity of h o l o m o r p h i c mappings, B u l l . A c . P o l . : M a t h . 2 8 ( 1 9 8 0 ) , 471-475. [8] K . R u s e k and T . W i n i a r s k i , Polynomial Automogphisms of C n, Univ. Iagell. Acta M a t h . 2 4 ( 1 9 8 4 ) , 1 4 3 - 1 4 9 . [9] S.Spodzieja, On c o m m u t a t i v i t y of the c o m p o s i t i o n of W h i t n e y operators, Bull. Soc. Sci. Lettres Lodz 39, N o . 1 3 ( 1 9 8 9 ) , I - 6 . [I0] Y.Stein, On lineir d i f f e r e n t i a l o p e r a t o r s r e l a t e d to the

315

Jacobian Conjecture, J.Pure Appl.Algebra 57(1989),175-186. [ii] Y.Stein, On the density of image of differential operators generated by polynomials, J.Analyse Math. 52(1989),291-300. [12] Y.Stein, Linear differential operators related to the Jacobian Conjecture have a closed image, J.Analyse Math. 54 (1990),237--245. [13] D.Wright, On the Jacobian Conjecture, Illinois J.Math.25

(1981),423-440.

On a subanalytic stratification satisfying a Whitney property with exponent 1

by K r z y s z t o f K u r d y k a Instytut Matematyki, Uniwersytet Jagiellofiski, Reymonta 4, Krakdw PL-30059 (Poland)

If A is a subanalytic and closed set in R n, then A satisfies following Whitney property (see [St]): every point of A has a neighbourhood U such that, if x and y are two points in A M U then there exists an are )~ joining x and y in A M U such that length~_< C I x - y I~ ,where C and o~ are constants depending only on U. Actually we have the same property for strata of a stratification obtained by projections (e.g. triangulations or cellular decompositions, see [Ha], [Lo3] ). In this paper we construct a stratification compatible with a given family of subanalytic sets such that on each stratum the above property holds with exponent o~=1 ( corollary B ). Actually the strata we obtain (theorem A) are L-regular in the sense of Parusifiski [Pall i.e. are of the form {(z',zn) E R n-1 x R; / ( x ' ) < x~ < g(x'),x' G A'} ,where A' is L-regular in R ~-1, f and g are subanalytic and continous in --~, A analytic in A', f(x') < g(x') for x' G A'. Moreover I dx, f 1< _ M,I d~,g I< M for each z' ~ A',where M is a constant depending only on n. Other strata are the graphs of aaaalytic functions (with bounded differentials) defined on sets of previous form. Our result is related to the papers of Parusifiski ([Pal],[Pa2l). We use only elementary facts on subanalytic sets i.e. subanalycity of tangent mapping and stratifications of mappings. Hence our result holds in semialgebraic and semianalytic case. This paper was inspired by a question of Prof. S.Lojasiewicz (corollary C), the author is very grateful, to him for interest and encouragement. The author thanks G.Jasi~ski for helpful remarks. The paper was written when the author was associate member of Dept. of Math. of Universit~ de Savoie at Chambfiry and wants to acknowledge people working there (particulary P.Orro) for friendly ambiance and hospitality. The author thanks also the referee for valuables remarks.

0 . P r e l i m i n a r y r e m a r k s . Let M be an analytic manifold. By a stratification of M we mean a locally finite family T of analytic, connected submanifolds of M such that: M = U { T : T E T} (disjoint union), moreover if S M(T \ T) ~ 0, where S, T E T, then S C (T \ T) and d i m s < dimT. We say that stratifcation T is subanalytic if all strata (i.e. members of :r) are subanalytic in M. Let A be a family of subsets of M, we say that stratification 7" is compatible with family ,4 if T M A ~ 0 implies T C A, for each T C T, A E A. Let C be a family of analytic, connected submanifolds

317

of M, subanalytic in M. We say that family C is stratifying if for every locally finite family A of subanalytic subsets of M there exists a subanalytic stratification T of M , compatible with .,4 such that T C C. It follows from the general m e t h o d of constructing of subanalytic stratification (see e.g. [DS],[Lo3]) that C is stratifying if and only if : (*)For every A analytic submanifold of M, subanalytie in M, there exists F subanalytic in M, closed, nowhere dense subset of A such that every connected component of A \ F belongs to the family C. We recall a definiton of angle between linear subspaces. 1 . D e f i n i t i o n . Let X be a linear subspace in R'`, let P be a line in R'`.\¥e define angle between P and X as

5(P, X ) = inf{sin(P, S); S is a line in X } where sin(P, S) is a sine of the angle between P and S. Let Y be a linear subspace in R'~,we put 5(]I, X ) = sup{5(P, X); P is a line in Y } If Y = 0 we put 5 ( 0 , X ) = 0. Function 5 takes values in [0, 1]. In general 5 is not symmetric, however we have: (1.0) if d i m X = d i m Y then 5(X, Y) = 5(]I, X). (1.1) if Y C X then 5 ( Y , X ) = O. (1.2) if dirnZ < dirnY < dirnX then 5(Z, X ) <_ 5(Z, Y ) + 5(Y, X). Let us denote by Gim the grassmanian space of all i-dimensional linear subspaces of R n equiped with the natural structure of real algebraic variety. Then we have (1.3) the mapping Gi,n x Gj,,~ 9 (Y, X ) ~ 5(Y, X ) E R is continous and semiatgebraie (see [Lo4]). 2 . R e m a r k . Let X denotes a hyperplane and P a line in R'`. Then for each rn > 0 there exists M > 0 such that if 5(P, X ) > rn then X is a graph of linear mapping ~: P± , P satisfying II ~ II< M ( P ± denotes the orthogonal complement of P). 3 . L e m m a . Given r,n C N, there exists constants c > 0 and rn > 0 such that for given X1, ...Xr hyperplanes in R n ,there exists a line P such that, if Y1, ...Y~ are hyperplanes verifying 5(Y/,Xi) < ~, i = 1, ...r, then

5(P, Yi) > rn f o r each i = 1, ..., r Proof. Let us take a metric d on the sphere S "-1 defined as follows : d(p, q) = 5(Rp, Rq) forp, q C S "`-1. Let us denote X~ = {p E S "`-1 : dist(p, Xi 7/5 ' ' - 1 ) < c}, where dist(p, Z) = inf{d(p, q) : q C Z}. It is enough and sufficient to prove the following fact : (3.1) Given r E N, there exists c > 0 and m > 0 such that the complement of Ui~=l x ~ in S n-1 contains a ball of radius rn (in metric d). We use induction on r to prove (3.1). The case r = 1 is obvious. Let us denote cr and m r corresponding constants in (3.1) for r hyperplanes. Let B(p, mr) be a ball in

318

S n-1 disjoint with each X ~ ' , i = 1, . . . r . P u t s~+~ = m~+~ = min{e~,m~}/3, then the x'~'+~ contains a ball of radius rn~+l Let us recall some known facts on subanalytic sets. Let F be an analytic submanifold and subanalytic subset of R n, dirnF = k, then the Gauss m a p p i n g

set B(p, m,) \

T :F g X l

, T=F • G~,,~

is subanalytic i.e. its g r a p h is subanalytic in R ~ x Gk,n. As usual T~F denotes the tangent space to P at x. Moreover if E is a subanalytic subset of Gk,~ then T - I ( E ) is subanalytic in R ~ (see e.g. [DW],[Lo2]). 4 . D e f i n i t i o n . Let F be a C 1 submanifold of R ~ and let e > 0. We say t h a t P is s - flat, if for each x, y • F we have 5(T=F, TyF) < ¢. If dimF = 0 we assume t h a t F is s-flat for every e > 0. 5 . P r o p o s i t i o n . Let A be locally finite family of subanalytic sets in R n. Then for given e > 0 there exists a subanalytic stratification T compatible with the family ~4, such that each stratum of ~r is s-flat.

Proof. We use condition (*) of remark 0 to prove the existence of such stratifications. For every k = 1, ...,n - 1 let us take a subanalytic finite partition of Gk,n into disjoint sets E/k, i = 1,..., rk such t h a t 5(X, Y) < ¢ for each X, Y • Eik. Actually we can take a stratification of Gk,n compatible with a finite covering by 5-balls of radius ¢.Let A be an analytic submanifold, subanalytic subset of R ~, d i m A = k. Let r : A ~ Gk,~ rk be corresponding Gauss mapping, then the set F = A \ Ui=l IntA(v--l(E~)) is closed and nowhere dense in A. Clearly every connected c o m p o n e n t of A \ F is s-flat. Hence by (*) of r e m a r k 0 the proposition follows. Remark 5.I. Actually we can require more for the stratifcation 2r. By a theorem of Stasica [St] (see also [KR]), for given M > 0, we can refine T in a such way that every s t r a t u m T of T , dirnT = k < n, is a g r a p h (in suitable coordinate s y s t e m in R n) of an analytic m a p p i n g ¢ : U ~ R n - k , where U is open in R k and [ dz¢ 1<_ M for each z • U. Suppose t h a t ¢ = ( ¢ l , . . . , ¢ n - k ) and put ~ = S n - k , t h a n T is also a g r a p h of :B * R , where B is a graph of (¢1, . . . , ¢ , - k - 1 ) : U ~ R n-k-1. Clearly B is an analytic submanifold of R n-1 , we have also [ d ~ [_< M for each x • B. 6. D e f i n i t i o n . W e say that subanalytic set A in R n is a 8-cell, if A is a point or in some coordinates in R n A is of one of the following forms

4) /~)

A={(x',x,~)eR A={(x',x,,)eR

n-1 x R :

'~-' xR:

h(x')=xn, x'•B}

f(x')<xn
x'eB}

where B is a s-cell in R n-1 ; f , g and h are subanalytic continous functions on the closure of B, analytic on B. Moreover we assume t h a t f ( z ' ) < g(x') for each x' E B. Let A be a s-cell of dimension k, then A is an analytic submanifold of R n h o m e o m o r p h i c to an open ball in R k. Moreover ~ \ A is h o m e o m o r p h i c to a sphere in R k.

319

7 . D e f i n i t i o n . We keep the notation of definition 6. We say that s-cell A in R " is L-regular with constant M > 0, if I d=,h t<_ M (resp. I d~,f I<_ M and I d~,g I< M ) for each x f C B, where B is a L-regular s-cell with constant M in R ~-1 . If A is a point we assume t h a t A is L-regular with every constant M > 0. Remark. T h e closure of our L-regular s-cell is a subanalytic version of the L-regular set of Parusifiski (comp. def.3.2 of [Pal]). Proceeding by induction n we get: 8 . P r o p o s t i o n . Let A C R ~ be a L-regular s-cell with constant M > O. Then for every x, y E A there exists a smooth curve ,~ joining x and y such that

length~ < C l x - y l , where C is a constant depending only on n and M . Now" we state our m a i n result. T h e o r e m A. Let `4 be a locally finite family of subanalytic sets in R ~. Then there exists a subanalytic stratification f of R ~, compatible with the family `4, such that every stratum of :T is a L-regular s-cell with a constant M , where M depends only on n. From t h e o r e m A and proposition 8 we get C o r o l l a r y B. Let `4 be a locally finite family of subanalytic sets in R ~. Then there exists a subanalytic stratifcation 7" of R'~,compatible with the family `4, such that , if x , y E T, where T E ~I-, then there exists a smooth curve ~ in T joining x and y, such that length£ < C I x - y l,

where C is a constant depending only on n. We state also an i m m e d i a t e consequence of corollary B, which seems to be of its own interest. C o r o l l a r y C. Let U be a subanalytic set, analytic submanifoId of R n. Then there exist M > 0 and a subanalytic stratification T of R ~, compatible with U such that~ if 9~ : U ..... ~ R is a differentiable function such that for some constant C > O, I d=~ I< C for every ~z C U, then I V ( x ) - ~ ( Y ) I<- C M I x - y l,

for every x and y belonging to the same stratum T 6 7-, T C U. Proof of theorem A. By (*) of r e m a r k 0 it is enough to prove the following: (**) Let A be a subanalytic set, analytic submanifold of R n, t h e n there exists F a subanalytic, closed, nowhere dense subset of A such t h a t every connected component of A \ F is a L-regular s-cell with constant M . Constant M will be determined later on. We use induction on n to prove (**). We m a y add to .4 a family of cubes [kl, 1 + kx] x ... x [kn, 1 + k,], kl E Z, ki = 1, ...,n. We call this new family .4'. Clearly every stratification compatible with `4f will have only b o u n d e d strata. Hence it is enough to prove (**) for A bounded. Suppose now t h a t

320

A is a bounded, subanalytic, analytic submanifold of R n, d i m A = k < n. T h e n by r e m a r k 5.1 for given M > 0 we have A=FUFIU...

UFs

where F is subanalytic in R n, closed in A, dim F < k. Each Fi is a connected component of A \ F, moreover in suitable coordinate system every Fi is a graph of an analytic, subanalytic function pi : Bi ~ R, where Bi is an analytic submanifold of R n-1 and I dx~i I<_ M for x E Bi. Now applying induction hypothesis in R n-1 we m a y suppose t h a t Bi is a s-cell L-regular with constant M. This ends the proof of (**) for A of dimension less than n. Suppose now that A is subanalytic, open, bounded subset of R n. T h e following l e m m a is crucial for the proof of (**) in this case. 9 . L e m m a . Let A be an open, bounded, subanalytic subset of R '~, let e > O. Then there exists finitely many open and disjoint s-cells A i , i = 1, ...,p, Ai C A such that P A i is closed and nowhere dense in A, i) the set A \ Ui=l i such that ii) for each i = 1,...,p there exists subanalytic subsets BI, .. ., B k, a) ki <_ 2n b) each B} is an e-flat analytic submanifold of R n, dirnB} = n - 1, c) ~I Ijk~ = l B j~ is an open, dense subset of-~i \ Ai.

Proof of the lemma 9. We prove the l e m m a by induction on n = dimA. The case n = 1 is obvious. Let us assume the l e m m a holds true for open, bounded, subanalytic subsets of R n - l . W e proceed by following steps: Step 1. By Prop.5 there exists a subanalytic stratification ~ of R n, compatible with A \ A, such t h a t all s t r a t a of 71 are e-flat. Step 2. By K o o p m a n - B r o w n theorem (see [Lol],[Lo3] ) we can find such coordinates in R " that, if we denote by 77 a projection 7r(xl, ..,Xn--l,Xn) = (Xl, ..,Xn-1) = X', then 77 is finite on each s t r a t u m T E 71, T C ~ \ A. By a theorem of H a r d t (see [Hal] ) we can stratify the projection 77 restricted to A, i.e. there exists a stratification ~ of R n compatible with family {T E 71 : T E A}, there exists a stratification $ of R n - l , such that i) each bounded s t r a t u m T of 2r2 is of one of the following f o r m ~)

r={(x',xn) T

= { ( xi, x ~ ) E

ER~:

xn=h(x'),

x'ES}

R n : f(x') <x~
where S is a s t r a t u m of ,5" , f ( x ' ) < g(x') for every x' E S. Functions f , g and h are subanalytic and analytic in S.

ii) if T is s t r a t u m of T2 of the type a ) and T C ~ , d i m T = n - 1, then T C T1 for some T1 E ~ , dirnT1 = n - 1 ( this follows from the proof the t h e o r e m of Hardt). Hence T is also e-flat. iii) if s t r a t u m S satisfies condition (s) (i.e. every point of ~ has a basis of neighbourhoods {U,}, such that U, M S is connected), then the function h (resp. f and g)

321

be extended continously on S (see [Lol],[Lo3] ). Notice that every s-cell satisfies condition (s).

can

Step 3. For each open s t r a t u m S E S, S C 7r(A) we apply induction hypothesis. ! Hence S is a union of open, disjoint s-cells A~, ..., Aq and a nowhere dense, closed subset of S. Every A} verifies conditions a), b), c) of ii) of lemma 9. Let us take a s-cell of the form Q={(x',xn) ER': f(x') < x , < g(x'), x' E A',} where f and g are those of Step 2. T h e boundary of Q is a union of the graphs of f and g (which are already e-flat by Step 2) and the set

b(Q) = { ( x ' , x . ) E R " : f ( x ' ) <_ x . < g(x'), x' e -A7i \ A~} Let us denote by B~' the sets, contained in the boundary of Ai, satisfying b) and c) of ii) in lemma 9. T h e n the sets (some of them perhaps empty) •

B}--{(x',xn):

I

f ( x ' ) < x , < g ( x ' ) , x' • B~ i}

are e-flat. ActuMly they are open in B~ ~ x R which are e-flat, since each B~' is e-flat by induction hypothesis. Clearly the union of all B} is dense in b(Q) \ (7 U ~) ( we identify function with its graph ). Hence making the induction step we get at most two sets more in ii) i.e. graphs of f a n d g. Thus our s-cell V satisfies a), b), and c) of ii). All s-cells constructed in the same way as Q i.e. { ~ r - I ( A ~ ) M T : i = l,...,q; T e T2, T e A ,

dimT--n}

gives our family of desired s-cells A1, ..., Ap. This ends the proof of lemma 9. Now we come back to the proof of (**) for A open and bounded in R n. Applying lemma 3 in R n for r = 2n we get corresponding positive constants e and m. By lemma 9 we can suppose that A is a s-cell satisfying a), b) and c) of ii) with e as above. Let us denote by B j , j = 1, ..., k, k _< 2n open sets in the boundary of A, satisfying b) and c) of ii). Let us take arbitrary points xj • B j , j = 1, ...,k and put X j = TxjBj. Since Bj is e-fiat, we have 5(Xj, Tu¢ By) < e for every yj G By. Hence by lemma 3 there exists a line P in R '~ such that 6(P, Tuj Bj) > m. We take this line as x,~ - axis, the orthogonal complement of P will be a (xl, ;..., x n - 1 ) -hyperplane denoted by R n-1. Locally each set Bj is a graph of an analytic mapping ~ : U --~ R , where U is an open subset of R '~- a. Moreover by Remark 2 there exists a constant M > 0 (depending only on m) such that ] d~,~ ]<_ M for each x ~ • U. Since in the claim of the theorem of Koopman-Brown the set of admissible projections is dense , so we can assume that canonical projection ~r : R n -~ R n-1 is finite on the boundary of A. Now we repeat the construction of Step 2 of the proof of lemma 9. Hence our A is a disjoint union of a nowhere dense, closed set F and sets of the form {(x',x,~) • R n : f ( x ' ) < x . < g(x'),x' • A'} where A I is an open and subanalytic subset of R ~ - l , f and g are analytic functions in A r such that I d=,f I< - M , I d=,g I< - M. By induction hypothesis we can assume that A' is a L-regular s-cell with constant M. This ends the proof of (**) for A open and bounded. Hence theorem A is proved.

322

Remark. Actually in the above step the th. of Koopman-Brown is superfluous, since the fact that 7r is finite on the boundary of A follows from Cor. 1.8 in [KR]. 1 0 . R e m a r k To obtain a semialgebraic version of theorem A we suppose that ,4 is a finite family of semialgebraic subsets of R n. The stratification we obtain will be also finite and semialgebraic. However we should change the definition of s-cell. We simply assume in definitions 6 and 7, case fl), that f or g may be identically equal to ec or to - o o . Clearly we can assume d , f = 0 (resp. dxg = 0) in that case. Notice that proposition 5 and the theorem of Hardt have semialgebraic versions, see respectively the proof in [Lo2] and chap.9 in [BCR]. Hence our proof holds in semialgebraic case.

References

[BCR] J.

[DS] [DW] [Hall

Bochnak, M. Coste, M-F. Roy; G6ometrie alg@brique r6elle, (Ergeb. der Math., Folge 3, Bd.12), Springer, 1987. Z. Denkowska, J. Stasica; Sur la stratifiction sous-analytique, Bull. Acad. Pol. Sci. S@r. Math., 30 (1982), 337-340. Z. Denkowska, K. Wachta; Sur la sous-analycit@ de l'application tengente, Bull. Acad. Pol. Sci. S6r. Math., 30 (1982), 329-331. R. Hardt; Stratification of real analytic mappings and images, Invent. Math.

28 (1975), 193-208.

[Ha2] R. Hardt; Triangulation of subanalytic sets and proper light subanalytic maps, Invent. Math. 38 (1977), 207-2170.

[KR] K. Kurdyka, G. Raby; Densit6 des ensembles sous-analytiques, Ann. Inst.

[Loll [Lo2] [Lo3] [Lo4] [Pail [Pa2] [St]

Fourier, 39 (1989), 753-771. S.Lojasiewicz; Triangulation of semi-analytic sets, Annali Scuola Norm. Sup., $6r.3, 8 (1964), 449-474. S.Lojasiewicz; Sur la semi-analycit6 de l'application tangente, Bull. Acad. Pol. Sci. S@r. Math. 27 (1979)525-527. S.Lojasiewicz; Stratifications et triangulations sous-analytiques, Seminari di Geometria (Bologna), (1986) 83-97. S.Lojasiewicz; Semianalytic and subanalytic geometry, book in prepartion. A. Parusifiski; Lipschitz properties of semianalytic sets, Ann. Inst. Fourier, 38 (1988) 189-213. A. Parusifiski; Regular projections for subanalytie sets, C. R. Acad. Sci. Paris, 307 S@rie I (1988) 343-347. J. Stasica; The Whitney condition for subanalytic sets, Zeszyty Naukowe Uniw. Jag. 23 (1982), 211-221.

UNE POUR

BORNE LE

SUR

LES

THEOREME REEL

DEGRES DES

ZEROS

EFFECTIF

Henri LOMBARDI Laboratoire de Math6matiques. UFR des Sciences et Techniques Universit6 d e Franche-Comt6. 25030 Besan¢on c6dex France R4sum4. Nous donnons les id6es et r6sultats essentiels d'un calcul d'une majorafion des degr6s pour le th6or6me des z6ros r6els effectif. Abstract We give the main ideas and results concerning a computation of a degree majoration for the real nullstellensatz.

1)

Introduction

N o u s rendons compte dans cet article d u calcul d ' u n e borne sur les degr6s a c c o m p a g n a n t la p r e u v e constructive d u th6or6me des z4ros r4el et d e ses variantes (cf. [Lom d]). Les preuves sans les majorations de degr6 p e u v e n t atre trouv6es dans [Lom b ] . Les r4sultats o b t e n u s

Une formulation g~n4rale d u th6or~me des z6ros r6el et de ses variantes p e u t ~tre la suivante (cf [BCR] th6or6me 4.4.2) : o n consid~re u n syst~me d'6galit~s et in~galit4s portant sur des polynomes de K[X] = K[X1,X2,...,Xn], oR K est un corps ordonn6 de cl6ture r6elle R ; ce syst~me d6finit une partie S semialg6brique d e R n ; le th6or~me affirme que S est vide (fait g6om6trique) si et s e u l e m e n t si il y a u n e certaine identit6 alg6brique construite ~ partir des p o l y n o m e s donn6s, identit4 qui d o n n e une p r e u v e d e ce fait g6om4trique. Calculer u n e b o r n e sur les degr6s p o u r le th6or~me des z6ros r6els consiste calculer u n e m a j o r a t i o n sur les degr6s des p o l y n o m e s i n t e r v e n a n t dans le r4sultat (l'identit6 alg6brique construite) ~ partir de la taille de l'entr4e (le syst~me de conditions de signes portant sur la liste de polynomes donn4e au d6part). Les param6tres qui controlent la majoration des degr6s dans le r6sultat sont en fait : le n o m b r e k de p o l y n o m e s dans l'entr6e, le degr6 d des polynomes dans Pentr6e, et le n o m b r e n de variables. Le calcul de majoration est obtenu en suivant pas a pas la p r e u v e constructive d'existence d e Pidentit6 alg6brique et en explicitant les majorations a chaque 6tape de la preuve. C'est une majoration primitive r6cursive, donn6e par une tour d'exponentielles : le nombre d'6tages dans la tour est n+4 et en haut de la tour on trouve : d.log(d) + loglog(k) + cte.

324

Ce r4sultat n'est pas trop mauvais, dans la mesure oh la principale responsabilit6 de l'explosion est support6e par l'algorithme de H 6 r m a n d e r , ~t la base de la p r e u v e effective. On peut esp6rer baser une autre p r e u v e effective sur des algorithmes plus p e r f o r m a n t s et n6anmoins de conception tr~s simple, et obtenir en cons6quence une majoration o~ le param~tre n interviendrait de mani6re m o i n s c a t a s t r o p h i q u e , sans t o u r d ' e x p o n e n t i e l l e s . I1 s e m b l e n 6 a n m o i n s i m p r o b a b l e d'obtenir d'aussi bonnes bornes que dans les meilleures versions effectives d u th6or~me des z6ros de Hilbert (cf. [He] ,[FG] et [Ko]). La preuve constructive du th4or~me des z6ros r4els

De mani~re g6n6rale u n ~th6or6me des z6ros~ affirme q u e certains faits ~g6om6triques~ ont une p r e u v e p u r e m e n t ~alg6brique~. Un exemple simple est fourni par la formule de Taylor. Par exempte p o u r un polynome de degr6 < 4 , on a l'identit6 alg6brique : (avec A = U - V) P(U) = P(V) + A.P'(V) + (1/2).A2.p"(V) + (1/6).A3.p(3)(V) + (1/24).A4.P (4) Cette idenfit6 alg6brique r e n d manifeste le fait g6om6trique suivant : si en un point v le p o l y n o m e P a toutes ses d6riv6es positives, alors p o u r tout u > v on a P(u) > P ( v ) . Ce fait g6om6trique, qui peut ~tre rendu manifeste par un tableau de variation, est 6galement clair par la formule de Taylor. C'est u n cas particulier d u l e m m e de Thorn, qui affirme (entre autres) que l'ensemble des points o~ un p o l y n o m e et ses d6riv6es successives ont chacun un signe fix6, est convexe. La construction d u nullstellensatz r6el utilise u n e version "identit4 alg6brique" de ce fait, donn6e par ce que nous appelons les formules de Taylor mixtes et les formules de Taylor g6n6ralis6es. L'id6e g6n6rale de notre p r e u v e constructive est la suivante. P o u r un corps o r d o n n 6 K il y a un algorithme de conception tr~s simple p o u r tester si u n syst6me de csg (conditions de signes g6n6ralis6es) portant sur ces p o l y n o m e s en plusieurs variables est possible ou impossible dans la cl6ture r6elle de K . C'est l'algorithme d e H 6 r m a n d e r (cf. la p r e u v e d u principe de Tarski-Seidenberg dans [BCR] chap. 1), appliqu6 de mani~re it6rative pour diminuer par 6tapes le nombre de variables sur lesquelles portent les csg. Si on r e g a r d e les a r g u m e n t s sur lesquels est bas6e la p r e u v e d'impossibilit6 (en cas d'impossibilit6), on voit qu'il y a essentiellement des identit6s alg6briques (traduisant ta division euclidienne), le th6or~me des accroissements finis et l'existence d'une racine p o u r un p o l y n o m e sur un intervalle oi~ il change de signe. Les ...-stellensatz r6els effectifs doivent donc pouvoir ~tre obtenus si on arrive "alg6briser" les arguments de base de la p r e u v e d'incompatibilit6 et les m6thodes de d6duction impliqu6es. Un pas important a d6j~ 6t6 r6alis6 avec la version alg6brique d u th4or6me des accroissements finis p o u r les p o l y n o m e s (cf. [LR]), qui a 6t6 ~ l'origine des formules d e Taylor mixtes et g6n6ralis6es. Un a u t r e pas a consist6 a t r a d u i r e sous f o r m e d e constructions d°identitds algdbriques certains raisonnements 616mentaires (du genre si A ~ B et B ~ C alors A ~ C ). I1 fallait en outre trouver une version "identit6 alg6brique" des axiomes d'existence dans la th6orie des corps r6els clos. C'est ce qui est fait a t-ravers la notion

d'existence potentielle.

325

Remarques sur Particle present N o u s intoduisons dans cet article une probl6matique o~ le role central est tenu par les constructions d'identit6s alg~briques (appel6es ~dncompatibilit~s fortes~) partir d'autres identit4s alg6briques, alors que dans les versions pr4c6dentes c'6taient plut6t les identit~s alg6briques elles-m~mes qui jouaient le role central. Ce c h a n g e m e n t de point de r u e a 6t6 motiv6 par le calcul de majoration luim~me. Ce qui, dans [Lom d], apparaissait sous l'appellation p e u plaisante d'~dmplication forte v u e c o m m e existence potentielle~, s'appelle d6sormais ~dmplication d y n a m i q u e ~ . Q u a n t aux anciennes implications fortes, elles ne jouent p r a t i q u e m e n t plus aucun role. N o u s d o n n o n s darts ce n o u v e a u cadre un t r a i t e m e n t unifi6 p o u r les p r e u v e s cas par cas 0~disjonction d y n a m i q u e ~ ) , l'implication (~dmplication dynamique~0 et l'existence 0~existence potentielle~). Significations de la preuve constructive pour diff6rentes 4coles philosophiques Bien que nous nous placions a priori dans un cadre constructif "~ la Bishop", tel que d6velopp6 dans [MRR] p o u r ce qui concerne la th6orie des corps discrets, comme nous ne pr6cisons pas le sens d u mot effectif ni celui du mot d6cidable, toutes les preuves p e u v e n t ~tre lues avec des lunettes adapt6es ~ la philosophie ou au cadre de travail de chaque lecteur particulier. Si on a d o p t e un point de vue "classique" par exemple, les procedures effectives r6clam6es dans la structure d u corps des coefficients par le math6maticien constructif p e u v e n t ~tre consid~r6es comme donn6es par des oracles. En cons6quence, les p r e u v e s fournissent u n e p r e u v e dans le cadre classique, et sans recours ~ l'axiome du choix, d u th60r~me des z~ros r~els dans un corps ordonn6 arbitraire. En fait les preuves donn6es fournissent des algorithmes uniform6ment primitifs r6cursifs, "uniform6ment" s'entendant par r a p p o r t ~ un oracle qui d o n n e la structure du corps des coefficients d u syst~me de csg consid6r6 : Si (Ci)i=l,...,m est la famille des coefficients et si P ~ 72 [(Ci)i=l,...,m ] l'oracle r6pond a la question ~ Quel est le signe de P((q)i=L...,m) ? ""

2)

Incompatibilit6s fortes

Incompatibilit6s fortes : d6finitions et notations N o u s consid6rons un corps ordonn6 K , et une liste de variables X1, X2..... Xn d~sign6e par X. N o u s notons doric K[X] l'anneau des polynomes K[X1,X2,._,Xn]. Etant donn~e une partie finie F de K[X] : nous notons F .2 l'ensemble des carr6s d'616ments de F . le monot'de multiplicatif engendrd par F est l'ensemble des produits d'616ments de FU {1}, nous le noterons M ( F ) . le c6ne positif engendrd par F est l'ensemble des sommes d'616ments d u type p.p.Q2 o~ p e s t positifdans K , P est dans M(F), Q est dans K[X]. Nous le noterons C~F) . enfin nous noterons I(F) l'id6al engendr6 par F .

326

D6finition et n o t a t i o n I : Etant donn6s 4 parties finies de K[X] : F>, F~, F=, F~, contenant des polynomes auxquels on souhaite imposer respectivement les conditions de signes > 0 , >/0, = 0 , :~ 0 , on dira que F = [F> ; F~ ; F= ; F~ ] est fortement incompatible dans K si on a une 4galit4 dans K[X] du type suivant : S + P + Z = 0 avec S c Yvf(F>U F#'2), P c Cp(F~U F>), Z c I(F=) Nous utiliserons la notation suivante pour une incompatibilit6 forte: ,~ [ S1 ) 0..... Si> 0, P1 )/0 ..... Pj,'/0, Z 1 = 0..... Zk= 0,N 1 • 0..... Nh:~ 0 ] ,], I1 est clair qu'une incompatibilit6 forte est une forme tr6s forte d'incompatibilit6. En particulier, elle implique l'impossibilit4 d'attribuer les signes indiqu4s aux polynomes souhait4s, dans n'importe quelle extension ordonn6e de K . Si on consid~re la cl6ture r6elle R de K , l'impossibilit6 ci-dessus est testable par l'algorithme de H6rmander, par exemple. Le th4or~me des z4ros r6els et ses variantes

Les diff6rentes variantes du th6or6me des z6ros dans le cas r6el sont cons6quence du th6or6me g6n6ral suivant : Th4or~me : Soit K un corps ordonn6 et R une extension r6elle close de K . Les trois fairs suivants, concernant un syst6me de csg portant sur des polynomes de K[X], sont 6quivalents : l'incompatibilit4 forte dans K l'impossibilit4 dans R l'impossibilit6 dans toutes les extensions ordonn4es de K Ce th4or6me des z4ros r4els remonte/~ 1974 ([Ste]). Des variantes plus faibles ont 6t6 4tablies par Krivine ([Kri]), Dubois ([Du]), Risler (IRis]), Efroymson ([Efr]). Toutes les preuves jusqu'~ ([Lom a]) utilisaient l'axiome d u choix. Degr6 d'une incompatibilit4 forte Si nous voulons pr6ciser les majorations de degr4 fournis par notre preuve du th6or6me des z6ros r6el, nous devons pr6ciser la terminologie. Nous manipulons des incompatibilit6s fortes 6crites sous forme paire, c.-~-d.: S + P + Z = 0

avec S c g , f(F>"2UF#'2), P c C ~ F ~ U F > ) ,

Zc/(F=)

(la consid6ration des formes paires d'implications fortes a pour unique utilit6 de faciliter un peu le calcul de majoration des degr4s). Q u a n d nous parlons de degr6, sauf pr4cision contraire, il s'agit du degr6 total maximum. Le degr~ d'une incompatibilitd forte est par convention au moins 4gal ~ 1, c'est le degr4 m a x i m u m des polynomes qui ~composenb~ l'incompatibilit6 forte. Par exemple, si nous avons une incompatibilit4 forte : ~, [ A > 0 , B > 0 , C > / 0 , D > , , 0 , E = 0 , F = 0 ] ~, explicit6e sous forme d'une identit6 alg6brique :

327

h k A2.B6 + C. ~ pi.Pi2 + A.B.D. ~ qj.Qj2 + E.U + F.V = 0 i=1 i=1 le degr6 de I'incompatibilit6 forte est : sup { d(A2.B6), d(C.Pi2) (i = 1,...,h), d(A.B.D.Q2) (j = 1,...,k), d(E.U), d(F.V) }. Le calcul de majoration Nous allons expliquer dans cet article comment peut ~tre men4 un calcul de majorations primitives r4cursives pour le th6or6me des z6ros r6els. Les d6tails des calculs sont dans [Lom d]. Les donn4es sont trois enfiers d, n, k qui majorent, dans un syst6me de csg incompatible H , respectivement les degr4s des polynomes, le hombre des variables et le nombre de csg. Le calcul doit aboutir ~ 3 fonctions primitives r6cursives explicites ~(d,n,k), ~(d,n,k) et ~(d,n,k) qui donnent des majorants pour, dans une incompatibilit6 forte ~, H ~, , respectivement le degr6 maximum, le nombre de termes dans la somme, et le nombre d'op4rations arithm6tiques dans K n6cessaires pour calculer les coefficients dans l'incompatibilit4 forte ~ partir des coefficients donn6s au d6part. En fait, chacun des th40r6mes ou propositions qui conduit ~ la preuve constructive du th4or6me des z4ros r6el peut ~tre accompagn4 d'une majoration primitive r6cursive du m~me type. Ces majorations s'enchainent les unes les autres, sans difficult4 majeure. Comme le calcul est tr6s fastidieux, nous nous en sommes tenus aux majorations de degr6s, laissant au lecteur courageux les deux autres majorations. On notera que l'usage de l'algorithme de H6rmander 'sans raccourci', ~ la base de notre m6thode, rend a priori les majorations obtenues sans int6r~t pratique. Constructions d'incompatibilit4s fortes D6finition 2 : Nous parlerons de construction d'une incompatibilit6 forte h partir d'autres incompatibilit6s fortes, lorsque nous avons un algorithme qui permet de construire la premi6re ~ partir des autres. I1 s'agit donc d'une implication logique, au sens constructif, liant des incompatibilit6s fortes. Notation 3 : Nous noterons cette implication logique (au sens constructif) par un signe de d6duction "constructif". La notation

($

$ et $ H2 $)

o.s $ H3 $

signifie donc qu'on a Tan algorithme de construction d'une incompatibilit4 forte de type H 3 ~ partir d'incompatibilit6s fortes de types H~ et H 2 Cela n'a d'int4r~t que lorsque les incompatibilit6s fortes d4sign6es en hypoth6se et en conclusion comportent des 616ments variables. Un exemple fondamental aidera a mieux comprendre.

328 Le raisonnement par s~parafion des cas (selon le signe d'un polynome) N o u s donnons ici un ~nonc~ d~taill~ des <
Proposition 4 : Soit H u n systbme de csg portant sur des polynomes de K[X], Q un ~l~ment de K[X] , alors: [ $ ( H , Q < O ) $ et ,], ( H , Q > O ) ,], ] ~ons ~, ( H, Q * 0 ) ,], (a) ~.~ $ H $ (a') [ J. ( H , Q~/0),]. ] De m~me : [ $ ( H , Q > O ) $ et ,[. ( H , Q= O) ,]. ] ~ons ,l, ( H , Q ) , , 0 ) ,~ (b) $ H ,1, (c) [ $ ( H , Q ~ = O ) $ et ~.(H, Q=O) J.] et ~, ( H , Q~<0) ~,] ~ons,~, H ,~, [$(H,Q>0)$ (d) Dans chacun de ces cas, notons d I e t d 2 les degr~s des deux incompafibilit~s fortes donn~es dans l'hypoth~se, le degr~ de l'incompafibilit~ forte construite est respecfivement major~ par : (Pa(dl,d2) = %,(dl,d 2) = d 1 + d 2 (pb(dl,d2) = dl.d 2 ~oc(dl,d2) = dvd 2 (Pd(dl,d2) = dl.d 2 + d 2 Ces 4 foncfions sont major~es par q)(d:,d 2) = d 1.d2 + d 1 + d 2 Enfin, p o u r d~montrer que H est fortement incompatible, on peut raisonner en s~parant selon les 3 cas Q > 0 , Q < 0 , Q = 0 , et en construisant une incompafibilit~ forte dans chaque cas. Ce qui renvient h affirmer : [,~(H,Q>0),~ et ~ ( H , Q < 0 ) et , ~ ( H , Q = 0 ) , ~ ] ~ons ~- H ,~ (e) Notons d 1 , d 2 et d 3 les degr~s des trois incompatibilit~s fortes donn~es dans l'hypoth~se, le degr~ de l'incompatibilit~ forte construite est major~ par : %(di,d2,d 3) = dl.ct2+ dl.ds+ d2.d3

3)

Versions alg~briques dynamiques de l'implicafion et d e la d i s j o n c t i o n

La v e r s i o n a l g 6 b r i q u e d y n a m i q u e

de l'implicafion

D6finition et notation 5 : Soient H I e t H 2 deux syst6mes de csg portant sur des polynomes de K[X] . N o u s dirons que le syst~me H I implique dynamiquement H2 lorsque, pour tout syst6me de csg H portant sur des polynomes de K[X,Y], on a la construction d'incompafibilit6 forte : .~, [ H2(X), H(X,Y) ] .1. ~ns "]- [ Hi(X), H(X,Y) ] .~ N o u s noterons cette implication dynamique par :

329

"(Hx(x) =~ H2(x) )" Lorsque le syst6me H 1 est vide, nous utilisons la notation " ( H 2 ( X ) ) ' .

Remarques : 1) On a trivialement l'4quivalence des affirmations : $ HI$ et "( H x ~ ( 1 = 0 ) ) " 2) La vision dynamique de l'implication correspond, dans les r6f6rences [Lore x] a l',dmplication forte vue comme existence potentielle),. En tant qu'implication forte ~statique,,, c'4tait une liste d'identit6s alg6briques. En tant qu'implication dynamique, cela devient un algorithme de manipulations d'identit6s alg4briques. Dans la mise en oeuvre concr6te d'algorithmes de construction du th4or~me des z6ros r4el, la vision dynamique est en fait beaucoup plus fructueuse que la vision statique. Certaines subtilit6s s'introduisent, comme le fait que deux implications qui ont la m@me signification peuvent avoir des dynamiques distinctes (c.-~-d. qu'elles se traduisent pas des algorithmes de manipulations d'identit6s alg6briques distincts), et ont alors des cofits (en termes de temps de calcul) diff6rents (pour plus de d6tails voir le paragraphe "Variations sur le th6me des implications dynamiques'). II apparait en fin de compte que les "bonnes notions" sont celles d'incompatibilit6 forte et d'implication dynamique, tandis que la notion d'implication forte serait plut6t un incident de parcours. Nous verrons un peu plus loin que les raisonnements cas par cas peuvent ~tre interpr6t6s par une autre "bonne notion", la version dynamique du ~ou~. La notation que nous utilisons iciest 16g6rement distincte de celle utilis6e dans les pr4c6dentes r6f6rences [Lom] pour noter les existences potentielles. Ceci nous permet de mieux insister sur la diff6rence de signification entre une implication forte ~statique~ et une implication dynhmique. Fonction-degr~ d'une implication dynamique Une implication dynamique *( H~ ~ algorithme fournissant la construction :

H2)*

signifie par d6finition un

$[H2,H] $ ~on,$[Hi,H]$ Chaque fois que nous 6tablissons une implication dynamique particuli~re, nous devons 6tablir des "majorations primitives r6cursives de degr6" pour cette construction d'incompatibilit6s fortes : le degr6 de l'incompatibilit6 forte construite est major6 par une fonction A(d,..;k,...) o~ d est le degr6 de l'incompatibilit4 forte initiale, k le nombre de csg dans H 2 etc .... (le pointvirgule isole les 'variables', qui d6pendent de l'incompatibilit6 forte initiale, des 'param~tres', qui ne d6pendent que de Hx et H 2 ). Nous disons qu'il s'agit d'une fonction-degr6 acceptable pour l'implication dynamique consid6r4e, ou encore, (par abus) nous parlons de la fonction-degr6 attach4e ~t l'implication dynamique.

Renforcement simultan4 de l'hypoth~se et de la conclusion dans les implications dynamiques Soient H I , H 2 , H3 des syst6mes de csg portant sur des polynomes de K[X].

330 Si on a l'implication dynamique

*(HI ~ H2)* on a 6galement l'implication d y n a m i q u e

° ( [ H 1, H 3] ~

[H2, H31 )°

et cette derni~re accepte la mSme fonction degr6 que la premi6re (simple constatation).

La transitivit6 des implications dynamiques La proposition suivante est imm6diate : il suffit d'enchainer les deux algorithmes de constructions d'incompatibilit6s fortes.

Proposition 6 : Soient H ~ , H 2, H 3 trois syst~mes de csg portant sur des polynomes de K[X]. Alors: [ ' ( H I ~ H 2)* et * ( [ H 1 , H 2] ~ H 3 ) ' ] impliquent *( H I ~ H 3)* Supposons que la premi6re implication dynamique admette comme fonctiondegr6 acceptable Al(d;p) off d est le degr6 de , ] , [ H 2, H ] ,~, et p repr6sente certains param6tres d6pendant de H 1 et H 2, supposons de mSme une fonction-degr6 acceptable A2(d;q) pour la deuxi6me implication dynamique, alors une fonction-degr6 pour I'implication dynamique construite est obtenue en composant les deux fonctions-degr6 pr6c6dentes : A(d;p,q) = Al(A2(d;q);p) La proposition qui suit est un corollaire imm6diat de ta pr6c6dente.

Proposition 7 • Soient H 1 , K 1 , K 2 ...... K n des syst6mes de csg portant sur des polynomes de K[X] . Alors : ['(H 1 ~

K1)','(H

I ::~ K2)" ...... " ( H I =~ K n ) ' ] ~ns "( HI ~ [ K 1 , K 2 , - - - ' , K n ] ) " En outre, une fonction-degr6 pour l'implication dynamique construite est obtenue en composant (dans un ordre arbitraire) les fonctions-degr4 des implications dynamiques de l'hypoth6se

Cas des implications dynamiques avec une seule condition de signe dans la conclusion Combin4e avec la proposition 7, la proposition qui suit permet de montrer l'6quivalence d'une implication dynamique avec la donn6e d'une liste d'incompatibilit6s fortes. Cette donn6e 6tait appel6e une implication forte dans les articles [Lom x].

Proposition 8 : Soient H 1 un syst6me de csg portant sur des polynomes de K[X] , Q un 616ment de K[X], (~ un 616ment de { >, < , =, > , <, ~ } et Cf l'~16ment oppos6, ators : ,], ( H 1 , Q G 0 ) ,], si et seulement si ° ( H I ~ Q (3" 0 ) °

331

Si d 1 est le degr4 d'une incompatibilit6 forte ,], ( H 1 , Q ~ 0 ) ,~, alors une fonction-degr4 acceptable pour l'implication dynamique est doon6e par (d;d 1) J ~ q~(d,dI) = d.d 1 + d + d 1 . (cf. prop. 4) Inversement si dQ est le degr6 du polynome Q et si A1 est une fonctiondegr4 acceptable pour l'implication dynamique °( H~ ~ Q ~" 0 )°, le degr4 de l'incompatibilit6 forte peut ~tre major6 par AI(2.dQ).

preuve> Dans le sens direct : soit H u n

syst~me de csg et une incompatibilit6 forte ,], ( H , Q a ' 0) ,~ de degr6 d , on peut construire l'incompatibilit6 forte ,], ( H I , H ) ,~ en raisonnant cas par cas. Dans te cas Q ~ 0 on utilise l'incompatibilit6 forte .J, ( H I , Q o 0 ) ,~ de degr6 d 1 et dans le cas Q G' 0 on utilise l'incompatibilit4 forte ~, H , Q o ' 0 ,], , on conclut en utilisant la proposition 4. R6ciproque : on a une incompatibilit6 forte sous forme paire de degr6 2.dQ : ~, ( Q o 0 , Q ~' 0),~, obtenue en lisant convenablement l'identit6 Q2+Q.(_Q) = 0, on applique alors la d6finition de l'imptication d y n a m i q u e en prenant pour H la seule condition Q 0 0. O v e r s i o n a l g 6 b r i q u e d y n a m i q u e d e la d i s j o n c t i o n D ~ f i n i t i o n et n o t a t i o n 9 :

Soient H 1 , H 2 . . . . . H k et K~, K 2 ..... K m des syst~mes de csg portant sur des polynomes de K[X] . Nous disons que le syst~me H~ implique dynamiquement la disjonction K I V K 2 V ... V K m lorsque, pour tout syst6me de csg H portant sur des polynomes de K[X,Y], on a la construction d'incompatibilit6 forte : {,~[ K~(X), H(X,Y) ]~, et ... et ,],[ KIn(X) , H(X,Y) ],~, } ~ons ,~[ HI(J0 , H(X,Y) ],~ NOUS noterons cette implication-disjonction dynamique p a r : " ( H i ( X ) ~ [K~(X) V K2(X) V ... V KIn(X)] ) ' . Lorsque le syst6me H 1 est vide, nous utilisons la notation " ( K I ( X ) V K2(X) V ... V Km(X))'. Enfin, la notation : " ( [ H I V H 2 V ... V Hk] ~ [ K I V K2V .... V Kin] )" signifie que chacune des implications-disjonctions dynamiques ° ( H i ( X ) ::~ [KI(X) V K2(X) V ... V Kin(X)] )° est v6rifi6e

(i = 1,...,k)

Remarque : Toute formule sans quantificateur de la thdorie du premier ordre des anneaux totalement ordonn6s dicrets/~ param6tres dans K est 6quivalente a une formule en forme normale disjonctive et donc ~ une formule d u type KI(X) V K2(X) V ... V Kin(X) oh les Ki(X) sont des syst~mes de csg portant sur des polynomes de K[X]. Les implications-disjonctions d y n a m i q u e s consituent une forme de raisonnement p u r e m e n t ~ddentit6 alg6brique~> concernant les formules sans quantifi-

332 cateur, oh la logique a 6t6 6vacu6e au profit d'algorithmes de constructions d'identit6s alg4briques. Fonction-degr4 d'une implication-disjonction dynamique

Une implication-disjonction dynamique " ( H I ( X ) ::~ [KI(X) V K2(X) V ... V KIn(X) ] )* signifie par d6finition un algorithme fournissant la construction : {~,[ KI(X), H(X,Y) ]~, et ... et ,~[ Km(X), H(X,Y) ],~ } ~ons ,~,[ HI(X), H(X,Y) ]~, Chaque fois que nous 6tablissons une implication-disjonction dynamique particuli6re, nous devons 6tablir des 'majorations primitives r6cursives de degr6' pour cette construction d'incompatibilit6s fortes : le degr6 de l'incompatibilit6 forte construite est major6 par une fonction A(d t .... d m) off d i est le degr6 de l'incompatibilit4 forte initiale n°i. Nous disons qu'il s'agit d'une fonction-degr6 acceptable pour l'implicationdisjonction dynamique consid6r6e. Exemples : La proposition 4 peut ~tre relue comme affirmant des disjonctions ou implications-disjonctions dynamiques : Proposition 4 bis : On a les implications-disjonctions dynamiques suivantes :

" ( Q e a 0 ==~[Q>O v Q < O ] ) * "( Q,, ~ O = ~ [ Q > O v Q=O]) ° "(Q~0 V Q=0) ° "(Q>O v Q~O) ° "(Q=O v Q)O v Q
(a) (a') (b) (c) (d) (e) q0x d6crites ~ la

La transitivit6 des implications-disjonctions dynamiques

L'6nonc4 le plus g6n6ral est le suivant : les implications-disjonctions dynamiques "( [ H 1 V H 2 V V Hk] ~ [K 1 V K2 V V Km] )° et "([ KIV K2V...VKm]~[LlV L2V...VLn])" impliquent : ...

...

"([ H~v H 2 V . . . V H k ] ~ [ L ~ V L2V...VL.])"

Cette transitivit6 s'obtient en enchainant les algorithmes de constructions d'incompatibilit6s fortes. Les fonctions-degr6 r6sultantes s'obtiennent donc par composition convenable des fonctions-degr6 initiales. On d6montrerait pour les implications-disjonctions dynamiques le principe de substitution analogue a celui d6montr6 pour les implications dynamiques (cf. infra proposition 15), par la m~me m6thode.

333

Cas avec une seule condition de signe Proposition 10:

Supposons que dans une implication-disjonction dynamique "( H I ::~ [ K 1 V K 2 V . . . V K m ] )" chaque syst6me K i du second membre soit une seule condition de signe Qi (~i 0, et notons Qi ~i 0 la condition de signe oppos6e. Alors on a l'implication-disjonction dynamique : ° ( H I ( X ) ~ [QI¢yl0 v Q2a2 0 v ... v QmOm 0] )° si et seulement si on a une incompatibilit6 forte : ,~[ HI(X),QI'C10,Q2-f20 ..... QmIm 0 ]•

(a) (b)

On obtient sans difficult4 les pr4cisions suivantes concernant les degr6s : Si (a) est v4rifi6 avec une foncfion-degr6 acceptable Al(dl ..... d m) et si chaque Qi a pour degr6 8 i alors on a une incompatibilit4 forte (b) de degr6 A1(2.81 ..... 2.8m). Si on a une incompafibilit6 forte (b) de degr6 8 , alors l'implication d y n a m i q u e (a) admet pour fonction-degr6 acceptable : (d 1 ..... d m) ~ * q0(d1 , q~(d2 .... , q0(dm, 8)...) (fonction q0 de la prop. 4) V a r i a t i o n s s u r le t h 6 m e d e s i m p l i c a t i o n s d y n a m i q u e s Pour de nombreuses implications de base, on a des algorithmes plus rapides, et moins cofiteux en degr6, que celui donn6 en appliquant les propositions 7 et 8, lorsqu'on veut les traiter en implications dynamiques.

Implications triviales et implications simples D6finition 11 : ( implications triviales ) Une implication HI(X) ~ H2(X) est dite triviale lorsque toute incompafibilit6 forte ~, [ H2(X) , H(X,Y) ] ~, fournit par simple relecture rincompafibilit6 forte $ [ H i ( X ) , H(X,Y) ] $ . L'implication dynamique "(HI(X) ~ H2(X))* accepte alors pour fonctiondegr4 : Ao(d) = d . Exemples : L'implicafion patibilit6 forte

[ A > 0, B > 0 ] ~

AB > 0 est triviale : dans l'incom-

.1, [ A B > 0 , H ] ,l, on relit chaque constituant AB (dans la partie .monoide>> ou dans la partie ,cone-) sous forme d u produit de A par B pour obtenir l'incompafibilit6 forte ~, [ A > 0 , B > 0 , H ] ,1, Notez que l'implication "contrappos6e" [ A > 0 , A.B < 0 ] ~ B < 0 n'est pas une implication simple. L'implication dynamique "([A>0,AB<0] ~ B < 0 )" peut 8tre obtenue par l'algorithme suivant : multipliez chaque terme de l'incompatibilit6 forte ,~ [ B < 0 , H ] ,~ par A et relire les termes off apparMt le produit A.B en consid6rant que AB est un seul bloc, provenant de l'hypoth6se AB _<0 : on obtient alors l'incompatibilit6 forte ~, [ A > 0 , A . B < 0 , H I ,1,

334

De m~me, l'implication B = 0 ~ A.B = 0 est triviale, tandis que la contrappos6e ne l'est pas. On a aussi l'implication triviale [ A > 0 , A ~ 0 ] ~ A > 0 , tandis que l'implication [ A > 0, A < 0 ] ~ A = 0 ne l'est pas. D6finitions 12: ( implications simples ) a) Une implication : HI(X) ~ T(X) = 0 est dite simple torsqu'elle est donn6e par une 4galit4 T = Y~Ni.V i o~ les N i sont les polynomes suppos6s nuls dans H~. On appelle degr6 absolu d'une telle implication simple l'entier : sup(d(Ni.Vi)) - d(T), et degr6 relatif le rationnel sup( d(NI.V i) ) / d(T) b) Une implication : Hi(X) ~ T(X) 5/ 0 est dite simple lorsqu'elle est donn6e par une 6galit6 T = Y. Ph.(Z Uh,j Uh,j 2 ) + Y. Ni.V i avec les m~mes hypotheses qu'en a), et off en outre tes Ph sont des produits de polynomes suppos6s > 0 , ou > 0, dans H~. (les Uh,j sont des positifs de K ). On appelle degr6 absolu d'une telle implication simple la diff6rence : sup(d(Ni.Vi) , d(Ph.Uh,j 2) ) -- d(T), et degr6 relatif leur rapport. c) Une implication : Hi(X) ~ T(X) > 0 est dite simple lorsqu'elle est donn6e par une 6galit6 T = S.R2 + Y. Ph.(Z Uh,j Uh,j 2 ) + ~ Ni.V i avec les m~mes hypoth6ses qu'en b), et off en outre S (resp. R ) est un produit de polynomes suppos6s > 0 (resp ¢ 0 ) dans H I . On appelle degr4 relatif d'une telle implication simple le rationnel : sup(d(S.R2), d(Ni.Vi) , d(Ph.Uh,j 2) ) / d(T) d) Une implication : Hi(X) ~ T(X) ¢ 0 est dite simple lorsqu'elle est donn6e par une 6galit6 T = S.R + ~ Ni.V i avec les m4mes hypoth6ses qu'en c). On appelle degr6 relatif d'une telle implication simple le rationnel : sup(d(S.R), d(Ni.V i) ) / d(T) e) Une implication Hi(X) ~ H2(X) est dite simple lorsque chacune des csg du second membre, Qi c~i 0, r6sulte de Hi(X) par une implication simple. On appelle degr6 relatif de cette implication simple le sup des degr6s relatifs des implications simples Hi(X) ~ Qi (3i 0. Lorsque le syst6me H2(X) ne comporte que des conditions de signes ferm6es ( = 0 , < 0, > 0 ) on appelle degr6 absolu de l'imptication simple, le sup des degr6s absolus des implications simples Hi(X) ~

Qi ~i 0.

I1 y a un algorithme particuli6rement simple p o u r expliciter l'implication dynamique correspondant a une implication simple donn6e d u type : H~(x) ~

T(X) = 0

Dans l'incompatibilit6 forte : ,], [ H(X,Y), T(X) = 0 ] ~, on remplace T par ~ N i . V i . Par exemple si T apparaissait sous forme T . W ,

on aura maintenant une

335

s o m m e Y. Ni.(W.V i) o~ chaque terme a un r61e a u t o n o m e dans la nouvelle implication forte :

~, [ H(X,Y) , Hi(X) ] ,1, On voit que le degr4 de cette derni~re a augment6 au plus du degr6 absolu de t'implication simple, et on en d4duit qu'it a 6t6 multipli4 au plus par le degr6 relatif de l'implication simple. Des consid6rations d u m~me genre s'appliquent aux autres cas d'implications simples et o n obtient : Proposition 13 : ( implications simples en tant qu'implications d y n a m i q u e s ) a) Une implication simple : Hi(X) ~ H2(X) , o~1 H2(X) ne comporte que des conditions de signes ferm4es, accepte p o u r fonction-degr6 : (d,8) ~ ~ d + 8 , o~ 8 est le degr6 absolu de l'implication simple. b) Une implication simple : Hi(X) ~ H2(X) accepte pour fonction-degr6 : (d;8") s * d.8" , oi~ ~' est le degr6 relate de l'implication simple. R e m a r q u e : Souvent, une implication simple a un degr~ absolu nul et u n degr6 relatif 4gal ~ 1, ce qui signifie que l'implication forte consid6r6e ne cofite rien p o u r ce qui concerne les degr6s. N o u s dirons indiff6remment 'implication simple de degr6 relatif 6gal ~ 1' o u 'implication simple qui ne cofite rien'. Dans une 4ventuelle mise en o e u v r e de l'algorithme, il est tou]ours plus 6conomique de traiter une implication simple en rant que telle. Trois e x e m p l e s :

Substitution d'dgaux : L'implicafion U = V ~ P(X,U) = P(X,V) est une implication simple qui ne coftte rien. (U et V sont ici suppos4es ~tre des variables et non des polynomes) Somme de deux positifs : L'implication [A > 0, B > 0 ] ~ A + B > 0 est simple de degr6 relate 5' = sup(d(A),d(B))/d(A+B) et accepte la fonction-degr6 : d~ ~ d.ff. L'implication [A > 0, B _ 0 ] ~ A + B _>0 est simple de degr6 absolu 8 = sup(d(A),d(B)) - d(A+B) et accepte la fonction-degr6 A : d t ....* d + 5. Point of~ un polynome unitaire a l e signe de son coefficient dominant : Soit Q un polynome, unitaire en la variable U distincte des Xi : Q(X,U) = U s + Cs_I(X).Us-1 + .... + CI(X).U + Co(X) Soit V(X) = s + Cs_I(X) 2 + .... + CI(X) 2 + Co(X) 2 . Alors on a des implications simples simultan6es qui ne coutent rien : [ ] ~ Q(X,V(X)) > 0 [ ] ~ Q(i)(x,v(x)) > 0 (d6riv6es par rapport ~ U) Signalons enfin quelques implications, qui sans &tre des implications simples, sont d'un traitement "rapide" en rant qu'implications dynamiques :

336

P r o p o s i t i o n 14 : (fonctions-degr4 de quelques implications particuli6res) a) L'implication [ A > 0 , A.B > 0 ] ~ (d;5)...~d+2.5

o~ 5 = d ( A )

B> 0

• M 6 m e chose avec = a la place de > .

b) L'implication [ A > 0 , A.B > 0 ] ~ (d;5,~'); .... ; sup( d.[i', d + 2.5 )

B> 0

"d+2.5

olh 5 = d ( A . B )

~ s u p ( d.5", d + 2.5 )

B> 0

accepte p o u r fonction-degr6 :

B> 0

accepte p o u r fonction-degr6 :

.

d) L'implication [ A > 0 , A.B > 0 ] ~ (d;5,5") ~

off 5 = d(A.B), 5' = d(A.B) / d(B).

e) L'implication [ A.B > 0 , A+B > 0 ] ~ fonction-degr6 : (d;5,5'), 8 = sup(d(A),d(B)). f) L'implication A 2k _<0 ~ (d;k):

[ A > 0 , B > 0 ] accepte p o u r

* d.3' + 25 off 5' = d ( A B ) / i n f ( d ( A ) , d ( B ) ) , A= 0

accepte p o u r fonction-degr6 :

" 2k.d

D e m ~ m e l'implication [ A > 0 , A < 0 ] ~ degr6 : d :

A = 0 accepte p o u r fonction-

," 2d.

g) L'implication (d;8').

accepte p o u r fonction-degr6 :

ot~ ~ = d ( A ) , 3' = d(A.B) / d(B).

c) L'implication [ A > 0 , A.B > 0 ] ~ (d;8):

accepte p o u r fonction-degr4 :

P(X,U) • P(X,V) ~ U :~ V accepte p o u r fonction-degr6 : ~ d.3' o£l 5' = d(P(X,U) - P(X,V)) / d(U - V)

Par e x e m p l e p o u r le a) : on multiplie, t e r m e a terme, l'implication forte p a r A 2, en p r e n a n t soin d e r e m p l a c e r les B.A 2 p a r (BA).A. Le p r i n c i p e de s u b s t i t u t i o n P r o p o s i t i o n 15 : O n consid6re des variables X1,X2..... Xn, UI,U2,...,U h, Z1,Z2,...,Zk, et des p o l y n o m e s P1,P2 ..... Pn d e K[Z]. N o t o n s P(Z) p o u r PI(Z) .... , Pn(Z) . Si on a

"(~,-~I(X,U) :~ ~--~2(X,U))"

(a)

alors o n a aussi ° ( H I ( P ( Z ) , U ) ~ H2(P(Z),U) ) ° (b) Si A 1 est u n e fonction-degr6 acceptable p o u r (a), u n e fonction-degr6 acceptable p o u r (b) est donn6e p a r :

d ~

~ g + Al(g + 8.d) of1 8 = sup(deg(Pi))

et X et g sont explicit6s dans la p r e u v e .

preuve> N o t o n s X = P(Z) p o u r : X 1 = P I ( Z ) ..... X n = Pn(Z). Soient Y1,Y2 ..... Yn d e s nouvelles variables. P o u r t o u t R f i g u r a n t d a n s H2(X,U) consid6rons P6galit6 qui p e u t ~tre o b t e n u e p a r d i v i s i o n s s u c c e s s i s v e s ties degr6s d e s R i sont tous inf6rieurs o u 4 g a u x au degr6 d e R) : R(X,U) = R(Y,U) + (X 1 - Y1) RI(X,Y,U) + ... + (Xn - Yn) Rn(X,Y,U) (1) En s u b s t i t u a n t Pi(Z) /~ Yi on obtient u n e 6galit6 : R(P(Z),U) = R(X,U) + (X 1 - P I ( Z ) ) R1(X, PfZ),U) + ... + (X n - P n ( Z ) ) Rn(X,P(Z),U) (2) Ces 6galit6s f o u r n i s s e n t u n e implication s i m p l e :

337

• ( [ H2(X,U), X = P(Z) ] ::# H2(P(Z),U) )" (3) d o n t le d e g r 4 absolu est major6 par ~t = s u p ( 0 , sup{ deg((X i - Pi(Z)) Ri(X,P(Z),U)) - deg(R(P(Z),U)) ; R figure d a n s H 2 , R(P(Z),U) ¢ cte } D e la me, m e manii~re, p o u r les R figurant d a n s H 1 o n a des 6galit6s : R(X,U) = R(P(Z),U) - (X1 - PI(Z)) RI(X,P(Z),U) - ... - (Xn - Pn(Z)) Rn(X,P(Z),U) (4) qui f o u r n i s s e n t u n e i m p l i c a t i o n s i m p l e • ( [ HI(P(Z),LO, X = P(Z) ] ::#

[ H I ( X , U ) , X = P(Z) ] ) •

(5)

d o n t le degr4 absolu est major6 par k = s u p ( 0 , sup{ deg((X i - Pi(Z)) Ri(X,P(Z),U)) - deg(R(X,U)) ; R figure d a n s H 1 } Par ailleurs l'implication d y n a m i q u e : • ( [ H I ( X , U ) , X = P(Z) ] ~

[ H2(X,U), X = P(Z) ] )•

accepte la m 6 m e f o n c f i o n - d e g r 6 q u e l'implication d y n a m i q u e En c o m p o s a n t (5), (6) et (3) o n obtient : • ( [ H I ( P ( Z ) , U ) , X = P(Z) ] ::~ Enfin, c o m m e les v a r i a b l e s l'implication dynamique :

X

H2(P(Z),U) ) •

ne f i g u r e n t pas d a n s

• ( H I ( P ( Z ) , U ) :::) [ H I ( P ( Z ) , U ) , X = P(Z) ]) )•

(6)

(a). (7) HI(P(Z),U),

on a

(8)

o b t e n u e en r e m p l a q a n t les X i par les Pi d a n s l'incompatibilit6 forte initiale. Elle accepte p o u r fonction-degr6 : d t * d.5 avec ~ = sup(deg(Pi)). En r6sum6, si a 1 est u n e fonction-degr6 acceptable p o u r (a), u n e f o n c t i o n - d e g r 6 acceptable p o u r ( b ) e s t d o n c : d. * ~,+Al(~t+&d) Q F o r m u l e s de T a y l o r m i x t e s

O n consid6re d e u x variables U et V e t o n p o s e A := U - V . O n consid6re u n p o l y n o m e P a coefficients d a n s u n corps o r d o n n 6 K o u plus gdndralement dans Un

commutatif A qui est une Q-alg~bre.

flnnffflu

Si degfP) x( 4 , o n a l e s 8 formules d e Taylor mixtes suivantes: P(U) P(V) = A.P'(V) + (1/2).A2.p'(v) + (1/6).A3.p(3)(V) + (1/24).A4.p (4) P(U) - P(V) = A.P'(V) + (1/2).A2.p"(V) + (1/6).A3.p(3)(U) - (1/8).A4.p (4) P(U) - P(V) = A.P'(V) + (1/2).A2.p"(u) - (1/3).A3.p(3)(V) - (5/24).A4.p (4) P(U) - P(V) = A.P'(V) + (1/2).A2.p"(U) - (I/3).A3.pC3)CU) + (1/8).A4.P (4) -

P(U) - P(V) = A.P'(LO - (1/2).A2.p'(v) - (1/3).A3.p¢3)(V) - (1/8).A4.p (4) PfU) - P(V) = A.P'CU) - (1/2).A2.p'(v) - (1/3).A3.p(3)0d) + (5/24).A4.P (4) P(U) - P(V) = A.P'(U) - (1/2).A2.p'0d) + (1/6).A3.p(3)(V) + (1/8).A4.P (4) P(U) - P(V) = A.P'CU) - (1/2).A2.p"(U) + (1/6).A3.p(3)(U) - (1/24).A4.p (4) C o m m e toutes les c o m b i n a i s o n s de signes possibles se pr4sentent, o n obtient : supposons que u et v attribuent la m ~ m e suite de signes (au sens large) p o u r les d6riv6es successives d ' u n p o l y n o m e P n o n c o n s t a n t de d e g r 6 < 4 , n o t o n s e 1 = 1 o u -1 selon que P'(u) et P'(v) sont tous d e u x > 0 o u tous d e u x < 0 , alors le fait q u e P(u) - P(v) a m S m e signe q u e e p ( u - v) est r e n d u 6vidnet

338

par l'une des formules ci-dessus, ce qui d o n n e l'implication sous f o r m e d ' u n e implication simple (u et v peuvent ~tre des 614ments de K mais aussi des variables, ou des polynomes) si u et v n'attribuent pas la m~me suite de signes p o u r un p o l y n o m e P de degr6 < 4 et ses d6riv6es successives, alors on a une identit6 alg6brique qui donne le signe de u - v a partir des signes des P(i)(u) et des ~(i) r (V) : la formule de Taylor mixte a utiliser est avec p(i) ( i = 0, 1, 2, ou 3) off i e s t le plus grand indice p o u r lequel tes deux signes ne sont pas identiques Plus g4n6ralement on a : Proposition 16 : (formules de Taylor mixte)

Pour chaque degr6 s , il y a 2 s- 1 formules de Taylor mixtes et toutes les combinaisons de signes possibles apparaissent. Formules de T a y l o r g6n6ralis6es (le l e m m e de T h o m sous f o r m e d'identit4s alg4briques) Le l e m m e de T h o m affirme (entre autres) que l'ensemble des points o~ un p o l y n o m e et ses d6riv6es successives ont chacun un signe fix6, est un intervalle. Une p r e u v e facile, p a r r6currence sur le degr6 d u p o l y n o m e , est bas6e sur le th6or~me des accroissements finis. Nous pouvons, grace aux formules de Taylor mixtes, traduire ce fait g6om6trique sous forme d'identit4s atg6briques, que nous appellerons des f o r m u l e s de T a y l o r g6n6raIis6es. Plut6t que de risquer un 6nonc6, nous d o n n o n s un exemple. Un exemple : Consid4rons le polynome g6n6rique de degr6 4 P(X) = co X~ + c I X3 + c2 X3 + c3 X2 + c4 X4 + cs Consid6rons le syst6me de conditions de signe portant sur le p o l y n o m e P e t ses d6riv4es successives par rapport a la variable X : H(U) : P(U) > 0, P'(U) < 0, P(2)(U) < 0, P(3)(U) < 0, P(4)(U) > 0. Consid4rons 6galement le syst6me de conditions de signe g6n6ralis6es obtenues en relachant toutes les in6galit6s, sauf la derni6re : H'(U) : P(LO > 0, P'(U) < 0, P(2)(U) ~ 0, P(3)(U) _<0, P(4)(LD > 0. Le lemme de T h o m affirme (entre autres) : [H'(U),H'(V), U
P'(Z) = P'(U) + P(2)(U).(Z - U) + 1/2 P(3)(Z).(Z - U) 2 - 1/3 P(4).(Z - U) 3 P(Z) = P(V) + P'(Z).(Z V) - 1/2p(2)(Z).(Z - V) 2 + 1/6 P(3)(V).(Z - V) 3 + ... 1/8 P(4).(Z - V)4

Po

n,

Dans ~) on remplace P(3)(Z) ,par son expression donn6e dans c0 et, 0n,,,,,0btient : !

[ ~,)

I

p(2)(Z) = p(2)(LD + p(3)(V).al _ p(4) [hi.a2 + 1/2 hi21

[

339

On obfient de la m~me mani6re, par substitutions : 'it)

P'(Z) = P'(U) + P(2)(U).A1 + 1/2 P(3)(V).A12-- P(4).[a12.A2/2 + A13/3]

et enfin 5') P(Z)

= P(V) - P ' ( U ) . A 2 - P ( 2 ) O d ) . [ • I . A 2 + 1/2A22] p(3)(V).[ A12.A2/2 + ~1.A22/2 + A23/6] + P(4).[A13.A2/3 + A12.Z~22/2+ A1.A23/2 + A24/8] -

Les 6galit6s (z), [Y), 7'), ~') donnent l'implication (1) sous forme d'une implication simple. La premi6re 4galit4 est une formule de Taylor ordinaire portant sur le p o l y n o m e p(3). Les trois derni6res peuvent ~tre r u e s c o m m e des formules de Taylor g6n6ralis6es portant sur les polynomes p(2), p, et P . Plus g4n4ralement, on obtient: Th6or~me 17 : (4vidence forte du lemme de Thorn) Soit T u n e variable distincte des C i . Soient P E K[C][T], de degr6 s e n T , °1, (~2..... a s une liste form6e de < ou >. On note H(C,T) ou H(T) le syst6me de csg : P'(C,T) c~ 0 ..... P(i)(C,T) c~i 0, ..., P(s)(C,T) crs 0 (les d6riv6es sont par rapport h T ). Soit H'(T) le syst6me de csg obtenu ~t partir de H(T) en relachant toutes les conditions de signe sauf celle relative h p(s). Soit HI(T) le syst~me de csg : P(s)(C,T) > 0, P(i)(C,T) )/0, i = 1, ..., s - 1 . Soient enfin trois variables U , V , Z distincte des C i . On a alors les implications dynamiques suivantes : "( [ H'Od), H'(V), U o 1 V ] ~ POd) > P(V) )" (a) "( [ HI(U), V > U ] ~ P(V) ) POd) )" (b)

"( [ H'(U), H'(V), U < Z < V ] ~

H(Z) )"

(c)

Ce sont des implications simples qui ne cofitent rien. preuve> L'implication d y n a m i q u e (a) r6sulte de formules de Taylor mixtes. L'implication d y n a m i q u e (b) r6sulte de la formule de Taylor ordinaire au point U. Les formules de Taylor g4n6ralis6es 6tablies pour l'implication dynamique (c) r6sultent des formules de Taylor mixtes. On constate qu'il s'agit d'implicafions simples qui ne cofitent rien (ceci parce que U, V, Z sont des variables et non des polynomes). Q

340

4)

Existences p o t e n t i e l l e s

N o t a t i o n s et d6finitions Elles sont tout h fait analogues ~ celles donn6es pour les implications dynamiques.

D6finition et notation 18 : Soient H I un syst~me de csg portant star des polynomes de K[X], H 2 un syst~me de csg portant sur des polynomes de K[X,TvT2,._,Tm] = K[X,T]. Nous dirons que les hypotheses H I autorisent l'existence des Ti vdrifiant H 2 lorsque, pour tout syst~me de csg H portant sur des polynomes de K[X,Y], les variables Yi et Tj 6tant deux ~ deux disfinctes, on a la construction d'implication forte :

$ [ H2(X,T) , H(X,~9 15 ~ons$ [ Hi(X) , H(X,Y) ] $ . d'existence potentielle des Ti vdrifiant les hypotheses H~

Nous parlerons 6galement

H 2 sous

Nous noterons cette existence potentielle par : "(H1CX) ::~ B T H2(X,T) ) ' . Lorsque le syst~me H 1 est vide, nous utilisons la notation * ( 3 T H2(X,T) )*. La notion de fonction-degr4 acceptable pour une existence potentielle peut ~tre elle aussi directement recopi4e du cas des implications dynamiques.

Remarques : 1) La notion d'existence potentielle est une notion d'existence faible. L'existence potentielle signifie qu'il n'est pas grave de faire comme si les T i existaient vraiment, parce que cela n'introduit pas de contradiction: on peut paraphraser la d6finition en disant : pour construire l'incompatibilit4 forte $ [ HI(X), H(X,Y) ] $ il suffit d'avoir construit $ [ H2(X,T) , H(X,Y) ]$ 2) On pourrait 4tendre la d6finition de l'existence potentieUe en remplaqant le syst&me de csg H2(X,T) par une disjonction de syst6mes de csg, comme on a fait avec la notion d'implicafion-disjonction dynamique.

Quelques r6gles de manipulation des 6nonc6s d'existence potentielle Transitivit4 La transitivit6 des existences potentielles est imm6cliate. Voici l'6nonc6 pr6cis6 en termes de fonctions-degr6 acceptables.

Proposition 19 : (transitivit6 dans les existences potentielles) On consid~re des variables X1,X2,...,Xn,TvT2,...,T m, U1,U2,...,Uk et des syst~mes de csg Hi(X), H2(X,T) et H3(X,T,U).

341

Les existences potentielles ° (HI(X) ~ 3 T H2(X,T) ) ° et ° (H2(X.T) ~ B U H3(X,T,U) ) ° impliquent l'existence potentielle : °(HI(X) ~ 3 T,U Ha(X,T,U) )" Supposons que la premiere existence potenfielle admette comme foncfiondegr6 acceptable Al(d;p) off d est le degr~ de l'incompafibilit~ forte ~, [ H2(X,T), H(X,Y) ] ,]. et p repr~sente certains param~tres d6pendant de HI(X) et H2(X,T), supposons de mSme une fonction-degr6 acceptable A2(d;q) pour la deuxi~me existence potentielle, alors une fonction-degr6 pour l'existence potentielle construite est donn6e par : A(d;p,q) = A1(A2(d;q);p) Preuves cas par cas Voici maintenant u n 6nonc6 corresponclant aux preuves cas par cas d'une existence potentielle, cons6quence imm6diate de la proposition 4 . Proposition 20 : (raisonnement cas par cas) Soit Q un polynome de K[X]. a) Pour d6montrer une existence potentielle "( [ HI(X), Q ~: 0 ) ~ B T H2(X,T) )" il suffit de d6montrer chacune des existences potenfielles " ( [ H I ( X ) , Q > 0 ] ~ B T H2(X,T))* e t ' ( [ H I ( X ) , Q ( 0 ] ~ B T H2(X,T))" Si A i (i = 1,2) sont les deux fonctions-degr6 des existences potentielles suppos6es, une fonction-degr6 pour l'existence potentielle d6duite est donn6e par : A1 + A 2 a'), b), c), d), e) : 6nonc6s analogues d6calqu6s de la proposition 4 Le principe de substitution Le principe de substitution pour les existences potentielles se d6montre comme pour les implications clynamiques. L'existence implique l'existence potentielle Un autre principe utile est le fait que l'existence implique l'existence potentielle. II s'obtient facilement : on remplace les variables T i ~existentielles~ par les polynomes concrets Pi qui r6alisent l'existence. On reconnait l~ une analogie formelle avec la r~gle d'introduction du quantificateur existentiel en calcul naturel par exemple (cf. [Pra]). Proposition 21 : ( l'existence implique l'existence potentielle) Soient PI,P2,...,Pm E K[X] et notons P(X) pour PI(X) ..... Pro(X). On a l'existence potentielle : "(H2(X,P(X)) ~ 3 T H2(X,T) )" . Si 8 majore les degr6s des Pi • l'existence potenfielle accepte pour fonctiondegr6 : (d;8) ~ ~ d.sup(1,8)

342

CoroUaire : (m~mes hypoth6ses)

Si *(HI(X) ::~ H2(X,P(X)) )" alors °(HI(X) ~

3 T H2(X,T)) °

Si A1 est une fonction-degr6 acceptable pour l'implication forte de l'hypoth6se, une fonction-degr6 acceptable pour la conclusion est donn4e par : (d;8) ~ , Al(d.sup(1,8)) off 8 majore les degr6s des Pi •

E.xistences p o t e n t i e l l e s f o n d a m e n t a l e s On sait d6montrer les existences potentielles correspondant aux deux axiomes existenfiels de la th4orie des corps r4els dos. Th4or~me 22 : (autorisation de rajouter l'inverse d'un non nul) On a l'existence potentielle de l'inverse d'un non nul. Ce qui s'6crit: " ( U ~ : 0 ~ 3 T I = U . T )* Soit ~ le degr6 de U, une fonction-degr6 acceptable pour l'existence potentielle est (d;8) : - d + d.3 + R e m a r q u e : La preuve de cette existence potentieUe recopie ce qu'on fait, dans la preuve d u th6or6me des z4ros de Hilbert, pour passer d u th6or~me des z6ros faible au th6or6me des z4ros g6n6ral (c'est le ~~, par exemple dans l'expos6 classique de van der Waerden). La notion d'existence potentielle de l'inverse d ' u n non nul est donc en filigrane dans les classiques. Th4or~me 23 : (autorisation de rajouter une racine sur un intervalle oh un polynome change de signe) Soit P(C,X) un polynome de degr6 s en X et de degr6 global 5. On a l'existence potentielle d'une racine sur un intervalle off ce polynome change de signe. Ce qui s'6crit, en notant P(X) pour P(C,X) : " ( [ P ( X ) . P ( Y ) ( 0 , X
343

5)

Majorations finales

Tableaux de H6rmander N o u s d o n n o n s ici quelques majorations directement li6es a l'algorithme de H6rmander lui-m~me (cf. [H6r] annexe, ou [BCR] chap. 1). Proposition 24 : (Tableau de H6rmander pour des polynomes en une variable) Soit K un corps ordonn6, sous-corps d'un corps r6el d o s R . Soit L = [P~, P2, ..., Pk] une liste de polynomes de K[Y]. Soit P la famille de polynomes engendr6e par les 416ments de L et par les op6rations P* ~ P ' , et (P,Q)* J Rst(P,Q). Alors : 1) T estfinie. 2) On peut 6tablir le tableau complet des signes pour T e n utilisant les seules informations suivantes : le degr6 de chaque polynome de la famille; les diagrammes des op4rations P 1 * P ' , et (P,Q) 1 * Rst(P,Q) (o1:1 deg(P) ~ deg(Q) ) dans T; et les signes des constantes de T . Si s majore les degr6s des Pi, le nombre de coefficients d'616ments de P , et donc aussi le nombre de points d u tableau de HSrmander est major6 par: (k+l) 2s Ualgorithme de H 6 r m a n d e r traite des polynomes en n variables, en 61iminant chaque variable l'une apr6s l'autre. A chaque 61imination d°une variable, le n o m b r e de p o l y n o m e s ~t consid6rer et leurs degr6s croissent de mani6re impressionnante. Ceci est pr6cis6 dans la proposition suivante : Proposition 25 : (Tableau de HSrmander param6tr4) Soit K un corps ordonnG sous-corps d'un corps r6el d o s R . Soit L = [Q1, Q2, -.., Qk] une liste de polynomes de K[X1, X2, .... Xn][Y]. On peut construire une famille finie y" de polynomes de K[X 1, X2, ..., Xn] telle que, pour tous x 1, x 2, ..., x n dans R , en posant Pi(Y) = Qi(Xl, x2, .... xn;Y), le tableau complet des signes pour L = [P1, P2 ..... Pk] est calculable a partir des signes des S(x1, x 2, .... x n) pour S E y-. Supposons que la liste L poss6de k 616ments de degr4 en X major6 par 8 et de degr4 en Y major4 par s . Consid6rons la famille G , form6e de tousles coefficients de t o u s l e s polynomes de tousles tableaux de H6rmander possibles, construits sur L , en remplaqant l'op6ration "reste" par l'op6ration "pseudoreste". Une famille F convenable peut ~tre extraite de G. Alors : le degr6 de chaque polynome de ~ et de chaque pseudo-division est major6 par : 8.(s+1)!, (sauf s i n = 0 , d o n c 8 = 0, et les degr6s sont major6s par s ). le nombre d'416ments de la famille

~

est major6 par :

(k+l) 2s

344

Men6 jusqu'au bout, cet algorithme p r o d u i t donc une explosion de degr4s obtenue en it6rant n-1 fois ( n 6tant le nombre de variables) la fonction s : ; s !. Ceci conduit a la majoration finale.

Nullstellensatz, positivestellensatz et nichtnegativestellensatz r6els effectifs Th4or~me 26 : Soit K un corps ordonn6, sous-corps d'un corps r6el d o s R . Soit H(X1,X 2.....Xn) un syst6me de csg portant sur une famille finie de polynomes de K[XI,X2,...,X n] . Ce syst6me est impossible dans R si et seulement siil est fortement incompatible dans K . En termes plus formalis6s : Si ~, H(XI,X2,...,X n) ~, (dans K ) , alors les csg H sont impossibles ~ r6aliser dans n'importe quelle extension ordonn6e de K . Si V x p x 2 , . . . , x n E R H(xl,x2,...,Xn) est absurde, alors : ~, H(X1,X 2..... X n) J, (dans K ). Pr6cis6ment, si k est le nombre de csg dans H(X1,X2,...,X n) et d le degr4 maximum, on peut calculer une implication forte ,], H(X1,X 2.....Xn) ~, (dans K ) de degr6 major6 par le nombre ~26(d,k,n) donn6 par la tour d'exponentielle a n+4 6tages •. d.lg(d)+lglg(k)+cte 22''

Remarque : La principale cause d'explosion des degr6s dans la majoration finale actuelle r6side dans l'utilisation de l'algorithme de H6rmander. On p e u t donc esp6rer am61iorer sensiblement ces majorations en se basant sur d'autres preuves, 616mentaires mais moins longues, d'incompatibilit6. R e m e r c i e m e n t s : Je remercie Marie-Franqoise Roy p o u r ses n o m b r e u x commentaires et suggestions. Bibliographie :

[BCR]

[Du] [Err] [1G]

Bochnak, Coste M., Roy M.-F. : G6om6trie Alg6brique r6elle. Springer-Verlag. A series of Modern Surveys in Mathematics n°11. 1987. Dubois, D. W. : A nullstellensatz for ordered fields, Arkiv for Mat., Stockholm, t. 8, 1969, p. 111-114 Efroymson, G. : Local reality on algebraic varieties, J. of Algebra, t. 29, 1974, p. 113-142. Fitchas N., Galligo A. : Nullstellensatz effectif et Conjecture de Serre (Th6or~me de Quillen-Suslin) pour le Calcul Formel. Math. Nachr. 149

p 231-253(1990) [He]

H e r m a n n Greta : Die Frage der endlich vielen Schritte in der Theorie der Polinomideale, Math. ANN. 95 (1926), 736-788

345

[H6r]

[Ko] [Kril [Lore a]

[Lom b] [Lore c]

[Lore d] [LR]

[MRR] Feral [Rts] [Ste]

H6rmander, L. : The analysis of linear partial differential operators, vol 2, Berlin, Heidelberg, New-York, Springer (1983). 364-367. Koll~ir J. : Sharp effective Nulstellensatz. I. AMS 1 p 963-975 (1988) Krivine, J. L. : Anneaux pr6ordonn6s. Journal d'analyse math6matique, t.12, 1964, p. 307-326 LombardiH.: Th4or~me effectif des z6ros r6el et variantes. Publications Math6matiques de l'Universit6 (Besanqon). 88-89. Fascictfle 1. Lombardi H . : Effective real nullstellensatz and variants, in ~MEGA 90~, mai 1991, chez Birkhafiser. (Version anglaise plus courte) Lombardi H. : Nullstellensatz r6el effectif et variantes. C.R.A.S. Paris, t. 310, S6rie I, p 635-640, 1990. Lombardi H.: Th6or6me effectif des z6ros r6el et variantes, avec une mat,oration explicite des degr6s. 1990. M6moire d'habilitation. Lombardi H., Roy M.-F. : Th6orie constructive 616mentaire des corps ordonn6s. 1989. Publications Math6matiques de Besanqon. Th6orie des Nombres 1990-91. Version anglaise moins d6taill6e ~Constructive elementary theory of ordered fields~ in ~MEGA 90~, mai 1991, chez Birkhafiser. R. Mines, F. Richman, W. Ruitenburg : A Course in Constructive Algebra. Springer-Verlag. Universitext. 1988. PrawitzD. : Ideas and results of proof theory. Proceedings of the second scandinavian logic symposium (juin 70). Studies in Logic and Foundations of Mathmatics n°63, 235-307. North Holland. Risler, J.-J. : Une caract6risation des id6aux des vari6t6s alg6briques r6elles, C.R.A.S. Paris, t. 271, 1970, s6rie A, p. 1171-1173. Stengle, G. : A Nullstellensatz and a Positivestellensatz in semialgebraic geometry. Math. Ann. 207, 87-97 (1974)

MINIMAL

GENERATION

OF B A S I C S E M I - A L G E B R A I C

OVER AN ARBITRARY

ORDERED

SETS

FIELD

M. MARSHALL AND L. WALTER If A is a ring (commutative with 1), F ( p ) denotes the residue field of A at a prime p c_ A and Sper A denotes the real spectrum of A [2], [3], [8], [9]. If A is an F-algebra and T C_ F is a preordering, SperTA := {Q e Sper A ! Q D_T}. This is a saturated set in Sper A in the terminology of [13]. Fix a real closed field R and a subfield F C_ R. Let P C Sper F be the ordering induced by the inclusion. Fix an algebraic set V = {x E R n I fi(x) = 0 for i = 1 , . . . , k} FiX, ,...,X.] where f l , . . . , f k e F [ X 1 , . . . , X ~ ] and let A = (/1,...,A) ' A semi-algebraic set S C V is basic over F (or simply F-basic ) if s = {~ ~ v 1g~(~) > o A . . . A g,(~) > 0 ^ h~(x) >_ 0 A - . . A h,(~) _> 0} where g~,..., g~, h i , . . . , h t G F [ X 1 , . . . , Xn]. If t = 0 (resp. s = 0), S is said to be basic open over F (resp. basic closed over F ). A semi-algebraic set is defined over F if it is a finite union of F-basic sets. Define s f ( V ) (resp. ~-F(V)) to be the least integer s _> 1 such that each S C_ V which is basic open over F (resp. basic closed over F ) is expressible as S = {X E V [ g l ( x ) > 0 ^ . . . ^ ~ ( x ) > 0} (resp. S = {x e V lgl(x) > 0 A . . . ^ ~ ( x ) > 0}) where g l , - . . , gs 6 F [ X 1 , . . . , Xn].

The object of this paper is to give bounds for s y ( V ) and ~F(V), thus generalizing results in [5], [16]. The result we obtain is stated in Theorem 1.2 below. In §2, we show, at least in the case V = R n, that the bounds in T h e o r e m 1.2 are best possible. 1. B o u n d s f o r s F ( V ) a n d g F ( V ) . We need the following notation: for any Q E S p e r F , let AQ : F --* N U o o be the associated real place and vq the valuation associated with AQ (see [10].) Define ~Q(F*) and e(Q) e {0,1} is defined as ~ ( Q ) := re(Q) + e(Q) where re(Q) :. . . .a. :. ~ 22vq(F.) follows: let qoQ : F -~ F(Q) U oo be the coarsest place such that AQ factors through ~Q and such that the induced place F(Q) -~ N U oc has 2-divisible value group. T h e n e(Q) :=

0

if F(Q) is hereditarily Euclidean

1

otherwise.

Here a field K is called hereditarily Euclidean if K and all its formally real (finite) algebraic extensions are Euclidean [11].

347

N o t e 1.1. An ordered field (K, P ) is hereditarily Euclidean iff P extends uniquely to each formally real finite extension of K. (The one implication is clear. For the other, let L be an extension of K and let Q 6 S p e r L extend P. If Q ¢ L 2, there exists a E Q, a ~ L 2. Then Q (and therefore, P) extends in two ways to L I v e ] . ) T h e o r e m 1.2. Let "~ := ~ ( P ) and d = dim V. Then

sf(V) <_d+-~ -gF(V) 0 or d > 0. K d = ~ = 0 then a much simpler argument shows s f ( V ) = ~F(V) = 1. A similar remark applies to Theorem 1.7 below. R e m a r k s 1.4. (1) If F is real closed then ~ = 0. In particular, if F = R we get the usual bounds for s ( V ) = s F ( V ) and "g(V) = XF(V) [5], [12], [16]. (2) If R = R then Ap is trivial so re(P) = O. (3) If re(P) = 0 then F ( P ) = F, ~ = e(P) E {0, 1} and ~ = 0 iff F is hereditarily Euclidean. For any Tychonoff closed X C_ Sper A, let do(X) denote the supremum of all integers l >_ 0 such that there exists Q~ e Z , i = O,...,l, with Q0 c . . . c Ql [3] and let d l ( X ) denote the Krutl dimension of the ring A~ n {suppQ t Q E X} [1], [4]. L e m m a 1.5. dim Y = do(SperpA) = dl(SperpA). Proof. Let F be the real closure of (F, P) in R. The natural map A -+ T[x~ ..... z,] ( f l ..... f k ) induces a homeomorphism Sper -~[xl .....x,] ~_ S p e r v A [2, 2.17] and therefore, the kernel ( f l ..... f k )

--

.....x~] ~ T[x~ ..... x,] is precisely the ideal n{supp Q ] of the composite map A --~ T[x~ (fl,...,f~) I(vn~") Q E SperpA}. Since F C_F is algebraic, A ~ T[x~/(vnY,),...,x.] is integral so d l ( S p e r p A ) = dim-~[zt,...,x,] z(vn~") = dim V n F'~ and by [3, 7.5.6], dim V N-F ~ = do(SpereA). It remains only to show that dim V = dim V n-~n. By [3, 2.3.6], V N ~ n is a disjoint union of semi-algebredc sets $ 1 , . . . , Sm where each Si is semi-algebraically homeomorphic to (0,1) a' C_ ~ d , and by [3, 2.8.9], d i m Y N-F n = m a x { d l , . . . , d m } . By the Transfer Principle [3, 5.2.3], V is the disjoint union of the semi-algebraic sets (S~)R,..., (Srn)R and each (Si)R is semi-algebraically homeomorphic to (0, 1) d' C R d~. Therefore, by [3, 2.8.9], dim V = m a x { d 1 , . . . , dm} = dim V n F=. This completes the proof. L e m m a 1.6. The map • : V -+ S p e r p A given by x ~-* P~ {f E F[X1,...,Xn] [ f ( z ) >_O} induces an isomorphism C ~-~ ~-1 (C) o£ ~he Boolean algebra of cons~ruc6ble se~s in SperpA onto ~he Boolean a/gebra of seml-algebraic sets in V which are defined over F. =

Proof. It is clear that the map C ~ (I'-I(C) is a surjectlve homomorphism. injectivity follows from the Transfer Principle [3, 5.2.3].

The

It follows from (1.6) that a F ( V ) = s(SperpA) and ~ F ( V ) = "~(SperpA) where s, are defined as in [13]. Thus, to prove (1.2), we need only show that s(Sper~,A) < d + ~

348

and X(SperpA) < d(d + 1)/2 + (d + 1 ) ~ , where d = do(SperpA) = dl(SperpA). fact, we obtain a slightly more general result.

In

1.7. Let A be a finitely-generated F-algebra, T C F a proper preordering and X = SperTA. Then

Theorem

(t)

do(X)

:

dl(X).

this e~e, we c m d ( X )

::

do(X) = d l ( X ) the di.~¢~io~ or X . mso,

(2)

s(X) <_ d + -~

(3)

~(X) < d(d + 1)/2 + (d +

l)m

where d : : d(X) and -~ := sup{~(P) I P 6 SperTF}.

Proof. (1) is true if T is an ordering (by (1.5) applied in the case R is the real closure of (F, T).) Also, it is clear that any chain of orderings in SperTA contracts to an ordering P 6 SperTF. Thus, do(SperTA) --sup{do(SperpA) [ P 6 SperTF} =sup{dl(SpervA) [ P 6 SperTF] <_d~(Sp~rTA).

The remaining inequality dl(SRerTA) < sup{dl(SperpA) I P 6 SRerTF} is a consequence of the following: L e m m a 1.8. Suppose A is a commutative ring, X C Sper A is Tychonoff dosed and

p is a minimal prime over •{suppP I P E X } . Then p = s u p p P for some P G X . Proof. For each f 6 A \ p , there exists P E X such that f ~ suppP. By the compactness of X in the Tychonoff topology, N

{P6X]ftsuppP}¢O"

f6_A\p

Thus, there exists P E X with suppP C_ p and by the minimality of p, we have

supp P : p. Before we can complete the proof of (1.7), we must generalize a result in [7] on the behavior of the stability index under field extensions. T h e o r e m 1.9. Let F C K be £elds, T C F a proper preordering, X = SperTK. Then

s(X) <_ trdeg K ] F + where ~ := sup{'~(P) [ P e SperTF}.

Proof. We use the fan characterization of s(X) [10]:

s(x) = ~p{log~ IVl I v c x is a fan}. Let V _C X be a fan. By Brgcker's theorem on trivialization of fans [10, 5.13j, there exists a valuation v on K fully compatible with V such that the induced fan V on the

349

residue field K is trivial. Let T1 = ~ K2T. By the Krull-Baer Theorem [10, 3.12], we have

,.

v(K*)

--

and Iog2[V I E {0, 1}. (Replacing V by a bigger fan if necessary, we may even assume we have equality.) Fix an ordering Q E V and let P = Q cl F. It suffices to show

_. v(K*) -dzm2v--(~l. ) + log2 IV I < trdeg KIF + -~(P).

(1)

Since T~ = ~ K2T C Q and Q is v-compatible, we have v(Tl*) = 2v(K*) + v(T*). Let W = {c~ e v(K*) 12~(~ e v(F*) for some r > 0}. Then

a. .(K*) .(K*) .. 2~(K*) + W zm2 v(Tl") = dim2 2v(K*) + W + a z m 2 2 ~ +'v(T-*)"

(2)

If ai E K*, ki C Z not all zero are such that ~ k l v ( a i ) C v(F*) then, dividing by the highest power of 2 common to the ki, we have ~ liv(ai) E W, where the li ~ Z and at ., v(K*) ., v(K*) least one of the Ii is odd. This shows azmz v(F.) --> azm22vUFa'yyw and hence, •

v(K*)

trdeg K[F > d z r n z ~

(3)

+ trdeg K I F

_.

~(g*)

>_e,m~

2v(-F6¥ w

--

2,(K*)+w

where F is the residue field of VIE. Also, 2v(g*)+v(T*)

(4)

,. 2v(K*) + W a,m: 2 ~ - ~ ~ - ~ )

~ ~--

w 2W+v(T*) SO we have

W W ,. v(F*) = dim2 2W + v(T*) < ~ i ' ~ -< a'm~ 27--~)

where the last inequality follows from Brbcker's Index Formula [7, 3.2]. Since Q is v-compatible, P is compatible with VIE so the place ,~p : F --+ ~ [.J oo factors through the place p : F -~ F U c~ associated with V[F. Thus, we have (5)

v( F * )

6"

zrn2 ~ )

,.

vp(F * )

< azm22vp(F,) =

re(P).

Combining (2)-(5), we have (6)

,.

v(g*)

azm2~

< trdeg K]F + re(P).

This leaves the case where IV I = 2 and we have equality at (6). To prove (1) in this case, we must show e(P) = 1.

350 N

w

Since we have equality at (6), we must have equality at (3) so F C K is algebraic. Fix a finite extension F1 of F such that F1 C_ K and the two orderings in V C Sper K remain distinct when restricted to F I . W'e also have equality at (5) so re(P) = a*rn2 " v(F*) Thus the place F ~ IR U eo 2v-ggp~. induced by factoring Ap through p has 2-divisible value group. By the definition of the place hop : F ~ F ( P ) U oe, p factors through hop inducing a place F ( P ) -+ F U ~ . Let F1 be a finite unramified extension of F(P) having F1 as residue field. Then F1 has at least two distinct orderings so F ( P ) is not hereditarily Euclidean and therefore, e(P) = 1. This completes the proof. We return to the proof of (1.7). By [13, 5.1],

s(X) = sup{s(X(p)) I p C_ A is a real prime with X(p) ~ (~}. Let p C_ A be a real prime with X(p) ¢ 0. Since X = SperTA, X(p) = SperTF(p) so we have

(*)

s(X(p)) < trdeg F ( p ) l f + ~ = dim A/p + m

by (1.9). Since dim A/p <_ d(X) = d, s ( X ) <_d +-~. This proves (2). Also, by [13, 5.3], d

_< ~0

where si = sup{s(X(p)) ] p C A is a real prime and d i m A / p <_ i}. By (*), si <_ i +-~ for 0 < i < d. Thus, d "g(X) < ~-~(i + "~) = d(d + 1)/2 + (d + 1 ) ~ i=0

which proves (3). This completes the proof of (1.7). 2. s r ( R n) a n d gF(Rn). One would like to show the bound for sF(V) given in §1 is the best possible. Although we are not able to do this in general, we can in the special case where V = R n. We use the following: T h e o r e m 2.1. For each P 6 Sper F and for each finite k <_~ ( P ) , there exists a finite extension K o f F such that s(SperpK) >_ k.

Proof. Let P denote the pushdown of P along the place hop : F -~ F ( P ) U oo. Fix a finite extension ~ of F ( P ) so that T extends at least 2 ~(P) ways to K ( K exists by w (1.1).) Let W be a group with v(F*) C_ W C_ ½v(F*) and [~(-[PU[ = 2k' where v is the valuation associated with hop. Choose a x , . . . ,ak • P* such that ½v(al),..., ½v(ak)is a Z2-basis of W modulo v(F*). Then v extends (in fact, uniquely) to F [ v # ~ , . . . , x / ~ ] and F[v/'~,... , v / - ~ ] / F is totally ramified of degree 2 k. Let K be a finite unramified extensi°n °f F [ x / ~ , - - . , x / ~ ] h a v i n g K as residue field. Since W C v(K*)and 1 ~ 1 = 2 k, v(K*) = W.

351

Let T = ~ K 2 P

andT

= ~'~-~2"~. Suppose x l , . . . , x s

E K * , p l , . . . , p 8 E P* are

2 v(a3) + v(yi), where x 2i p i e T* . Sincev(K*) -- W,v(zd=Ei=1 j=l mij E {0, 1} and y~ E F*, and therefore, there exists u~ E K* with v(ui) = 0 such t h a t ~,2~rnll "" "akm i k YiPi" 2 2 E p* and assume v(ql) _< v(qi) x~pi = ~i~l Let qi . .aTil . . a m,, k YiPi such t h a t t =

for all i. T h e n t = qlu, where u = £ u~qi/ql. Clearly v(u) > O,-~ E -T = E-K2-fi and i=1

since - 1 ~ T , we must have v(u) = 0. Thus, v(t) = v(ql) = v(x~pl) and if v(t) = 0 then v(ql) = 0 s o t = ~ l g E P . T = T. This s h o w s T i s the pushdown o f T and v ( T * ) = 2 v ( K * ) + v ( P * ) = 2 W + v(F*) = v ( F * ) so I = v(K*) .(F*) = 2 k. It follows t h a t s( SperpIf) > k + s( Sper-p If) k k + e( P ), which completes the proof. Let the notation be as in T h e o r e m 1.2. Suppose, for each finite k < ~ , there exist primes P0 D . . . D Pd in A such t h a t the natural m a p s ,ki : A/pi ---+ A/pi-1 extend to discrete rank 1 places Ai : F ( p i ) ---+ F ( P i - 1 ) U oo and such t h a t s(SperpF(po)) >_ k. Then, we have a fan Z0 in SperpF(po) of order 2 k and inductively, a fan Zi in SperpF(pi) of order 2 i+k obtained by pulling back Zi-1 C SperpF(pi-1) along Ai, i = 1 , . . . , d. It follows t h a t SF(Y) k s(SperpF(pd)) k s(Zd) = d + k. For example, such a sequence of primes exists if V = R " (so d = n and A = F [ X 1 , . . . , X , ] . ) Namely, one can t a k e p d = (0), Pd-1 = ( Z d ) , . . . , p l = ( Z 2 , . . . , Z d ) , P0 = (f, X 2 , . . . ,Xd) where f E FIX1] such t h a t F [ X l ] / ( f ) ~- If, K as in (2.1). Thus we have the following: Theorem

2.2. s y ( R ~) = s ( S p e r p F ( Z l , . . . , X , ) ) = n +-~.

T h a t is, the bound for s f ( V ) given in (1.2) is best possible, at least if V = R ". The analogous problem for -~F(V) is more complicated. Motivated by Scheiderer's construction in [16] we look for a sequence of prime ideals P0 _D . . . __DPd in A such t h a t A~ : A/pi --+ A/pi-~ extends to a discrete rank 1 place ~i : F(p~) --* F ( p i - 1 ) U oo, i _> 1, and spaces of orderings

Zi @ Z~ C_ SperpF(pl),

i >_ O,

such t h a t Zi, Z~ are fans of order 2 i + ~ and Zi is the pull-back of Z~_ 1 along ,ki if i > 1. (We assume ~ < oo.) T h e n we take i

r,= U(zj e z;). j=0

i

s,=

U({P,} e z.b j=o

where Pj is a fixed element of Zj. To obtain Z~, i >_ 1, we want to choose additional primes qi-1 D Pi, i = 1 , . . . , d , with qi-1 ~ P0 such t h a t the natural m a p pi : A/pi --* A/qi-~ lifts to a discrete rank 1 place pi : F ( p i ) ~ F ( q i - 1 ) U oo and s(SperpF(qi-1)) = i - 1 + ~ . T h e n Z~ can be obtained by pulling back a fan of order 2 i - 1 + ~ in SperpF(qi-1) along pi. By our construction, the places pi, /~i a r e independent so Zi, Z~ lie in distinct Brgcker classes and therefore Zi U Z~ = Zi @ Z~ holds automatically if i >_ 1 (see [6], [14, 3.2].) We use the notation and theory in [13] to show t h a t Y/ is s a t u r a t e d and Si is basic closed in Y/. Let 2 i = {a E A I a > 0 on Yi}. T h e n a 2 E Ei for a E A \ P0. Thus,

352 if Q E cl(Yi), Q ~ ~, then supp(Q) D_ q j-1 for some j _< i so supp(Q) ¢ Po, and consequently Q ~ U(E2). This proves that Yi is closed in U(E 2) = Sper E~-IA. Since Yi(Pj) = Zj • Z} is saturated in SperpF(pj) for all j < i, this implies Yi is saturated (see [13, 2.3].) Also, it is clear that Si is closed in Yi and S(pj) = {Pj} @ Z} is basic in Y,(pj) = Zj ® Z} for all j < i so, by [13, 3.21, s~ is basic closed in Yi. The argument in [16] shows that at least ~ = 0 ( J + m) = i(i + 1)/2 + (i + 1)N inequalities (>) are required to describe Si in Yi. Namely, i + ~ of these inequalities, say f~ >_ 0 , . . . , fi+m >_ 0, are required just to describe {Pi} = Si A Zi in Zi = Yi N Zi (since Zi is a fan of order 2i+m.) Moreover, each fk is strictly negative at some Q c Zi and, of course, fk _> 0 at Q' where Q' D Q is the specialization of Q in Z~_ 1 (since Z~_ 1 c_ Si.) Thus fk = 0 at Q', that is, fk c Pi-1. This is true for k = 1 , . . . , i + ~ . Thus, none of the inequalities fl >_0,..., fi+-~ >_0 can contribute to the description of Si-1 = Si 71Yi-1 in Yi-1 = Yi C?Yi-1. Therefore, by induction on i, ~ j =i--1 0 ( 3 - + ~ ) additional inequalities i

are required for the description of Si-1 in Yi-1 so, altogether, ~ j = 0 ( ) + ~ ) inequalities are required to describe Si in Y/. Thus g(Yi) _> i(i + 1)/2 + (i + 1)~. In particular,

"g(SperpA) > 2(Yd) >_d(d + 1)/2 + (d + 1)~. To be able to apply this in the case V = R n, we assume

(,)

There exists a finite extension L of F with Zo • Z~ C_SperpL where Z0, Z~ are fans of order 2 m.

Assuming this, we can realize the above set-up in A = / ~ [ X I , . . . , Xd] as follows: take Pa = (0), Pa-1 = ( X a ) , . . . , P l = ( X 2 , . . . , X a ) , P0 = (f, X2,...,Xa) where f E F[X1] is such that F[X1]/(f) ~ L. The choice of qi-1, i = 1,...,d, is fairly arbitrary, for example, we could take qa-1 = (Xa + 1 ) , . . . , ql = (-X'2 + 1, X 3 , . . . , Xd) and qo = (I(X~ + 1 ) , X 2 , . . . , X d ) . The point is, with this choice of qi, qi-x ~ P0, r(qi-1) F(X1,...,XI_I) for i _> 2, and F(q0) ~ L so s(SperpF(qi_l)) = i - 1 + ~ , i = 1 , . . . , d . This proves: Theorem

2.3. If (F, P) satisfies (*) then 5F(R") = n(n + 1)/2 + (n + 1)~.

If (*) does not hold, we are still able to get a somewhat weaker result by replacing L by the field K constructed in (2.1) and taking Z0 = ~ (so we only require that SperpF(po) contains a fan Z~ of order 2m) and taking i

Yi = U ( Z j ® Z}), j=l

i

Si = U ( { P j }

® z;).

j=l

i Then, starting our induction at i = 1 instead of i = 0 yields-g(Y~) _> ~j=1(2 +-~) = i(i + 1)/2 + i ~ . Thus we have:

Theorem

2.4. For any F, n(n + 1)/2 + nm < -gF(Rn) < n(n + 1)/2 + (n + 1)~.

353

Remarks 2.5. (1) If ~ -- 0, (F, P) is hereditarily Euclidean so cannot satisfy (*) but, nonetheless, by (2.4) the conclusion of (2.3) is valid. (2) We have been assuming ~ < co but this is unnecessary. If ~ -- co, then 7F(R n) SF(R ~) = n + ~ = co, so the conclusion of (2.3) is true in this case as well.

>__

(3) For 0 < ~ < co, (F, P ) may or may not satisfy (*). Fixing an integer k, 0 < k < co, we have the following examples: (i) Take F to be the iterated formal power series field ~ ( ( t l ) ) . . . ((tk)) and denote by A : F --* R U co the naturally associated discrete rank k place. Any ordering P E Sper F is compatible with ), and ~ ( P ) = k. But (F, P ) does not satisfy (*) since, for any finite extension L of F , L has at most 2 k orderings.

(ii) Take F to be the rational function field R ( t l , . . . , tk) and take P E Sper F to be the restriction to r of any ordering on R ( ( t l ) ) . . . ((tk)). Again, ~ ( P ) ---- k, but now, according to [15, §4], (F, P) does satiny (*). Of course, having (*) fail does not necessarily imply that the conclusion of (2.3) fails. We know of no example where the conclusion of (2.3) fails. REFERENCES 1. C. Andradas, L. BrCcker, J.M. Ruiz, Minimal generation of basic open semi-analytic sets, Invent. Math. 92 (1988), 409-430. 2. E. Beeker, On the real spectrum of a ring and its application to semi-algebraic geometry, Bull. Amer. Math. Soc. 15 (1986), 19-61. 3. J. Bochnak, M. Coste, M.-F. Roy, Gdomdtrie Algdbrique RdeUe, Ergeb. der Math. 3 12, SpringerVerlag, Berlin Heidelberg New York, 1987. 4. L. BrScker, On the stability indez o] Noefherian rings, preprint. 5. _ _ , On basic semi-algebraic sets, Expositiones Math. (to appear). 6.--, Uber die Anzahl der Anordnungen eines kommutativen KSrpers, Arch. Math. 29 (1977), 458-464. 7. _ _ , Zur Theorie der quadratischen Formen iiber formal reelen KSrpern, Math. Ann. 210 (1974), 233-256. 8. M. Knebusch, C. Scheiderer, Einfiihrung in die reele Algebra, Fiedr. Vieweg & Sohn, Braunschweig/Wiesbaden, 1989. 9. T.Y. Lam, A n introduction to real algebra, Rocky Mountain J. Math. 14 (1984), 767-814. 10. _ _ , Orderings, valuations, and quadratic forms, CBMS Regional Conf. Ser. in Math. no. 52, Amer. Math. Soc., Providence, R.I., 1983. 11. _ _ , The theory of ordered fields, Ring Theory and Algebra III (ed. B. McDonald), Lecture Notes in Pure and Applied Math., col. 55, Dekker, New York, 1980, pp. 1-152. 12. L. MahC, Une dgmonstration Jlementaire du thdor~me de BrJcker-Scheiderer, preprint. 13. M. Marshall, Minimal generation of basic sets in the real spectrum of a commutative ring, preprint. 14. J. Merzel, Quadratic forms over fields with finitely many orderings, Ordered Fields and Real Algebraic Geometry (eds. D. Dubois and T. RCcio), Contemporary Math., col. 8, Amer. Math. Soc., Providence, R.I., 1982, pp. 185-229. 15. C. Scheiderer, Spaces of orderings of fields under finite extensions, Manuscripta Math. 72 (1991), 27-47. 16. _ _ , Stability index of real varieties, Invent. Math. 97 (1989), 467-483. DEPARTMENT OF MATHEMATICS AND STATISTICS, UNIVERSITY OF SASKATCHEWAN~ SASKATOON, CANADA STN 0W0

CONFIGURATIONS

O F AT M O S T 6 L I N E S O F R P a

V. F. MAZUROVSKII Ivanovo Civil Engineering Institute

An unordered (n; k)-configuration of degree m is defined to be an unordered collection of rrz linear k-dimensional subspaces of R P n. We associate with each configuration its upper and lower ranks, i.e. the dimensions of the projective hull and intersection respectively of all the subspaces of the configuration. The combinatorial characteristic of a configuration is, by definition, the list of upper and lower ranks of all its subconfigurations. Two configurations are said to be rigidly isotopic if they can be joined by isotopy which consists of configurations with the same combinatorial characteristics. It is obvious that the property of being rigidly isotopic is equivalence relation. The equivalence class of a configuration by this relation is called its rigid isotopy type. The space SPC~k of unordered (n; k)- configurations of degree rn is naturally isomorphic to m-th symmetric power of Grassmanian manifold G,,+l,k+~. A configuration is said to be non-singular if all its subspaces are in general position. The set GSPC~k of non-singular configurations is an open subset of manifold SPCnm,~,in Zanski topology. The set of all non-singular configurations of the same rigid isotopy type forms a connected component of GSPC~,,k (in strong topology). These connected components are called cameras of manifold SPC~k. In the paper [6] O. Ya. Viro enumerated the cameras of the spaces SPC~,,1 for m _< 5 and showed that non-singular (3; 1)-configurations of degree ___5 are determined up to rigid isotopy by the linking coefficients of the lines of a configuration. In [4] the author enumerated the cameras of SPC~,1 and proved that non-singular configurations of 6 lines of R P a are not determined up to rigid isotopy by the linking coefficients. The main purpose of the present paper is to describe the mutual position of the cameras in SPC~,1 for ra < 6. In particular we give a detailed proof of classification of non-singular (3; 1)-configurations of degree 6 up to rigid isotopy (in [4] this proof was only outlined). The author is grateful to O. Ya. Viro for posing the problem and fruitful discussions. § 1. BASIC CONSTRUCTIONS 1.1. L i n k i n g coefficients. In the next two sections we describe the constructions of O. Ya. Viro (see [61, [7]). Two disjoint oriented lines L~, L~ in the oriented space R P a have linking coefficient Ik(L~, L~) equal to +1 or - 1 (here the doubled linking coefficient of cycles L~ and L~ in the oriented manifold NP a is considered). Let L = {L1,L2,La} be an unordered non-singular configuration of three lines of the oriented space R P a, and let L~, L~, L~ be the same lines equiped with some orientations. The product lk(L~,L~) x

355

Ik(L~,L~)lk(L~,L~) denoted by Ik(L1,L2,La) does not depend on the choise of the orientations of Li,i = 1,2, 3, is preserved under isotopies of L, and changes under reversal of the orientation of R P a. Unordered non-singular (3; 1)-configurations L = ! {L1, L 2 , . . . , Lm} and L' = {L1, L 2I . . . L~} are said to be homology equivalent if there exists a bijection q0 : L ---+ L' such that for a fixed orientation of R P a Ik(Li,Lj,Lk) = Ik(~(Li), v(Lj), v(Lk)) with any i,j,k = 1 , 2 , . . . , m , i C j , i # k, j # k. 1.2. C o n s t r u c t i o n o f t h e j o i n o f t w o c o n f i g u r a t i o n s . Let A = { A 1 , . . . , Am} be an unordered configuration of k-dimensional subspaces of R P ~, and let B = { B 1 , . . . , Bm} be an unordered configuration of/-dimensional subspaces of N P ~. We suppose that R P ~ and N P s are imbedded into I~P n+8+1 as disjoint linear subspaces. If n and s are odd we suppose, in addition, that N P n, NP~, and NP "+~+1 are oriented, and linking coefficient of the images o f R P n and R P ~ in R P n+s+l equals +1. Let f : { 1 , . . . ,m} ---* {1,... ,m} be some bijection, and Ci be the projective hull of the images of Ai and Bf(i) in ~P'~+~+~,i = 1 , . . . , m . It is clear that C = {C1,---,Cm} is a ( n + s + l ; k + l + l ) configuration of degree m. The configuration C is called the join of A and B. A configuration is called an isotopy join if it is rigidly isotopic to the join of some two configurations. 1.2.1. L e m r n a . The mirror image of an isototy join is also an isotopy join. 1.3. D e g e n e r a t i o n a n d p e r t u r b a t i o n . Let s : [0,1] --* SPCnm,k be a path with the beginning at a point A such that the restriction s][0,1) is a rigid isotopy of A. If configurations A and A' = s(1) have distinct combinatorial characteristics, then s is called a degeneration of configuration A, and path s -1 is called a perturbation of A'. Two subspaces Ai and A 1 of a configuration A = { A 1 , . . . , Am} are said to be contiguous if either they coincide, or there exists a degeneration such that (1) its restrictions on the configurations { A 1 , . . . , Ai-1, Ai+l,..., Am} and { A 1 , . . . , A j-l, A i + a , . . . , Am } which are obtained by removing the elements Ai, Aj from A respectively are rigid isotopies, and (2) the subspaces corresponding to the subspaces Ai and Aj coincide in the result of this degeneration. 1.4. A d j a c e n c y g r a p h . A configuration is said to be 1-singular if all configurations rigidly isotopic to it form a codimension 1 subset in the configuration space. The set of all 1-singular configurations of the same rigid isotopy type is called a wall. Two 1singular configurations are said to be p-equivalent if they belong to walls which separate the same cameras. The set of all 1-singular comCigurations of SPC~, k will be denoted by GI SPCr,n,k. The mutual position of the cameras in the configuration space can be described by means of the adjacency graph (see [2]), whose vertices and edges are in one-to-one correspondence with the cameras and walls respectively, and two vertices representing some cameras are connected by an edge if and only if these cameras are adjacent to the wall corresponding to this edge. It may happen that the beginning of an edge coincides with its end, as in the following cases: a) if the configuration space has a boundary and the wall is contained in it; b) if the wall is a one-sided subset of the configuration space; c) if the wall is a two-sided subset, but has the same camera adjacent at each side.

356 Each of these cases corresponds to a loop in the adjacency graph. In cases b) and c) the wall is called inner one. 1.5. A f f i n e (3; 1)-configurations. By an unordered afffine (3; 1)-configuration of degree m we mean an unordered collection of m lines of R 3. The canonical imbedding R3 ~ ~ p 3 induces a map of the set of a/fine (3; 1)-configurations of degree m into SPC~, 1. This map is called the projective completion, and the image of an atone (3; 1)-configuration K under this map is called the projective completion of K . A rigid isotopy of an affine (3; 1)-configuration is an isotopy of this configuration which induces a rigid isotopy of its projective completion. Let Oxyz be the canonical Cartesian coordinate system in R 3. The planes of R 3 defined by the equations z = const will be called horizontal planes. The common line of the projective completions of horizontal planes is called the horizontal line of infinity. Consider the affine (3; 1)-configuration S which consists of the following lines f y = 0 [ z 1' {x----0 z=l'

{ y=x

z=-l'

(y=--x

Thepairsoflines {y=0 z=l'

z=-l"

{x=O z=l

and { y = x

z=-l'

y = - x will be called the frames of configuration S. Let S be the projective comple-

x = --1 tion of S. Consider a configuration obtained by adjoining to S several pairwise disjoint lines which have no common points with :~. Such configuration is said to be framed. The subconfiguration S of a framed configuration will be called the skeleton, the other lines will be called free lines. We call the sliding translation the a ~ n e transformation given by the formulas x ~ = x + az, y' = y + bz, z t -- z, where a and b are some numbers. In what follows we suppose that X3 is canonically imbedded into ~ p 3 . § 2. JOIN CONFIGURATIONS OF LINES OF R P 3

2.1. Classification of non-singular isotopy join configurations of 6 lines of R P 3 up to rigid isotopy. Let L 1 and L 2 be two oriented disjoint lines in R P 3 with positive linking coefficient. Let A j be some pairwise different points of LJ, j = 1,2, i = 1, 2 , . . . , m, such t h a t the increase of index i agrees to the orientation of L j. Consider a permutation a of degree m. The lines passing through points A~ and Aa( 2 0 form a nonsingular join (3; 1)-configuration of degree m, which will be denoted by jc(a). It is easy to see that up to rigid isotopy this construction provides all isotopy join configurations of m lines of R P 3. It is also clear that the map Sm ~ rco(GSPC~,I ) which assigns to a permutation a C Sm the rigid isotopy type of jc(a) is well defined. We denote this map by >r. In what follows ( a l , a 2 , . .. am) denotes permutation ( 1 '

0-1

2

...

0"2

• . •

rn) am

"

2.1.1. L e m m a . 1) Let # and ~ be permutations of degree m preserving the natural cycle order. Then ~ ( , . 0 - . v) = ~(0-). 2) x(0- -1) = ~:(0-). 3) x ( ( m , m - 1 , . . . , 2 , 1 ) - a - (re, m - 1 , . . . , 2 , 1 ) ) = x(0-) .

357

4) Suppose that permutation a ---- ( a l , . . . , a i - 1 , a i , . . . , a j , a j + l , . . . , a m ) satisfies the following condition: for any integer p, with rain ak < p < max ak, there exists an i<_k<_j -- -- i<_k<_j index l(i <__l < j) such that cq = p. Leta= min.ak,(Ti,...,rj)---(j--i,...,2,1).(al--a+l,...,aj--a+l)(j--i,... 2,1), i
a)

jc(1,2,3,4,8,6), jc(1,2,3,4,6,5), jc(1,2,3,5,6,4), jc(1,2,4,3,6,5), jc(1,2,4,6,3,5), jc(1,2,5,6,3,4); b) mirror configurations jc(1, 2, 3, 6, 5, 4), jc(1,3, 5, 2, 6, 4), jc(1,2, 4, 6, 5, 3); c) the mirror images of the configurations of a): jc(6, 5, 4, 3, 2, 1), jc(5, 6, 4, 3, 2, 1), jc(4,6,5,3,2, 1), jc(5, 6, 3,4, 2, 1), jc(5,3,6,4, 2,1), jc(4,3,6,5,2,1). Proof of Theorem 2.1.2 is reduced to sorting all permutations of $6. First we split $6 into union ofdisjoint classes of permutations with homology equivalentcorresponding join configurations. Then, using Lemma 2.1.1, weprove that elements of each of these classes define rigidly isotopic non-singular join configurations. 2.1.3. R e m a r k . A similar consideration shows that any non-singular isotopy join (3; 1)-configuration of degree < 5 is rigidly isotopic to one of the following pairwise non-isotopic non-singular join configurations:jc(1), jc(1, 2), jc(1, 2, 3), jc(3, 2, 1), jc(1,2,3,4), jc(1,2,4,3), jc(4,3,2,1), jc(1,2,3,4,5), jc(1,3, 4,5,2), jc(1,4;5,2,3), jc(1, 3, 5, 2, 4), jc(3, 2, 5, 4, 1), jc(2, 5, 4, 3, 1), jc(5, 4, 3, 2, 1) (compare with Viro's classification in [6]). 2.2. Classification of 1-slngular isotopy join (3; 1)-configurations of degree _< 6 up to rigid isotopy. Consider again two oriented disjoint lines L 1 and 132 in R P a with positive linking coefficient. Let A], • .., A,~_ 1 1,Am 1 be different points of L 1, and 2 A~,... Am_ 1 be different points of L 2 such that the increase of the lower indices of the points agrees to the orientations of L 1 and L 2. Consider a map f from { 1 , . . . , m - 1, rn} onto { 1 , . . . , m - 1} and connect points A~ and AI(i ) 2 by lines, i = 1,.. ., rn - 1, m. The obtained 1-singular join (3; 1)-configuration of degree m is denoted by sjc(f). The set

358

of all surjections of { 1 , . . . , m - 1,m} onto { 1 , . . . , m - 1} will be denoted by Sin. It is clear that the map ~: S m --+ rco(G1SPC~,I) which assigns to a map f e Sra the rigid isotopy type of sjc(f) is well defined. Since a surjection f can be represented by the table

(

) ..."" f(mm-l-1)

f(m)m ) w e s h a l l a l s ° u s e t h e s y m b ° l ( f ( 1 ) ' ' ' ' '

,f(m-1),f(m))

to denote f. 2.2.1. L e m m a . 1) If # and lJ are permutations of degree m and m - 1 respectively preserving the natural cyclic orders, then k(#. f . ,) = k(f); 2) k ( ( m , m - - 1 , . . . , 2 , 1 ) - f . ( m - 1 , . . . , 2 , 1 ) ) = ?4(f); 3) Suppose that surjection f = ( f ( 1 ) , . . . , f(i - I), f ( i ) , . . . , f(j), f ( j + 1 ) , . . . , f(m)) satisfes the following conditions: (1) the dements f(i),.., f ( j ) are parwise different, and (2) for any integer p, with min f(k) <_ p <_ ir~xjf(k), there exists an index l i
_

_

(i <_1 <_j) such that f ( t) = p. Let a = min f(k), ( ~ i , . - . , aj) = (j - i , . . . , 2, 1). (f(i) - ~ + 1,..., f ( j ) - ~ + 1) iSk<_j •(j - i , . . . ,2, 1), and g = (g(1),... ,g(i - 1), g(i),...,g(j), g(j + 1 ) , . . . , g ( m ) ) , where g(s) = f(s) for any s = 1 , . . . , j - 1 and any s = j + 1 , . . . , m and g(k) = ak for any k = i, i + 1 , . . . , j - 1,j, where ak = ak + o~- 1. Then &(g) = &(f). Proof is analogous to the proof of Lemma 2.1.1. 2.2.2. L e m m a . The result of any perturbation of 1-singuIar join (3; 1)-confguration is a non-singular isotopy join (3; 1)-cont~guration. 2.2.3. T h e o r e m . Two 1-singular isotopy join confgurations of < 6//nes of IRP 3 are rigidly isotopic if and only if they are p-equivaJent. a) Any 1-singular isotopy join (3; 1)-comqguration of degree 2 is rigidly isotopic to the join confguration sj c(1, 1). b) Any 1-singular isotopy join (3; 1)-cor~guration of degree 3 is rigidly isotopic to the join confguration sjc(1, 1, 2). c) Any 1-singular isotopy join (3; 1)-confguration of degree 4 is rigidly isotopic to one d the following 3 pairwise non-isotopic join cont]guratlons: sjc(1, 1,2, 3), sic(l, 2, 1, 3), sjc(3, 2,1,1). Onty sjc(1, 2,1, 3) is a mirror configuration. d) Any 1-singular isotopy join (3; 1)-cont~guration of degree 5 is rigidly isotopic to one of the following 8 pairwise non-isotopic join confgurations: 1) sjc(1,1,2,3,4), sjc(1,2,1,3,4), sjc(1,3,1,2,4), sjc(1,1,3,4,2); 2) the mirror images of the configurations from 1): sjc(4, 3, 2, 1, 1), sjc(4, 3, 1, 2, 1), sjc(4,2,1,3,1), sjc(2,4,3,1,1). e) Any 1-singular isotopy join (3; 1)-configuration of degree 6 is rigidly isotopic to one of the following 31 pairwise non-isotopic join comqgurations: i) sjc(1,1,2,3,4,5), 8jc(1,2,1,3,4,5), sjc(1,1,2,3,5,4), sjc(1,2,3,1,4,5), 8jc(1,1, 3,4,5,2), sjc(1,1,2,4,5,3), sjc(1,3,1,2,4,5), sjc(1,2,1,3,5,4), sjc(1,3,4,1,5,2), sjc(1,1,3,2,5,4), 8jc(1,2,4,1,3,5), sjc(1,2,5,1,3,4), sjc(1,3,1,4,2,5), sjc(1,3, 1,4, 5, 2); ii) mirror contlgurations sic(l, 1, 3, 5, 2, 4), sjc(1, 2, 3, 1, 5, 4), sjc(1,2, 4, 1,5, 3); iii) the mirror images of the comfigurations from i): sjc(5,4,3,2, 1, 1), sjc(5,4, 3,1,2,1), sjc(4,5,3,2,1,1), sjc(5,4,1,3,2,1), sjc(2,5,4,3,1,1), sjc(3,5,4,2,1,1),

359

sjc(5,4, 2, 1, 3, 1), sjc(4, 5,3, 1, 2, 1), sjc(2,5,1,4,3,1), sjc(4,5,2,3,1,1), sjc(5,3,1,4,2,1), ~jc(4,3,1,5,2,1), sjc(5,2,4, 1,3, 1), sjc(2,5,4,1,3,1).

Proof of Theorem 2.2.3 is reduced to sorting all elements of Sm for rn < 6. At the beginning, using Lemma 2.2.2, Remark 2.1.3, and Theorem 2.1.2, we split ~5,~(rn _< 6) into union of disjoint classes consisting of maps with p-equivalent corresponding 1singular join configurations. Then, using Lemma 2.2.1, we prove that elements of each of these classes define rigidly isotopic 1-singular join configurations. 2.3. A d j a c e n c y g r a p h o f SPC~, 1 for rn < 5. 2.3.1. L e m m a . Any 1-singular (3; 1)- configuration of degree ~_ 4 is an isotopy join.

Proof. It is sufficient to prove the lemma for the case of 1-singular (3; 1)-configurations of degree 4. Let K = {K1, K2, K3, K4} be a 1-singular (3; 1)- configuration. W'e assume that K3 Cl 1(4 7t 0. Let Q be the quadric containing disjoint lines K1,/(2, and K3. Up to a small rigid isotopy of/(4 it can be assumed to intersect Q in two different points A and B. Consider the generatrices L 1 and L 2 of Q which pass through points A and B respectively and belong to the family of generatrices dual to the family containing K1, K2, and K3. It is clear that K i f l LJ 7t 0 for any i = 1, 2, 3, 4 and j = 1,2. 2.3.2. L e m m a . If there exists a line intersecting all lines of a given 1-singular (nonsingular) (3; 1)-configuration K, then K is an isotopy join.

Proof. (This proof is due to 0. Ya. Viro). Let line L 1 intersect all lines of the given (3; 1)-configuration K. Consider a projective transformation of ]RP 3 which preserves the orientation of R P 3 and puts L 1 on the horizontal line at infinity. Let K ~ be the image of K under this transformation. It is easy to see that the afl~ine parts of lines of K I lie on horizontal planes. Let line L 2 be transversal to the horizontal planes and, in the case when K is a 1-singular configuration, intersect the two lines of K ~ meeting each other. Obviously, every other line of K ~ can be translated (in its horizontal plane) so as to intersect L 2. After that we shall obtain a join configuration K ' , which is evidently rigidly isotopic to the given configuration K. 2.3.3. L e m m a . Any 1-singular (3; 1)-configuration of degree 5 obtained by a perturbation of a framed configuration is an isotopy join.

Proof is reduced to sorting all possible locations of the free line of a framed colffiguration and using Lemma 2.3.2. 2.3.4. L e m m a . Any 1-singular (3; 1)-configuration of degree _< 6 either is an isotopy join, or can be obtained by a perturbation of a framed cor~guration.

Proof uses induction on the number of the lines of the configuration. Lemma 2.3.1 provides the basis of the induction. Assume that the statement is true for 1-singular (3; 1)-conflgurations of degree < 6. Let K ---- {KI, K2,/(3,/(4, Ks,/(6 } be a 1-singular (3; 1)-configuration. Assume that K1 A K3 ~ O. If lines K1 and /(3 of K are contiguous, then K is an isotopy join due to the inductive hypothesis and Lemmas 2.2.2 and 2.3.3. Let lines K1 and K3 be not contiguous. Then there exists a subconfiguration K " of K which is a 1-singular mirror (3; 1)-configuration of degree 4. We can assume, without loss of generality, that

360

K'" = {/~1, K2, K3,/x~4}. Let Q be the quadric containing the disjoint lines K2, Ks, and /(4. Up to a small rigid isotopy of K1 it can be assumed to intersect Q in two different points A and B. Let L 1 and L 2 be the two generatrices of Q which belong to the family dual to that containing K2, K3, and K4 and pass through points A and B respectively. It is clear that Ki A L j ~ (g for any i = 1,2,3,4 and j = 1,2. Let K1 gl/(3 = A. Consider a projective transformation of N P 3 which preserves the orientation of R P 3 and puts L 2 on the horizontal line of infinity. After that we shall have the following situation in N 3 (canonically imbedded into N P 3) : the images of lines K1 and Ks have a single common point, the images of the lines K1, K2, /(3, and I~24 lie on horizontal plazles and intersect the image of L 1. Consider a sliding translation such that the image of L 1 is perpendicular to a horizontal plane. Let K~ be the image of the affine part of Ki after the transformations given above, where i = 1, 2 , . . . , 6. It is easy to see that lines K~ and K~ have a single common point, lines K~, K~, K~, and I~'~ lie on horizontal planes and intersect a vertical line (i.e. a line perpendicular to a horizontal plane). We can assume, without loss of generality, that K~ lies over K~, and line h~ lies over KI and K~. Consider the orthogonal projection of the affine configuration K ' = {K~, K~, K~, K~, K~, K~} onto the plane perpendicular to K~. If this projection differs from the cases a) and b) below, then it is clear that configuration K' can be obtained by a perturbation of a framed configuration.

0

0 ......

S ~ K/'$ Kz t \

Kx

,/K;, ..........

Kta

In cases a) and b) it can be proved, using Lemma 2.3.2 and sorting all possible locations of K~ and K~ in respect to K~ and K~, that K' is an isotopy join. The next theorem is a consequence of Lemmas 2.3.4 and 2.3.3. 2.3.5. T h e o r e m . Any 1-singular (3; 1) - comqguration of degree _< 5 is an isotopyjoin. 2.3.6. R e m a r k . Due to Theorem 2.3.5 and Lemma 2.2.2 any non-singular (3; 1) configuraton of degree _< 5 is an isotopy join. Together with Remark 2.1.3 it yields the classification of non-singular (3; 1) - configurations of degree _< 5 up to rigid isotopy obtained by O.Ya.Viro in [6]. In what follows [K] denotes the rigid isotopy type of configuration K. The following statements are consequences of Theorems 2.3.5 and 2.2.3. 2.3.7. T h e o r e m . Two 1-singular conIigurations of<_ 5 lines o f N P 3 are rigidly isotopic if and only if they are p-equivalent. Any 1-singular (3; 1) - configuration of degree _< 5 is rigidly isotopic to one of the pairwise non-isotopic join comqgurations given in Theorem

z2.s a)-d).

361 2.3.8. T h e o r e m . a) The adjacency graph of SPC2 : has one vertex and one edge-loop, which corresponds to the one-sided inner wall. b) The adjacency graph of SPCg,: is shown on the diagram below.

[sic (*,i,~)]

ti~(i,~,~)]

IL

[]c(~,a,i)]

c) The adjacency graph of SPC4,: is the graph presented below. The loop corresponds to the one-sided inner wail.

[s]c(~,2,i,l)] I

[j~O,~,~,~)]

t~(~,~,~,i)]

d) The adjacency graph of Spch,1 is the following.

T [jc(~,2,s,~,5)]

[S]¢'L~ 'T~'['i 'i)] [ []c(5,t~,~,2,i)]

362

§ 3. KAUFFMAN POLYNOMIAL OF NON-SINGULAR CONFIGURATIONS OF LINES OF ]t~P 3

3.1. Polynomial invariant of non-singular (3; 1)-configurations. Yu. V. Drobotukhina in [1] defined an analogue of Jones p o l y n o m i a l for links in R P a. T h i s p o l y n o m i a l was defined b y m e a n s of t h e s t a t e m o d e l analogous to K a u f f m a n ' s m o d e l [3] for the J o n e s p o l y n o m i a l of links in S 3. This c o n s t r u c t i o n yields the b r a c k e t e d K a u f f m a n p o l y n o m i a l of links in R P 3. This p o l y n i m i a l is not an isotopy invariant of links in ]RP 3, since it is not p r e s e r v e d u n d e r tile R e i d e m e i s t e r m o t i o n f~l of link d i a g r a m . Nevertheless, it is an rigid isotopy invaxiant of non-singular configurations of p r o j e c t i v e lines, since in the process of rigid isotopy this R e i d e m e i s t e r m o t i o n does not occur. This p o l y n o m i a l will be called the K a u f f m a n p o l y n o m i a l of a non-singular configuration of lines of R P 3.

3.2. Insufficiency of the linking coefficients. 3.2.1. Theorem. The non-singular (3; 1) - configuration M, afBne part of which is shown on d i a g r a m 1 in Appendix, and its mirror i m a g e are homology equivadent, but not rigidly isotopic.

Pro@ T h e configuration M a n d its m i r r o r i m a g e can be distinguished by K a n f f m a n p o l y n o m i a l , which for M is equal to - A 15 -t- 6A 11 + 6A 9 - 5A 7 - 6A 5 + 10A a + 16A + A - I - 10A -3 + 10A -7 + 5A - 9 , a n d for its m i r r o r i m a g e is equal to 5A 9 + 10A 7 - 10A a + A + 16A -1 + 10A - a - 6A -5 - 5A -~ + 6A -9 + 6A -11 - A -15. 3.2,2. Theorem. The non-singular (3; 1)-configuration L, aft/he p a r t of which is shown on d i a g r a m 2 in Appendix, and n o n - s i n g u l a r join (3; 1)-configuration jc(1, 2, 5, 6, 3, 4) are homology equivalent, but not rigidly isotopic.

Pro@ T h e s e two non-singular (3; 1) - configurations have different K a u f f m a n p o l y n o mials: for L it is A 17 - 5A 13 + 15A 9 + 10A 7 - 13A ~ - 12A a + 15A + 22A -1 _ A - a _ 1 2 A - ~ + A -7 + 8A -9 + 3A -11, a n d for jc(1, 2, 5, 6, 3, 4) it is A 13

+ A 11 + 4A 7 + 7A 5 + 3A 3 + 2 A - ' + 5A -3 + 3A -5 + 2A -9 + 3A - 1 I + A -13.

We d e n o t e t h e m i r r o r images of the configurations L and M by L ' a n d M I respectively. 3.2.3. Corollary. isotopy joins.

T h e non-singular (3; 1) - configurations M , L, M ~, a n d L t are not

363

§ 4 . CONFIGURATIONS OF 6 LINES OF N P 3 4.1. S i n g u l a r f r a m e d c o n f i g u r a t i o n s o f 6 lines o f N P 3. Consider the projection of the skeleton of the affine part of a framed configuration onto a horizontal plane / \

/ /

/ \

/

/

\ \ \

Here the projection of the frame in the horizontal plane z = 1 is shown by the continuous lines, the projection of the frame in the horizontal plane z = - 1 is shown by the dash lines. The frames of the skeleton divide the horizontal planes z = 1 and z = - 1 into four open domains, which we denote as it is shown below

Xt

Xt

Z=-£ We denote by {Xl, Yj} the union of all lines of R 3 which intersect the open domains Xi and Yj of the horizontal planes z = 1 and z = - 1 , where i,j = 1,2,3,4. It is easy to show that {XIYj} N {XkYt} # O for any i,j, k, l = 1, 2, 3, 4. Thus there exists a degeneration of any afflne framed configuration of 6 lines which keeps the skeleton fixed, and such that the images of t h e free lines of the configuration after this degeneration have a single common point and do not intersect the skeleton. T h e result of such a degeneration will be called an affine singular framed configuration of 6 lines. T h e free lines of an affine singular framed configuration are said to intersect in the middle part if their point of intersection lies between the horizontal planes z = 1 and z = - 1 . Let Z - l , 1 = {(x,y,z) E N 3 1 - 1 < z < 1}, and let i,j,k,l be some fixed elements of the index set such that {XiYj}n{XkYt}NZ-I,1 # 0. Consider the space X consisting of ordered configurations (L1, L2) of two lines of N 3 such that L1 C {XiYj}, L2 C {XkYt}, and La and L2 intersect at a point of Z - l , a .

Space 2( with the naturM topology is arcwise connected. Proof. L e t / 3 = {XiYj} f~ {XkYt} fl Z-a,1. It is clear that B is an open convex subset of

4.1.1. L e m m a .

R a. Fix some point b E/3. Let .Tb be the set of all configurations of 2( the lines of which intersect at b. It is easy to see that the subset .Tb of X is arcwise connected. Consider the m a p p : 2( ~ / 3 which assigns to a configuration of X the point of intersection of its lines. It is not difficult to prove that the bundle (X',p,/3) is topologically trivial. Since

364

the fibre and the base of this bundle are arcwise connected, the total space 2( is also arcwise connected. 4.1.2. L e m m a . Any atone singutar framed comqguration of 6 lines is rigidly isotopic to an a/~ne singular framed configuration of 6 lines the free lines of which intersect in the midd/e part.

Pro@ If the free lines of an atone singular framed configuration of 6 lines do not intersect in the middle part, then we consider a projective transformation which preserves the orientation of II{Pa and moves the plane of infinity onto the horizontal plane z = 0. Introduce the following notation: the symbol [XiYj] denotes the set of all affine singular framed (3; 1) - configurations of degree 6 the free lines of which intersect in the middle part and one d them is contained in {XiYj}, where i,j E {1,2,3,4}. The projective (3; 1) - configuration of degree 6 will be called a singular framed configuration if it is the projective completion of an afflne singular framed configuration of 6 lines. Two projective configurations are said to belong to the same coarse projective type if either they belong to the same rigid isotopy type, or one of t h e m is rigidly isotopic to the mirror image of the other. 4.1.:3. L e m m a .

Any singular framed cont~guration of 6 Iines of N P s belongs to the

coarse projective type of the projective compietion of a configuration of [X1 }1"1]or [X1Y2]. Pro@ It immediately follows from s y m m e t r y of the skeleton. 4.2. Classification o f 1-singular (3; 1) - configurations o f degree 6 up to rigid i s o t o p y . We shall denote a free line of an affine framed (3; 1) - configuration of degree 6 intersecting domain Xi of the horizontal plane z = 1 and domain Yj of the horizontal plane z = - 1 , where i,j E {1,2,3,4}, by XIY/. 4.2.1. L e m m a . Any 1-singular (3; 1) - configuration of degree 6 either is an isotopy join, or belongs to the coarse projective type of a 1-singu/ar cont~guration which is obtained by a perturbation of the projective completion of one of the followlng three a n n e framed con~gurations of 6 lines:

,,J/ ,--....

365

¢)

Proof. We sort all possible cases, using Lemmas 2.3.4, 4.1.1, 4.1.2, 4.1.3, 2.3.2. and 1.2.1. 4.2.2. L e m m a . Any 1-singular (3; 1) - configuration of degree 6 either is an isofopy join, or belongs ~o the coarse projective type of fhe projecfive completion of one of the following three aiFme configurations of 6 lines:

t) I~ ~

4

:K

~X~, 2~.~ ~L" ~,,

~L~~ ~,(~~'~,/.~"~,~

366

0

Z~4t >

(The notation z~cl > z~c3 > zlc~ =

> Z

"-

z~c4 m e a n s

that horizontal line 1Ci lies over

horizontal tine K3, line K3 lies over intersecting horizontal lines }C2 and K4). Proof. Perturbing the framed configurations of Lelnma 4.2.1 we obtain twelve 1-singular configurations. Three of them are represented above. The free lines of the other nine configurations either can be moved so as to intersect the line I xY = 0 ' or can be put onto k

horizontal planes: hence, these configurations are isotopy joins due to Lemma 2.3.2. To complete the proof we use Lemma 1.2.1. 4.2.3. L e m m a . 1-singular (3; 1) - configurations 2C, £, and M of Lemma 4.2.2 are not isotopy joins and belong to different coarse projective types.

Proof. K belongs to the inner wall which separates the camera containing jc(1, 3, 5, 2, 6, 4). Configuration L~ belongs to the wall separating the cameras containing jc(1, 2, 4, 6, 3, 5) and L. A4 belongs to the wall separating the cameras containing jc(1, 3, 5, 2, 6, 4) and M. Indeed, similar to Lemma 4.2.2, one can see that after one (or both, in the first case) of the two possible perturbations of these configurations we obtain an isotopy join: its rigid isotopy type can be calculated using Theorem 2.1.2. Therefore, due to Theorem 2.2.3, K, £ and M can not be p-equivalent to a 1-singular isotopy join (3; 1)-belong to different coarse projective types. 4.2.4. L e m m a .

1-singular (3; 1) - cont~gurations lC, f., and A4 of Lemma 4.2.2 are not

mirror. Proof. ]C is not mirror, since its non-singular subconfiguration consisting of lines •I,

]C3, ]C5, /C6, is not mirror. 3/I is not mirror, since, due to Theorem 3.2.1, it is not p-equivalent to its mirror image, f is not mirror for the same reason. As a consequence of Lemmas 4.2.2, 4.2.3, and 4.2.4 we obtain the following theorem.

367

4.2.5. T h e o r e m . Any 1-singular (3; 1) - configuration of degree 6 is either an isotopy join (see Theorem 2.2.3), or rigidly isotopic to one o£ the following six pairwise nonisotopic configurations: 1) configuration ]C, 2) configuration L, 3) configuration .MI, 4) the mirror image of ~, 5) the m/rror image of f~, 6) the mirror image of .A/(. 4.2.6. C o r o l l a r y . There exist exactly 37 rigid isotopy types of 1-singular (3; 1) configurations of degree 6. 4.3. C l a s s i f i c a t i o n o f n o n - s i n g u l a r c o n f i g u r a t i o n s o f 6 l i n e s o f R P 3 u p t o r i g i d

isotopy. Any non-singular (3; 1) - configuration o f degree 6 either is an isotopy join, or belongs to the coarse projective type of the projective completion of one of the following two atone configurations: 4.3.1. L e m m a .

%., i)

ii)

Ms/

M~

Li

/Lw ZL~.~'ZLz,>ZL~) ZL~

Ha ZML>ZM~) ZMz) Z M~

(It is easy to see that the configurations of i) and ii) are rigidly isotopic to L and M respectively (see, respectiveIy, dia.2 and 1 in Appendix)). Proof. Perturb the 1-singulax (3; 1) - configurations of L e m m a 4.2.2 and use Lemmas 2.3.2 and 1.2.1.

L e m m a . Non-singular (3; 1) - con~gurations M, L, and L' are not homology equivalent. As a consequence of Lemmas 4.3.1, 4.3.2, Theorems 3.2.1, and Corollary 3.2.3 we obtain the following theorem.

4.3.2.

4.3.3. T h e o r e m . Any non-singular (3; 1) - configuration o£ degree 6 is either an isotopy join (see Theorem 2.1.2), or rigidly isotopic to one of the following four pairwise non-isotopic configurations:

368

1) 2) 3) 4)

configuration L, configuration M, the mirror image of L, the mirror i m a g e o f M .

4 . 3 . 4 . C o r o l l a r y . T h e r e exist exactly 19 rigid isotopy types of non-singular (3; 1)configurations of degree 6.

Non-singular configurations of 6 lines of R P 3 are rigidly isotopic if and only if their Kauffman polynomials are equal. To prove this theorem it is sufficient to show that the configurations of each of the 19 rigid isotopy types o f non-singular (3; 1) - configurations o f degree 6 have different Kauffman polynomials. The complete list o f these polynomials can be found in [5].

4.3.5, Theorem.

4 . 4 . A d j a c e n c y g r a p h o f SPC6,1. D e n o t e t h e m i r r o r images of the 1-singular (3; 1) configurations ]C, f , a n d A4 b y / C , f e a n d A/I t respectively. Theorem. Adjacency g r a p h o f SPC~, 1 is shown on diagram 3 in Appendix. All loops of the graph correspond to one-sided unner walls.

4.4.1.

Proof. T h e t h e o r e m follows f r o m T h e o r e m s 4.3.3 a n d 4.2.5. REFERENCES [1] Yu. V. Drobotukhina, An analogue of the Jones polynomial for links in ~p3 and a generalization of the Kauffman-Murasugi theorem, Leningrad Math. J. 2 no. 3 (1991). [2] S. M. Finashin, Configurations of seven points in ~p3 Lect. Notes in Math. 1346 (1988), 501-526. [3] L. Kauffman, State models and Jones polynoraial, Preprint, 1986. [4] V. F. Mazurovskii, Configurations of six skew lines, J. Soviet Math. 52 no. i (1990), 2825-2832. [5] V.F. Mazurovskii, Kauffman polynomials of non-singular configurations of projective lanes,Russian Math. Surveys 44 no. 5 (1989), 212-213. [6] O. Ya. Viro, Topological problems concerning lines and points of three-dimensional space, Soviet Math. Dokl. 32 no. 2 (1985), 528-531. [7] O. Ya. Viro and Yu. V. Drobotukhina, Configurations of skew lines, Leningrad Math. J 1 no. 4 (1990), 1027-1050.

O

Ovq

O

~.

O

o

Lid

C~ a~ C~O

0

0

Y

=, 0

371

Dctt,z,~,~,s,6)]

)]

t~3 )]

~L']/

T DIA. 3. Adjacency graph of

SPC6~,I.

E x t e n s i o n s of R o k h l i n c o n g r u e n c e for c u r v e s o n

surfaces Grigory Mikhalkin Abstract The subject of this paper is the problem of arrangement of a real nonsingular algebraic curve on a real non-singular algebraic surface. This paper contains new restrictions on this arrangement extending Rokhlin and Kharlamov-GudkovKrakhnov congruences for curves on surfaces.

1

Introduction

If we fix a degree of a real nonsingular algebraic surface then in accordance with Smith theory there is an u p p e r b o u n d for the total Z2-Betti n u m b e r of this surface (the same thing applies to curves). T h e most interesting case according to D.Hilbert is the case when the u p p e r b o u n d is reached (in this case the surface is called an M-surface). Let A be a real nonsingular algebraic projective curve on a real nonsingular algebraic projective surface B. If A is of even degree in B then A divides B into two p a r t s / 3 + and B_ (corresponding to areas of B where polynomial determining A is positive and negative). Rokhlin congruence [1] yields a congruence modulo 8 for Euler characteristic X of B+ provided t h a t (i) B is an M-surface

(ii) A

is an M-curve

(iii) B+

lies wholly in one component of B

(iv) rk(in.:

Ha(B+; Z2) ~ Ha(B; Z:)) = 0

(v) if the degree of polynomial determining A in B is congruent to 2 modulo 4 t h e n all c o m p o n e n t s of B containing no components of A are contractible in Pq 1 K h a r l a m o v - G u d k o v - K r a k h n o v [2],[3] congruence yields congruence m o d u l o 8 for x(B+) under a s s u m p t i o n s (i),(iii),(v) and either a s s u m p t i o n t h a t A is an (M-1)-curve and rk(in.) = 0 or a s s u m p t i o n t h a t A is an M-curve and rk(in.) = 1 a n d all c o m p o n e n t s of A are Z2-homologically trivial. Recall t h a t Rokhlin congruence for surfaces yields

1Remark. Paper [1] contains a mistake in the calculation of characteristic class of covering Y ---*CB. It leads to the omission (after reformulation) of (v) in assumptions of congruence. The proof given in [1] really uses (v).

373

congruence modulo 16 for x(B) so a congruence modulo 8 for x(B+) is equivalent to a congruence modulo 8 for x(B-). One of the properties of M-surfaces (similar to the property of M-curves) remarked by V.I.Arnold [4] is t h a t a real M-surface is a characteristic surface in its own complexification (for M-curves it means that a real M-curve divides its own complexlfication since a complex curve is orientable).We shall say that a real surface is of a characteristic type if it is a characteristic surface in its own complexification.Note that the notion of characteristic type of surfaces is analogous to the notion of type I of curves. Consider at first the weakening of assumption (i) in Rokhlin and Kharlamov-GudkovKrakhnov congruences.Instead of (i) we can only assume that B is of characteristic type. According to O.Ya.Viro [5] there are some extra structures on real surfaces of characteristic type ,namely, Pin_-structures and semiorientations or relative semiorientations (semiorientation is the orientation up to the reversing). In this paper we introduce another structure on surfaces of characteristic type - - complex separation which is also determined by the arrangement of a real surface in its complexification. T h e complex separation is a natural separation of the set of components of a real surface of characteristic type into two subsets.Note that the set of semiorientations of a surface is an affine space over Z2-vector space of separations of this surface. We use the complex separation to weaken assumption (iii). In theorem 1 instead of (iii) we assume only that B+ lies in components of one class of complex separation. The further extension ,theorem 2, can be applied not only for curves of even degree but also sometimes for curves of odd degree. Another weakening of assumptions in theorem 2 is t h a t components of a curve are not necessarily Z2-homologically trivial. These extensions can be applied to curves on quadrics and cubics. T h e o r e m 1 together with an analogue of Arnold inequality for curves on cubics gives a complete system of restrictions for real schemes of flexible curves of degree 2 on cubics of characteristic type (see [6]).An application of theorem 2 to curves on an ellipsoid gives a complete system of restrictions for real schemes of flexible curves of degree 3 on an ellipsoid and reduces the problem of classification of real schemes of flexible curves of degree 5 on an ellipsoid to the problem of the existence of two real schemes (see [7]). An application of theorem 2 to curves on a hyperboloid extends Matsuoka congruences [8] for curves with odd branches on a hyperboloid (see [9]). Applications of theorem 2 to empty curves on surfaces give restrictions for surfaces involving complex separation of surfaces. Restrictions for curves of even degree on surfaces can be obtained also by the application of these restrictions for surfaces to the 2sheeted covering of surface branched along the c u r v e , if we know the complex separation of this covering. This complex separation is determined by complex orientation of the curve.For example in this way one can obtain new congruences for complex orientations of curves on a hyperboloid. These applications and further extension of Rokhlin congruence for curves on surfaces will be published separately in [9]. For example the assumption that the surface and the curve on the surface are complete intersections is quite unnecessary ,but this assumption simplifies definition of number c in formulations of theorems. The formulations of these results were announced in [7] as well as the formulations of results of the present paper. T h e a u t h o r is indebted to O.Ya.Viro for advices.

374

2

N o t a t i o n s and formulations of main t h e o r e m s

Let the surface B be the transversal intersection of hypersurface in Pq defined by equations P j ( x 0 , . . . , xq) = 0,j = 1 , . . . , s - 1 ; C B and R B be sets of complex and real points of B; let A be a nonsingular curve on B defined by an equation Ps(x0,..., xq) = 0 ,where Pj is a real homogenous polynomial of degree mj j = 1,... ,s, q = s + 1; CA and R A be sets of complex and real points of A. Let conj denote the involution of complex con]-r as - Ii

jugation. Set c to be equal to ~ . I f m, is even then denote {x e RB[ + P,(x) >__0} by B+ and set d to be equal to rk(in. : Hi(B+; Z2) --* H I ( R B ; Z2)). A real algebraic variety is called an (M-j)-variety if its total Z:-Betti number is less by 2j then total Z2-Betti number of its complexification (Harnack-Smith inequality shows that j > 0). Let A be an (M-k)-curve. Let DM be the operator of Poincar6 duality of manifold M.We shall say that B is a surface of characteristic type if [RB] = DCBW2(CB) E H2(CB; Z2) (as it is usual we denote by [RB] the element of H2(CB; Z2) realized by R B ) . We shall say that (B, A) is a pair of characteristic type if [RB] + [CA] -4- DcB(w2(CB)) = 0 E H2(CB; Z2).It is said that A is a curve of type I if [RA] -0 E H~(CA; Z2).It is said that A is of even(odd) degree if [CA] = 0 E H2(CB; Z2) (otherwise). As it is usual we denote by a and X the signature and the Euler characteristic.By /~(q) we mean Brown invariant of Z4-valued quadratic form q.

Theorem 1 If B is a surface of characteristic type then there is defined a natural separation of surface R B into two closed surfaces B1 and B2 by the condition that B j , j = 1,2, is a characteristic surface in CB/conj.Suppose that m, is even, B+ C B1, every component of R A is Z2-homologous to zero in R B a n d / f rn8 -- 2 (rood 4) then suppose besides that B2 is contractible in R P q. a) I f d + k = O t h e n x ( B + ) ~ _ c b) I f d + k = l

(rood8).

thenx(B+)=c4-1

c) I f d + k = 2 and x(B+) -- c + 4 orientable.

(mod8). (rood8) then A is of type I and B+ is

d) If A is of type I and B+ is orientable then x(B+) -~ c

(rood 4).

Theorem 2 If (B, A) is a pair of characteristic type then there is defined a natural separation of surface R B \ R A into two surfaces B1 and B2 with common boundary R A by the condition thai Bj U CA/conj is a characteristic surface in C B / c o n j , there is defined Guillou-Marin form qj on HI(Bj O CA/conj; Z2) and x(Bj) - c + x ( R B ) - a ( C B ) 4 + fl(qj)

3

(mod S).

Proof of theorem 2

Consider the Smith exact sequence of double branched covering ~r : C B --* C B / c o n j

--~ H3(CB/conj, R B ; Z2) ~ H2(RB; Z2) @ H2(CB/conj, R B ; Z2) -~ H2(CB; Z2) ~

375

Let ¢ denote the composite homomorphism

H2(CB/conj, RB; Z~) o~d H2(RB; Z2) @ H2(CB/conj, RB; Z~) -% H2(CB; Z2). Let j denote the inclusion map (CB/conj, +) -+ (CB/conj, RB).Recall that ¢. 0 j . is equal to Hopf homomorphism rc~. It is easy to deduce from the exactness of the Smith sequence that ¢ is a monomorphism. Indeed, ~rl(CB) = 0 hence ~rl(CB/conj) = 0 and Ha(CB/conj; Z~) = 0. Therefore boundary homomorphism 0 : H3(CB/conj, R B ; Z2) --* H2(RB; Z2) is a monomorphism.Therefore,since 0 is the first component of 73, Ira73 N ({0} + g2(CB/conj, RB; Z2)) = 0 and ¢ is a monomorphism. It is easy to check that

:r*w2(CB/conj) = D ~ [RA] + w2(CB). Thus 7r:(DcB/~o,jw2(CB/conj)) = [CA], therefore,because of the injectivity of ¢,we obtain that

j.DcB/co,jw2(CB/conj) = [CA/cony, RA] e H2(CB/conj, RB; Z2). It means that there exists a compact surface Ba C R B with boundary RA such that B1 U CA~cony is a characteristic surface in CB/conj. Surface RA is homologous to zero in CB/conj since RA is the set of branch points of :r. Set B~ to be equal to the closure of ( R B \ B1). Then B2 U CA/cony is a characteristic surface CB/conj,

B1 U B2 = RB,B1 n ]32 = OB1 = OB2 = RA. Let us prove the uniqueness of pair {B1, B2 }.It is sufficient to prove that the dimension of the kernel of inclusion homomorphism H2(RB; Z2) -+ H2(CB/conj; Z2) is equal to t. This follows from the equality dimH3(CB/conj, RB; Z2) = 1 that can be deduced from the exactness of the Smith sequence. We apply now Guillou-Marin congruence [10] to pair (CB/cony, Bj U CA/conj),j = 1,2 a(CB/cony) =- [By U CA/cony] o [By U CA/cony] + 2/3(qj) (mod 16). Hirzebruch index theorem gives an equality a(CB/conj) = °(CS)-x(l~).To finish the 2 proof note that [Bj U CA/cony] o [Bj U CA/cony] = 2c - 2x(Bj) (the calculation is similar to Marin calculation in [11]).

4

Proof of the theorem 1

Pair (B, 0) is of characteristic type since A is of even degree in B. Thus the first part of theorem 1 follows from theorem 2 - - there exist a natural separation of B into two surfaces ]31 and B2 such that B1 and B2 are characteristic surfaces in CB/conj. Let V denote B+ U CA/cony.Let W denote V U B1. Recall that ]3+ N B2 = 0 thus VnB~ =0. Lemma I IV] = 0 E H2(CB/conj; Z2)

Proof. Since A is of even degree in B, there exists a 2-sheeted covering p : Y -+ C B branched along CA. Involution cony can be lifted to involutions T+ and T_ : Y

376

Y since C A construction points of T+ Consider

is invariant under conj. It is easy to see using the straight algebraic of p that T+ and T_ can be chosen in such a way that the set of fixed is p-l(B:t:), p the diagram Y • CB

Y/T_

CB/co j.

This diagram can be expanded to a commutative one by map p' : Y/T~: --~ C B / c o n j . It is easy to see that p' is a 2-sheeted covering branched along V. Therefore [V] = 0 E

H2(CB/eonj; Z2). Using Lemma 1 we see that W is a characteristic surface in C B / c o n j as well as B2.We apply Guillou-Marin congruence to these two surfaces :

a(eB/eonj)

==_[W] o [W] + 2~(qw) = 2c - 2x(B+) - 2x(B2) + 2~(qw)

a ( C B / c o n j ) =_ [B2] o [B21 + 2fl(qB,) - - 2 x ( B 2 ) + 2fl(qB2)

(mod 16)

(rood 16)

,where qw and qs2 are GuiUou-Marin forms of W and B2. Therefore

x(B+) =--.c + fl(qw) - fi(qB2) L e m m a 2 Vx E Ha(B2; Z2), qB2(X) -- q w ( x ) = ~ 0

(

(rood S).

if x is contractible in R P q if x is noncontractible in R P q .

Proof. It follows from the definition of Guillou-Marin form that values on x of qB2 and qw are differed by linking number of x and V in C B / c o n j that is equal to linking number of x and C A in C B . The last linking number can be calculated from the straight construction of a 2-sheeted covering branched along CA. It was shown in [12] that Brown invariant of form q on the union of two surfaces with common boundary is equal to the sum of Brown invariants of restrictions of q on these surfaces in the case when q vanishes on the common boundary. Now theorem 1 follows from this additivity of Brown invariant and the classification of low-dimensional Z4valued quadratic forms (see[12]). Indeed, since every component of R A is homologous to 0 in R B , fl(qw) = fl(qw]cA/co,~j) + fl(qwIB+) + fl(qWtB2)" Lemma 2 shows that under assumptions of theorem 1 ~(qw IB2) = fl(qB2)" To complete the proof note that ranks of intersection forms on Hi(B+; Z2) and H l ( C A / c o n j ; Z2) are equal to d and k respectively.

References [1] V.A.Rokhlin. Congruences modulo 16 in sixteenth Hilbert problem. Functsional'ni Analiz i ego Prilozheniya. 1972. Vol. 6(4). P. 58-64 [2] V.M.Kharlamov. New congruences for the Euler characteristic of real algebraic manifolds. FunctsionM'ni Analiz i ego Prilozheniya. 1973. Vot. 7(2). P. 74-78 [3] D.A.Gudkov and A.D.Krakhnov. On the periodicity of Euler characteristic of real algebraic (M-1)-manifolds. Functsional'ni Analiz i ego Prilozheniya. 1973. Vol. 7(2). P. 15-19

377

[4] V.I.Arnold. On the arrangement of the ovals of real plane curves, involutions of 4-dimensional smooth manofolds and the arithmetic of integral quadratic forms. Functsional'ni Analiz i ego Prilozheniya. 197t. Vol. 5(3). P.1-9 [5] O.Ya.Viro. The progress in topology of real algebraic varieties over last six years. Uspehi Mat. Nauk. 1986. Vol. 41(3). p. 45-67 [6] G.B.Mikhalkin. Real schemes of flexible M-curves of virtual degree 2 on cubics of type I tel. Diploma paper. Leningrad. 1991. [7] G.B.Mikhalkin. Congruences for real aldebgaic curves on an ellipsoid. Zapiski Nauchnyh Seminarov LOMI. 1991. Vol. 192, Geometry and Topology 1 [8] S.Matsuoka. Congruences for M- and (M-1)-curves with odd branches on a hyperboloid. Preprint. 1990. [9] G.B.Mikhalkin. The complex separation and extensions of Rokhlin congruence for curves and surfaces (to appear). [10] L.Guillou and A.Marin. Une extension d'un theoreme de Rohlin sur la signature. C.R. Acad. Sci. Paris. 1977. Vol. 285. P. 95-97 [11] A.Marin. Quelques remarques sur les courbes algebriques planes reeles. Publ. Math. Univ. Paris VII. 1980. Vol.9. P.51-86. [12] V.M.Kharlamov, O.Ya.Viro. Extensions of the Gudkov-Rokhlin congruence. Lect. Notes Math. 1988. Vol. 1346. P. 357-406

Complexit6

de la construction

des s t r a t e s h m u l t i p l i c i t ~ c o n s t a n t e d~un e n s e m b l e a l g 6 b r i q u e d e C ~

T.Mostowski

E. Rannou

R6sum~: Nous d@crivons dans cet article un algorithme qui construit la partition d'un ensemble alg@brique de C '~ en ensembles o~t la multiplicit@ est fix@e. L'ent%e de l'algorithme est constitu@ par un nombre fini de polyn6mes d@finissant l'ensemble alg~brique. Chaque strafe de la partition sera d6finie par l'algorithme ~ l'aide d'une formule du langage du premier ordre des corps algdbriquement clos sans quantificateurs. Cet algorithme est d6crit par un %seau arithm~tique, de complexit@ sequentielle polynomiale en la somme des degr~s des polyn6mes d'ent%e et simplement exponentielle en le nombre de variables. Une consequence imm@diate de cet algorithme est la possibilit@ de construire des ~quations polynomiales d@finissant l'ensemble des points singuliers d'un ensemble alg@brique de C n avec une complexit6 s@quentielle simplement exponentielle. Le coeur de l'algorithme est constitu@ par une g@n~ralisation ~ plusieurs variables d'une m6thode d'Hermite, qui permet de compter le hombre de points d'un ensemble alg6brique de C ~ de dimension z~ro. Cette m~thode est particuti~rement simple ~ mettre en oeuvre. Les auteurs sont tr&s reconnaissants g Mme M-F. Roy qui, pendant plusieurs discussions, a suggdr6 le probl~me et les id~es principales de la solution. Ils remercient le rapporteur pour ses critiques constructives tr~s utiles.

1.Introduction: 1.1.Notatlon

de base:

Soit V C C n une ensemble alg6brique d6fini par des polyn6mes f l , ..., L de C[X1, ..., Xn]. 8 Soit D := }-~-i=l degfi. Nous noterons V d la composante @quidimensionnelle de dimension d de V. Soit x un point de V. Nous d6finirons ta multiplicit6 ra,(V) de V en x par:

~(v) ::~ m~(V~) d:O et, pour d=O~...,n nous poserons:

379

m ~ ( V d) := ~L a n V a - ~H a n V a + 1

p o u r L d u n n-d-plan tel que ~L a f3 V d soit fini et maximal, et H a un (n - d)-plan passant p a r x tel que ~H a N V a soit fini et maximal. En effet, rnx(V d) est ddfini (dans [M], chapter 5, section 5A, p a r exemple) c o m m e ~L a r3 V d r3 B~, oh B , est une boule de centre x et de rayon e, e est assez petit, et L d est un (n - d)-plan qui passe assez pr6s de x et tel que ~L d N V a soit fini et maximal. R e m a r q u o n s que cette derni6re condition est satisfa]te si et seulement si L d coupe V a transversalement et[,a nr-da C C ~, ohLd,r/d sont les el6tures de L d, V d dans CP ~ . Si H d passe par x, alors Hal3 V a = {x, x a , . . . , xk } est m a x i m a l si et seulement si ~a Cl~a C C '~ et H a coupe V d traasversalement en chaque xi. Si Be, B,,i sont les boules de centres z, xi et de rayon e, et si L d est parall61e ~ H a, coupe V a transversalement et passe assez pr6s de x, alors, p o u r chaque i, L a fq V a f3 Be,i contient exactement un point, et (L d f'l V d) \ Be C U i ( L d n V d n Be,i). On a done

~(L a N y a) = ~(L a N y d 71B~) T k = m x ( V a) + ~(H d fl V a) - l, et il suffit de r e m a r q u e r que ~(L d fl V a) ne d6pend pas de L d. Nous appelerons la quantit6 ~L a n V d degr6 de 1'ensemble alg6brique V d que nous noterons degV d. Soit Vk (resp. Vd) l'ensemble des points de V (resp. de V a) o9, la multiplicit6 de V (resp. V d) vaut k.

1.2.Rdsultats: Le but est de construire les ensembles Vk = {x E V I m d V ) = k} avec une eomplexit6 s6quentielle simplement exponentielle. Nous d6montrerons done le r6sultat suivant:

Thdor~me

1.2.1: Soit V un ensemble algdbrique de C n ddfini par des polyn6mes f l , . . . , fs h coefficient dans un anneau fl. Soit D := ~ i = 1 degfi. Soit Vk = {x E V I

mdv)

= k}.

Alors il existe un r~seau arithm~tique sur Pi qui construit avec une complexit~ sdquentielle D n°(1) un entier kma~ < D n°°) et des formules du premier ordre du Iangage des corps algdbriquement clos 0 1 , ..., Okrna ~ tels que: Vk = 0 pour k > kma~ et Vk = {x E C n I Ok} sinon. On en d6duit le corolla]re suivant:

Corollaire 1.2.2: Soit V un ensemble algdbrique de C n dgfini par des polynSmes 8 f l , . . . , fs a coe.O~cient clans un anneau fl. Soit D := ~i=1 degfi.

380

Alors il existe un rdseau arithmdtique sur t2i qui construit avec une complexitd sdquentielle D n°(l) des polynSmes gl, ...,gt de C[X1,...,Xn] ddfinissant l'ensemble des points singuliers de V par: Vsing -- {gl = 0, ...,gt = 0}

Nous a v o n s Ysing = LJk>lYk. Pour obtenir les polyn6mes gl,..., gt, il suffit d'utiliser le thfior~me 1.3.3.(ici V~i~g = V~i~g). Remarque: Le re@me r@sultat peut @tre obtenu sur un corps Mg4briquement clos de caract6ristique 0.

1 . 3 . 0 u t i l s utilisds: Nous utiliserons les r~sultats suivants pour construire l'algorithme annonc4. Elimin_ation ,rapide" des qt~antificateurs:(Fitchas/Galligo/Morgenstem[FGM]) Soit d2 une formuIe prdnexe du premier ordre du langage des corps aIgdbriquement clos d n variables et h r blocs de quantificateurs definie par des pol~nomes ?t coej~cient dans un anneau ~. Soit D Ia somme des degrds des polynSmes apparaissant dans la formule ~.

Th ~o r ~ m e 1 . 3 . 1 :

Alors il existe un r~seau arithm6tique sur [4 qui construit avec une complexitd s~quentielle D n°(~) une formule q! sans quantificateurs dquivalente it d2.

A priori l'6timination des quantificateurs est doublement exponentielle en n (cas oh r = n)(voir [H]). Cependant dans de nombreuses situations significatives le nombre de blocs de quantificateurs est ind6pendant de n. Dans ce cas, l'41iminations "rapide" des quantificateurs permet d'obtenir des algorithmes simplements exponentiels. Le calcul de la multiplicit4 d'une vari4t4 V en un point x se fera en sommant les multiplicit6s des composantes 4quidimensionnelles V d de V en x. Aussi le r@sultat suivant nous sera d'une grande utilit@: D4comp0sition d'une vari4t4 V en composantes fiquidimensionnelles V d (Giusti/Heintz[GH]): Th~or~me 1 . 3 . 2 : Soit V un ensemble alg~brique de C n d~fini par des polyn6mes f l , . . . , fs. Soit D := ~i"=1 degfi. Alors il existe un rdseau arithmdtique qui construit avec une complexitd sgquentielle 0 °("2) des polynSmes f~,..., fs°o, ..., fd, .--, f,~, ..., f ? , ..., f , : ddfinissant les composantes dquidimensionneIles V d de V et vgrifiant: * V d = {x e Cn l f d . . . . . fsdn = 0 } pour d = O , . . . , n * E d.= l E i s=d l degfi d <- DO(~ ~)

381

Cloture de Zariski d'un constructible (Heintz): Th~or~me 1 . 3 . 3 : Soit ~ une formule sans quantificateur du langage des corps algdbriquement clos ddfinissant sur C'* un ensemble constructible V. Soit D Ia somme des degrds de tout les polynSmes apparaissant dans ~.

Alors il existe un rdseau arithmgtique qui construit avec une comptexitd sdquentielle D n°(~) des polyn6mes fl,..., fs ddfinissant la cloture de Zariski "V de V: Y={xeCnlfl ..... f~=0} I1 semble qu'il n'y ait pas de r6f6rence pr6cise pour ce r6sultat. En voici une d6monstration, aimablement fournie par le rapporteur. (i) Soit Y = {fl = 0,...,fp = 0, g # 0}, d'apr~s la d6finition de degr6 de [H], on a deg(V) < O n. (ii) En utilisant un raisonnement similaire £ [HI Remark 4 + Prop. 3, on montre qu'il existe une famille de polyn6mes gl,..-, gn telle que V = {gl = 0, ..., gn -= 0} et deg(gi) _< D n (voir aussi [CGH] Prop 1.2 ). (iii) Etant donn~ que g.gi(x) = 0 (1 < i < n) si f l ( x ) - f~(x) = 0, on a

g.gi E R a d ( f l , . . . , f p ) , et , comme deg(g.gi) < D n+l, (g.gi) D°("2~ e ( f l , . . . , h ) et il existe des representations (g.gi) D°("2) = A1fi + ... + Apfp avec deg(A/ _< D °(n2) Voir [DFGS] Remark 7. (iv) Soit B l'ensemble des polyngmes F E k[X1,...,Xn] tels que g D ° ( ~ b F = AIF1 + ... + ApFp, avec deg(Ai ___ D °(n2), deg(F) _< D °(n2), B e s t un k-espace vectoriel de dimension _< D O(n~). On peut en calculer une base en r6solvant un syst&me d'6quations o~ les inconnues sont les coefficients des polyn5mes F, A1,..., Ap. Si H1,..., HN en est une base, clairement V = {HI = 0, ..., HN = 0}.

2.G~n~ralisatlon d'une m~thode d'Hermite permettant de compter le cardinal d'une vari~t~ alg~brique de dimension zero: 2.1.Principe

de la m~thode:

La m6thode consiste g g6n6raliser au cas de plusieurs variables la m6thode d'Hermite qui permet de compter le nombre de racines distinctes d ' u n polyn6me d'une variable.(L'interpr6tation dans le cas r6el clos de cette g6n6ralisation sera expos6 par Pedersen et Roy dans Mega92 ~ para~tre) Soit K un corps alg~briquement clos de caract6ristique 0. Soit p u n ideal de K[X1, ...,Xn]. Supposons que l'ensemble z(p) des z~ros de p dans K n soit de dimension z6ro.

382

Posons z ( p ) = { a l , . . . , a , } . Notons A l'anneau de coordonn6es suivant: A = K[X1, ..., X , ] / p . P a r hypoth~se, A est noetherien et artinien, e'est un K-espace veetoriel de dimension finie que nous pouvons 6trite sous la forme:

8

A = l~k=l A~, off p o u r k = 1, ..., s, nous avons: A~ = K[X1,...,Xn]/3~ l'ensemble alg6brique {ak}-

qui v6rifie ~

= m~ k id6al m a x i m a l associ6 £

Nous d6finissons la multiplicit6 n~ h de p au point ak par: n~ k := dimKA~ k = dimKK[X1, ...,Xn]/2c, k. Soit 11,..., lq une base de A comme K-espace vectoriel. Soit Q E A. Consid~rons la forme bilin6uire symdtrique de Aq x Aq dmls K d6finie par: B ( p , Q)(al,..., aq) := ~ ; = 1 nak Q(ak )(al I] (ak) + ... + aq lq(ak ))2 =

On

Eqj=I(E =i n

Q(ak)li(ak)lj(ak))aiaj

a alors:

Proposition

2.1.1:

Le rang de la matrice de B ( p , Q ) est dgal au hombre de point,s

de z(p) n'annulant pas Q. Preuve: I1 suffit de m o n t r e r que les formes lin6aires ( a l l l ( a k ) ind6pendantes. Soient A1,..., A, E K tels que ~ = 1

+ ... +

aqlq(ak))k=l,...,, sont

Akli(ak) = 0 p o u r i = 1, ..., q.

C o m m e (h)~=l,.. ,q est une base de A = K[X~,..., X , ] / p , la forme lin6aire

A

=

K[Xl,...,Xn]/p e

~ I

K )

8

est identiquement nulle. Mais la forme lin6aire

, P

'

K

E;=I

) kP(O k)

est identiquement nutle sur p, elle est donc identiquement nulle sur K[X1,..., X , ] . Soient ;3k l'id6al r6duit associ6 £ l'ensemble alg6brique { a l , ..., a k - 1 , a k + l , . . . , a , } p o u r k = 1, ..., s.

383

Nous avons £ cause du Nullstellensatz 3k # VfP-. I1 existe donc un polyn6me Pk e 3 k \ v ~ pour chaque k = 1, ..., s. Nous avons done, pour chaque k = 1,...,s:

~;'=1 )~k,Pk( ak' ) =

Ak Pk( ak ) = 0

==~ Ak = 0 pour k = 1, ..., s. Par consequent les formes lin~aires ( a~ l~ ( ak ) +... + aq lq( ak )) k=L...,8 sont ind~pendantes. Remarque: Darts le cas o~ K est un corps r~el clos, la signature de ia matrice de B(p, Q) est ~gale £ Ia difference du hombre de z~ros de z(p) rendant Q positif et du hombre de z~ros de z(p) rendant Q n~gatif (voir [PRS]). Ii nous reste £ exprimer la matrice de B(p, Q) de mani~re £ pouvoir effectuer son calcul de maniSre agr6able. C'est l'objet du r6sultat suivant.

Proposition 2.1.2:

Soi, h e A =

K[X1,...,X,]/p.

Soii m h t'endomorphisme suivant: mh:

A P

~ A ~ , hP

Alors:

les ( h ( a k ) ) k = l ..... s sour les valeurs propres de mh avec les multiplicitds (n,~k)~=l ..... 8. Preuve: Nous avons: $

A -- 1]k=l A,~k avec, pour k = 1, ..., s, nc,, := dimKA,~ k = d i m • K [ X 1 , ..., X , ] / 3 ~ , k oh A,~, = K [ X 1 , . . . , X , , ] / 3 ~ , k et ~ = m~, ideal maximal associ~ £ l'ensemble alg~brique {ak }. I1 suffit donc de montrer que les (A,~,)k=l ..... 8 sont les sous-espaces propres associ~s au valeurs propres (h(ak))k=l,...,s. Comme h -

h ( a k ) E ma~, il existe un entier nk tel que (h - h ( a k ) ) n~ E :ta~.

Soit P E K[X~, ...,X,]. Soient I d l'identit~ de K[X1, ...,Xn] et I d A celle de A Nous avons (ink -- h ( a k )Id)"k P E 3 ~ Done A~,, est nilpotent par m h -- h ( a k ) I d A . On en d6duit imm~diatement

Corollaire 2.1.3:

trace(mh ) = ~=1

na, h ( a k ).

384

On peut donc 6crire la forme bilin6aire B(O, Q) de la mani6re suivante: B(p, Q)(al, ..., aq) := ~ q , j = l ( ~ k = l na~Q(ak)l,(ak)lj(ak))aiaj q = Ei,j=I

trace(mQld~ )aiaj

2.2.Complexit~ de la m6th0de: Pour mettre en oeuvre cette m6thode permettant de compter le nombre d'616ment d'un ensemble alg6brique de dimension z6ro, nous avons besoin: * d'une base (/1,-.-, Ip) de d = K[X1,..., Xn]/P * d'effectuer des calculs dans A, e'est h dire modulo p. Le calcui d'une base de Gr6bner de p remplit ces besoins. Tout les autres calculs sont ensuite des constructions d'Mg6bre lin6aire pour lesquels nous disposons d'Mgorithmes polynonfiaux, ou des r6duction modulo la base de Grbbner de p. Les calcuts de base de Grbbner ayant 6t6 paxticuli6rement d6velopp6s ces derni6res ann6es, l'impl6mentation de la m6thode d'Hermite g6n6ralis6e ne pose pas de probl6me majeur. Pour construire notre algorithme de partition en strate de multiplicit6 donn6e, nous aurons besoin d'une version param6tr6e de la m6thode d'Hermite g~n6ralis~e. I1 nous faudra construire une subdivision de l'ensemble des param~tres en sous-ensembles sur lesquels la base de Gr6bner de p prend une forme donn6e. Il nous faudra en particulier connMtre l'ensemble des param6tres off la dimension de l'ensemble alg6brique z(p) n'est pas de dimension z6ro. Nous utiliserons le r6sultat suivant, qui r6sulte facilement de [DFGS]:

Proposition 2.2.1: Soient fl,...,f, E K[Y1,...,Ym, X1,...,Xn]. Soient Yl,-.., Ym des dldment8 de K, Pyl,...,v,, d6signera l'id6al de K[X1,..., Xn] engendr6 par f l , ...,ft o~z les Y1,...,Ym ont dtd spdcialis~s en Yl, ...,Ym. Alors il existe un r~seau arithmdtique qui construit, ave un complexitd D (n+m)°O), un entier r ~_ D (n+m)°°) et une partition de K n ( K - 1 , K o , K 1 , ...,Kr) d6finie par: Ki -- {(yl,---,ym) E K m ]dim z(pyl ..... y~) = 0 et rang B(p w .....y,,, 1) = i} pour i >_ 0 et g - 1 = {(Yl,...,Ym) e ~ m I dim z(pv, .....v~) > 0} v6rifiant: K,- # O. 3.Ddmonstration

du

th6orhme

principal:

Dans le th6orSme 1.2.1, nous voulons construire les ensembles Vk = {x Z V I m~(Y ) = k} La multiplicit6 m , ( V ) de l'ensemble alg6brique V au point x est d6finie par:

385

n

m.(v) := F., m*(v~) d----O

oh V a d4signe la eomposante 4quidimensiormelle de dimension d de V. P o u r d=O,...,n nous aeons: m x ( V a) := d e g V a

-

~H d n v ~ + 1

pour H a un n-d-plan passant par x tel que ~H d fl V a soit fini et maximal, d e g V d 6rant d6fini par la quantit6 ~ L a O V d pour L d u n n-d-plan tel que ~ L a N V a soit fini et maximal. Dans un premier temps, nous d6composons V en vari6t6s 6quidimensionnelles V a l'aide de l'algorithme de Giusti et de Heintz (Th6or6me 1.3.2). Nous obtenons donc avec une complexit6 s6quentielle D °(n2) des polyn6mes:

S o, ..., So0, ..., St, ..., SL, ..., S;', ..., S:o v ~ a n t : * V d = {x e C n l f (

..... Sd, = 0 } pour d = O , . . . , n * ~ d'*= l ~ i =~d1 degS~ <_ DO(n~).

P o u r chaque d -- 0, ..., n,nous allons construire les ensembles Vkd = {x E V d I m = ( V a) = k} Fixons k E {0, ..., D "°(~) }. * Calculons t o u t d ' a b o r d le degr6 d e q V a de Vd: Soit (ai,j)i,j E Ma,.+I(C) l'ensemble des matrices £ d lignes et n + 1 colonnes £ coefficients dans C. Soit L(a~, i)i,j l'ensemble alg6brique d6fini par les 6quations affines: al,lX1 ...

+ -Jr

... ...

+ ..]-

al,nXn ...

= ~-

al,n+l ...

ad,lX 1

-Jr

...

.ar

ad,nX n

---- ad,n+ 1

, -t-

,,, ...

, +-

fL, al,nXn

...

+

...

+

...

ad,lX1

+

...

+

aa,nXn

Soit P(~,i)~,i l'id6al engendr6 par: s~ al,lXl

- -

--

al,n+l, ad,n+l

E n appliquant la version param6tr6e de la m6thode d'Hermite g6n6ralis6e (proposition 2.2.1) £ cette situation, nous obtenons avec une complexit6 ( D ° ( n * ) ) ('*+d('*+1))°(1) = D '~°(~) u n entier r > 0 tel que:

Kr = {(a~,~)~,i ~ Ma,.+I(C) I dim z(P(a,.~),,~ ) = 0 et ,'ang B(p(a,,~),,~ , 1) = r} #

386

et {(ai,j)i,j E K/]d,n+l(C) l d i m

z(p(~,,j),,, ) =

0 et r a n g B ( p ( a , j ) , j , 1) > r} ----0.

Pour (a~,j)~,j fix6, L(a~j)~j est soit vide, soit sous-espace affine de C n de dimension sup6rieure ou 6gale £ n - d. Si d i m L(~j)~,i > n - d, alors L ( ~ j ) , j M V d est soit vide, soit irdini. Par consdquent K r 17 {(ai,j),,j E Md,~+l(C) I d i m

L(~,,i),, j > n - d} = O

En appliquant la proposition 2.1.1 de la m6thode d'Hermite g6n6ratis6e, nous avons: { L n-d-plan affine ] ~L M V d = r} # 0 et { L n-d-plan affine I r < ~L fq V d < ~ } = O.

Par cons6quent d e g V d = r. * Calculons les ensembles Vd = {x ~ V d I m r ( V d) = k}: Soit (bi,j)i,j E Md+I,,~(C). Soit H(b~j),j l'ensemble alg~brique d~fini par les 6quations affines: bl,l(Xl -bd+l,1) ... b d , l ( X l -- bd+l,1)

+ ... + ... "4- ...

+ + +

bl,n(Xn-bd+l,n) ... bd,n(Xn - bd+l,n)

+ + +

... ... ...

+ + +

=

0

=

0

Soit q(b~,j),.i l'id6al engendr6 par: fd bl,l(Xl-bd+l,1) ... bdj(Xl--bd+l,1)

fd bl,n(Xn-ba+l,n), bd,,(X,~'"'bd+l,n).

E n appliquant la version param6tr6e de la m6thode d'Hermite g6nfralis6e (proposition 2.2.1) £ cette situation, nous obtenons avec une complexit6 ( D ° ( n 2 ) ) (n+n(d+l))°(t) = d T.(d K 1d , ' " , K r~] d D n°(1) un entier r d < D n°O) et une partition de Md+l,n(C) ( k'_l,,~0, v6rifiant: K/a = { ( b i j ) i j e Md+l,n(C) l d i m z(q(b,,~),,j) = 0 et r a n g B(q(b,,i),.i,1) i>0 et K - 1 = {(bi,j)i,j e P~d+l,n(C) I d i m z(q(bij )ij > 0}.

= i} pour

En utilisant le m~me argument que pour le ealeul du degr6 nous avons que V~ est l'ensemble des x = ( x l , ..., Xn) E V d qui v6rifient :

387

3(bi,/)ij E Md+a,.(C) tel que: ;V1 ~-- b d + l , 1 , . . . , X n ~- bd+l, n

(b~j)~,i e K dde g V C l + l - k

et

v(c~,i)~,i E M~+~,.(c) Ceci constitue une formule du premier ordre du langage des corps a]g~briquement clos £ 2n(d + 1) variables, £ 2 blocs de quantificateurs et de degr6 totM au plus D'*°°).D ~°~) = D ~°(~) (ou plus exactement 2 formules £ n(d + 1) variables et 1 seul bloc). I1 suffit alors d'utiliser pour chaque 1 < k < r d l'61imination "rapide" des quantificateurs (thror&me 1.3.1) pour obtenir des formules sans quantificateurs drfinissant les Vd de degr~ total au plus (D2(d+a)"°(~)) ("°(~)) = D "°(~). Au total, pour chaque 0 < d < n , avec une complexit6 (n + 1).D~°°).D ~°u) = D ~°u), nous obtenons une partition (Vd)k=l .....degYd de V d. Afin d'obtenir la partition (Vk)k=l ..... k,~o= de V, il suffit d'dcrire les Vk sous la forme:

(ko,...,k,,.,) EN"

d=o

ko+...+k~=k

Nous pouvons ehoisir pour km.x l'entier degV = ~.'d=o deg Vd (kin,,= <_ D"°(~)). I1 y a, au plus pour k _< k.~.~ fixr, k , ~ k0 + ... + k, -- k.

= D "°u) n+l-uplet (k0, ..., k~) E N "+1 vrrifiant

Par consrquent chaque Vk est drfini par une formule du premier ordre du langage des corps algrbriquement clos sans quantificateur de degr6 total au plus Dn°(~).(n + 1).D ~°0) = D n°(1).

R~f$rences: [CGH] L.Caaiglia, A.Galligo, J.Heintz Equations for the projective closure of an aJ~ne algebraic variety, £ parMtre dans AAECC 7, Toulouse 1989 [DFGS] A.Dickenstein, N.Fitchas, M.Giusti, Sessa The membership problem for unmixed polynomial ideals is solvable in single exponential time, £ parMtre dans AAECC 7, Toulouse 1989 [FGM] N.Fitchas, A.Galligo, J.Morgenstern Precise sequantial and parallel complexity bounds for the quantifier elimination in algebraically closed fields, £ para/tre dans J. Pure Applied Algebra

388

[GH] M.Giusti, J.Heintz Algorithmes "disons rapides" pour la ddcomposition d'une varidtd algdbrique en composantes irr~duetibles et Lquidimensionnelles , Effective Methods in Algebraic Geometry (MEGA 90), 169-191, Birkhauser (1991) [H] J.Heintz Definability and fast quantifier elimination in algebraically close fields Theoret. Comp. Sci. 24 (1983)239-277 [MR] T.Mostowski ,E.Rannou Complexity of the computation of the canonical Whitney stratification of an algebraic set in C n , AAECC 9 Springer Lecture Notes in Computer Science 539 [M] D.Mumford Algebraic Geometry I: Complex Projective Varieties, Springer 1976 [PRS] P. Pedersen, M-F. Roy, A. Szpiglas Counting real zeros in the multivariate case, soumis £ MEGA 92

T. M. : University of Warsaw, Institute of Mathematics, ul. Banacha 2, 00-913 Warszawae, Poland E. R. : IRMAR (unit6 associ6e CNRS 305), Universit6 de Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France

REAL PLANE ALGEBRAIC CURVES WITH MANY SINGULARITIES E.I.Shustin (Math. Dept., Samara State University, 443011 Samara, USSR)

ul. Acad. Pavlova 1,

In this article we give the construction of real plane algebraic curves of a given degree with prescribed singularities. In particular, we construct real curves of degree V77 with ~ 4 ~ / ~ real cusps. Introduction Let ~ ~p2 be an isolated singular point of the curve , and ~ C ~ p ~ be the sufficiently small closed ball centred at Z . The topological type of the pair C ~ ~)~F~ we call the type of singular point ~ ~ F (or singularity). The topological type of triad ( ~ D ~ 29 ~ F j ~ D / ~ ~ 2 ) is called the real type of singular point ~ p ~ ( or real singularity). It is well-knov~ ~7~ that for any integers ~ p ~90 satisfying O ~ ~ ~ ~ M 4 - 0 ~ M 4 - ~ / ~ there is an irreducible curve of degree ~ in ~ p a with ~ nodes. The same for real curves [2], although in this case the relation of numbers of knots (type 3f 2 - ~ a ~ ) , of single points (type 5~2+~2--O) and of imaginary nodes is unknown in general (Viro [11] constructed curves with only single points). For other singularities there are results on complex curves: some constructions [3~ with singularities number, which is a linear function of the curve degree, and upper bounds of singularities number by a quadratic function of the curve degree [5, 6~. Here we'll give the complete solution of the problem on nodal curves (sec. 2), the construction of real cuspidal curves (sec. 3), the construction of real curves with arbitrary singularities (sec 4). Our constructions are based on the theory of gluing algebraic curves and on the independence of singularities deformations. From now on the term "curve" means the polynomial ~-- and the locus of points defined by F = ~ . we use below the joint numbering of formulae and statements. I. Preliminaries 1.1. Th~ ~l_ui~g_o_f~i~g~lar curves. Further on we'll use the suggested in ~8~ modification of the Viro method ~11-13]. Recall certain definitions (see [8, 11-13~)o

390 Let ~ be an isolated singular point of a curve F ~ ~ p 2 . If ~ is zero-modal we put ~ C 2 ) --- ~ ( ~ ) - - ~ , where ~ is a Milnor number. If the modality of Z is positive we put e ( ~ ) equal to the sum of orders of all infinitely near points in full resolution of 2 , except the nodal points of union of exceptional divisor with the proper transform of F . Let F ( ~ # ~ ) be a real polynomial with the non-degenerate Newton polygon Ah The section F ~° of F ( ~ ) on edge ~ c z A is the sum of items from F , corresponding to the edge j~ . If each section F J° hasn't any multiple factors (except ~ j ~ ), we'll call F g ~ ~) peripherally non-degenerate. Let ~ ( F ) denote the singularities collection of the curve F \ ~ ~ ~ - - 0 } . Remind of the definition of the chart of a peripherally nondegenerate polynomial ~ ( ~ j ~ ) [11-13]. Introduce the covering

E '. E

\ o ) z ,,

E

=

e. f

Put ~ £ = R a × (R~7) a • There is a sufficiently gon z ~ z C ~ such that (i) there is a homotety ~.~Z__# ~ Z , which z~" into Zh , (ii) the pair ( 7 ~ Z 2 E-q(F)/q "~/u~:X ) , --_ /klx ( ~ ~ l ~ ) z is a strict deformation retract

large polytransforms where Z of the pair

E-'(F))

.. (iii) E - f ( F ) ~ V(~') x ( ~ i ~ ) 2) -~ ~ , where V ( ~ is a set of vertexes of /hi , (iv) for each edge 2 1 ~ /%1 , the set E ' ¢ ( F ) ~ ( f ~ is, firstly, homeomorphic to a disjoint union of intervals, which correspond one-to-one to local branches of the curve F determined by a relevant edge ~ ~ /k , ~nd, secondly, isotopic to

( 7J9

E - ' ( F . P ) n (,p')~(lRil:--{)z,.)

in

.9"× [ I R ~ , ) ~

.

Then there is a set ~ ~ Z~ such that (a) F" is isotopic to E - ' ( F ) in (z~'\ ~/(za'))x([~P,~if) ~ by an isotopy, which is invariant with respect to the action of the complex conjugation and the automorphism group of the covering ~ , ( b ) ~ D (~'~¢ ([~r'2~.)~) = E-f(F,P) I,') (j~,~, (IR~E])') for every edge ~ z ~ A t . At last, put

2

391 The chart of the polynomial

F

is a pair

Let

be peripherally non-degenerate real polynomials with Newton polygons

Z k 1 ~ , , .2 Z h N - Assume (i) ZZ~ U . , . U Z~ N -- Zh (ii)

Zh~ • Zhg

is a convex polygon,

is empty, or a common vertex, or a com-

mon edge of ZZ~) Zhg if 2[9~ ~ , (iii) there is a convex continuous function

~--~ ~

, li-

near on any /k ~ ) 4_< K_K ~J , and non-linear on any union I. 1.2. Remarks. We observe that the adjacency relation for Zhl~ ,,.;

Z~lq

j%

is like one for

~

~I)'--~

/%

Z ~ t~,.. tJ Z ~ N --- Zh = ~/J~Z~ ~ ( ~ / 7 ) from (1.1.1) that

F~P = ~L

mon edge. In particular,

, if F~ 0~O

Define the gluing of charts of

2

Z~/j

)

, and that

. Also it follows

#--z~#z~

is a com-

zhe --- F~ ~ ] ~ FI~,..~

~ zZ~

F~J

.

by

It should be noted that the singularities set of


~F.1),,.,,~

•

consist of onwith the chart


/-

and Newton polygon Z~ . be an oriented graph of adjoining the polygons

ZhA/

without cycles.

1.1.4. Theorem [8]. Assume that for any kl= ~ ,..; ,/%/ curve F~ \ { X ~ - -

where

~

OJ

runs through all edges of Z ~

pond to arcs of ~ lynomial F

coming to Zl~

with Newton polygon Z~

< F., > # . . . I T

the

i s irreducible, and

, which don't corres-

Then there exists a real poand the chart

F ,,/.>

1.1.5. R~merks.

(1) Proofs of theorems 1.1.3, 1.1.4 E8] impl y that local branches of curve f with centres on coordinate ax-

392 es correspond one-to-one to local branches of Ft~,,,; F,v determined by edges of Z~ . (2) In [8 ] it is shown that we can smooth an arbitrary subset of S C F ) in theorems 1.1.3, 1.1.4, while all other singularities are retained. 1.2 Singularities deformation. Let ~l~p ~ * 9 = S 4 ( ~ 4 + 3 ) / ~ j be the space of real curves of degree ~99 in plane. Let F ~ ~ p ~ be a reduced curve and S£ M S ( F ) -- 5,/bJ 5 2, where 51 ~ 5a = ----~ and 51, 5 z are invarlant with respect to complex conjugation COMj . If ~ = { 0 ) O ) in some affine coordinate system { x 1 ~ ) ~ then F C x ~ I ~ ) denotes the polynomial, which defines the curve ~ in this affine plane• The minimal order /< of a not vanishing K-jet of polynomial F[-xj ~/~ ) is called the order OV-~/= (~-) of F at the point ~ . Put

~O(~) = ~/~P ~ [t?,P'~I

o~(~)

~ ozd Fc~)-~].

Let T ( ~ ) ~ E65~}[I::.)~ mean the tangent cone at F to the germ of locus of curves 5m ~ ~ P ~ with a singular point in some sufficiently small neighbourhood of 1.2.1. Lemma (see [9]). For any 2 ~ 5 6 ~ (F) ,

T(~)

= ( ~ ~/~p'~ I

a E

1.2•2• Lemma (see [9]). Linear varieties

intersect transversally in ~ P Put

n4£

= O~Z,~F (2:) -- 2 ~ 2 ~- S ¢

1.2.3. Coro!lary. %vith properties

(i)

For any polynomials

J~P~n4~

P/9~-jet of

&

(x,~),

~e

5f)

~S~

(iii) P~ ~ ~ 6 5~ ~ there exists the close to F following conditions : (1)

~

qb(:~l~

curve

)

are sufficiently close to zero; ~ ~ ~P n satisfying the

P~ (x~)~@51,, neighbourhood of 5 4

is equal to

(2) ~ is non-singular outside some 1.2.4. Theorem [2]. If ~ ~ P ~ is a nodal curve, then there exists the anyhow close to F curve ~ b ~ ~ n with nodes in neighbourhoods of every point ~ ~ ~ ~ and non-singular outside these neighbourhoods.

393 2. Construction of nodal curves Here we consider curves with ~ ( / : 3 consisting of only nodes. Any gluing of such curves is allowed by theorem I. 1.3. The curve ~ with Newton polygon z~ and ~ { ~ ) , consisting of d~ knots, ~ single points and ~ g imaginary nodes, we'll denote as F ~ z~ ~ j ~/ c~ or F ~ ~ / ¢ c~ . If F~ {~/~ 6~ is irreducible and has the degree ~ , then according to the PlUcker formula [14]. 2.2. Theorem. For any non-negative integers &~ ~/ C satisfying (2.1), there is an irreducible curve ~ 6 {&]/~/ C ~ of degree ~ . Proof. According to theorem 1.2.4 it is enough to study the case ~x W- ~ -~- ~ C -: { M 4 - 4 ~ 4 - ~ / ~ . We'll use induction in ~ . All nodal curves of degree ~ Z/ are well-known [15], therefore we suppose ~ ~ ~ . I step. Assume ~ = O ) 6L ~ 2 ~ - ~ . Then

-,- ,I. (c

7-.) =

therefore there is a curve q) ree M4--~ . Using theorem from curve ~O U g ~ t] t ~ ginary conic with C~/ {] ~ p Z 2 step. Assume ~= O) (2 ~ ~ . Since -I- 2. ( c

3.)

~ ( ' ~ / O / g- - 2 ~ + ~ of deg1.2.4 we obtain the desired curve C2, where C 2 is an ima-__ ~ . ~ - 3 -~C--< ~m,!-- ~ . Then

=

.,

there is a curve q9 ~ {61-"/~ ,O, C - ~ 4 + 3 ) of degree M~-~ . Also there is a real conic C 2 meeting @9 exactly at two real points, because g l - { $ "/ . Using theorem 1.2.4 we smooth one of real points in ~ ~ ~ 2 and obtain the desired curve /c ~ {~/ % C ) . 3 step. Assume ~---- O ; C ~ PM-2.3. Lemma. ;or any ~ ~ O there exist curves ~ ¢ ; Z~¢~ (~C9-{3/~2

O2

(0; ~p.f-'f~, {9,.v~,'O]

(9.) ]

with Newton triangles and

/ (O; @3

{ ((9,'0)/ ((9,-~+~2 (~,1 /JJ

resp ec t ively. Proof. We get the curves ~ by gluing the curve ~ F~, where ~ ( ~ ) = ~ - Dg ~ { 5 ~ + ~ ~ ( ~ , ~ is a union of different real straight lines in general position, with curves

,

394

g~'=~H*~fczD+z ~+~

g+= ~(4*~c~3)÷ 4 m

Let H = ~-~ (~-43 (2d-2~)'..." (:3~-144 + ~g ) o There are real straight lines Lfj L a meeting H exactly at ~ [ C / ~ and ~ [ ( d t ~ / ~ _ ] imaginary points respectively. The gluing of 7 ~ _ 4 and H gives a curve ~ E ~ E ~ 4 4 - 3 ) ~ N 4 - ~ p / ~ ; O 1 0 3. Smoothing two real points in ~ / O (L~ U I-2) we get the desired curve F from the curve ~ U L ~ L 2 / _ z " 4 step. Assume ~ ~ j c ~ M4- ~ . Since ,~ . ( t - ~ ) -,- ~- (c - ~ . 3 ) = (~-3.)(~-~)/2 .. there is a curve ~b E {G2 g- % d - ~ t 4 + 3 ) of degree /~/?-2. Let L be a general imaginary straight line. Then we obtain the desired curve F from the curve %~ U L ~ d D ~ Zby smoothing two conjugate points in ? D ( L U co~' L ) . 5 step. Assume ~ ~ ~ ~< M4-~,. C ~ ~ta-c/ 2.4. Lemma. For any triangle U- with integral vertexes

there i~ a c ~ v e

{ E T(O,

~, O )

, and al~o, if ¢, is even,

a curve S E-T"fd~ I O / 9 / ~ ) " Proof. A curve with Newton triangle V- can be obtained from a curve with Newton triangle ~ fO/'2)~ (~-4,' ~ ~" ~ ~ by birational transformation remaining ~ (~3 (or ~ {~ ). Then is an image of

where

P~,.¢ ( g ) ---" ~-O.S6 ~ 0

polynomial, and, ~

~T~g~ ~ )

is the Chebyshev

is an image of

2.5. Remark. It should be noted that sections of polynomials /~ ~ on edge [ ( p g ? - p ; g ) 2 { ~ - f , . d p ~ ] of Newton triangle m are products of real factors, linear in ~ . Introduce the following polygons: ( i ) quadrangle ~E" =" { {~]0),, (~,1 .~)/ (JIM-'l] "/.,)2 ('IM/ OJ ~,; (ii) triangles T ~

-- {<0} ~ - ¢ - ~ )

}

{O;

I,'14-¢Ji,

¢=o,<..-,~-v,. (ii±) tri~gles T¢-= {{~-¢-¢,'03; 6~-¢; ~)~ ~-¢-z; ¢,~-z) 3, or= o,'/, - - - ~ ~ - g .

(~,~,~; n~- ¢-©,

The corresponding curves from lemma 2.4 we denote by

395

To get the desired curve F we have to glue curves 7~q ~~- f into a curve of type f C ( ~ - ~ ) / 2 ) 6~) O 2 and then to complete the construction as in 3 step. 6 step. Assume ~ ~ /4,1- ~ C ~ /4,?- ~ . 2.6. Lemma. :For any non-negative integers ~ j ~ satisfying

j

there is a curve of type ~ g - ~ ( ~ l ~ ) • Proof. According to lemma 2.4 there is a curve ~ ~I-~O a ~I-~ 0 ~ • The statement follows from the method to substitute (A) a single point for a knot, or (B) two single points for two imaginary nodes. Realize the Cremona transformation supposing that o ~ g T ( ~ ; O ; O ) = ~ - ~ , and. / O : 1 : O ) ( O ; O ~ I~ are single points of ~ . According to [14~ the proper image ~ of ~ has order ~ - 4 ' at 14;0 ; O~ , touches the line ~ = O and meets the line ~ / = O at two imaginary points, We have to turn the line ~ / = O round the point (~:O;O~ so that a pair of imaginary intersection points with ~ substitutes for a pair of real these, or we have to substitute lines ~= O) ~ = O for a pair of imaginary conjugate lines through ( ~ : ~ : ~ . And then we realize the transformation It is easy to see that the first operation is (A), the second operation is (B) Now we'll construct a curve ~ ~ ( ~ ~ ~ ) by the following algorithm. Put

~,,= ~'~ ~ -, c ~

F ,= ~ 2

~z~-~

,,

c~ = ~-~,

~= -t.

The algorithm step is as follows: (I) if ~ > ~,~a-,~-~~ ~C'~ or ~ < 0 then stop, (2) if ~ is even and ~ C ~ then glue F / with ~ ( ~ ' ¢ ~ and decrease ~/ by ~ / ~ , other~ wise glue ~ with I [ ~ £ 3 and decrease by ~ (3) substitute F ~ for the obtained curve, ~ for - ~ , and ~ for -~ , and then return to (S). After stopping algorithm we get: If ~ = ~ then ~ is the gluing of F / and a curve

F " ~ ~c'~[cb 6" c? zf ,~, ;> O )

"

l ' q - .~d' ~ ~ -

£

then F

i s the g l u i n g

396

't ec rve E ~ " ( i , i , l - y _ --

~, ~ g'j

Let

E =

q

E COjg~I

O)

"-<

#' - ! e

,, ~

dao

ve

c').

q,~ ~_c'; #'+ ~ c ' ~ ~ - ~ ,

g ' ~ (~-.~J/2#

. According to lemma 2.4 there is a curve with Newton triangle

q~

-I Cos ~ - ~ - ~ ) . , {~.~ ; v~-~J,, <"q+t, ~ - q - I J j , where ~ = ~ - ~ " o According to 2.5 the section of the polynomial ~ on the edge ~ f O / ~ - ~ - ~ 3 , {~T/ ~~3 ] has two real factors ~ ~ 55~ ~ , ~ e ~ 5~ ~ ° Therefore the curve

-~-~

with Newton triangle is of the type ~ s

% = ~ . ~ <6~ - ~ ' - + ( ~

~ {0]1411 -~.-2-), 6~. ~x" ~-~'*)., EL:); ~il-~,3J O ) 0.) . Now put ~'~-O. . . . . c ~ - ~ , +:z~J + "iJ >

where ~ 2 ~ ,~-~-4 (, ~.~15__~7-,~'~W.q ) is the section of the polynomial ~ on the edge [ { ~ ] ~ - - $ ] , {~+~j ~l-~.-fJ]. Gluing curves ~ ## ~#y ~mi ~s and a curve of type ~i-{~-~ -~gl) O) ~ we obtain a curve with Newton polygon 11{O/O).. 60/ ~4-~)~ (¢k;~-~), {~4,.o) ], According to 1.1.5(I) a section of a polynomial ~i, on the edge {O/~-~) , {~ ," ~ - ~ 3 ] is, evidently, a product of linear real factors. Hence a curve ~ ~ ~ I ~, d ) can be obtained by gluing ~ ~i with a relevant curve ~ % ~ ~ - * from proof of lemma 2o3. Analogously one has to construct the desired curve ~ ~ 1 ~# ~) in cases and 3. ,,,Construction,, ,,o,,<,,,cuspidal ,,, curves. The number ~ of ordinary cusps (i.e. of type S2#-DKJ--O ) of a complex curve of degree V~ does not exceed 5 - ~ z //~6 [5, 6]. On the other side there are curves of degree ~ " ~ with ~ a l z / cusps ~6~. Namely, let ~ f ~ l ~ , ;"~L'Z.) be the sextic curve dual to a non-singular cubic. It has 9 cusps. Then the curve ~ ~ ) 5 ~ ) .,~d'Z~") has ~ : 2 = ~'2//~/ cusps. However in this case at most ~ cusps are real, and also

397

it is unknown what numbers

~E

~'0 .;

~2/z/)

of cusps can be

realized,

We consider the problem to find the number = ~ ] /~ I for each /VE KO/ ~ ] there is a real curve of degree ~ with /V real cusps as only its singularities 3, It is well-knovm that ~('4)= ~(~)-O~ ~ ( . 3 ) = ~) ~ ( Y ) =

"~(~)

= 3, ~ ( 5 ) =

5- [4].

3. I. Theorem.

If

~

~

&

then

Proof. We'll use below the construction 1.1.4, therefore, according to 1.1.5(2), it is enough to construct a curve of degree V~

with P ~ real cusps. 3.3. Lemma. There exists a curve ~D

<'r=.l(o,.o).. (o,.3),(3,.o.).,

(3;33 ~

with Newton quadrangle and s ( ~ . ) consis-

ting of 4 real cusps. Proof. Let C q be a quartic with 3 real cusps. Let the points (O: ~: O) ,~ ( 0 : 0 : ~) lie on Cg ,and the straight line Xo = 0 touch C 4 at a real point. Then (see ~14~) ~ is a transformation (2.7) image of C ~ • 3.4. Lemma. There exists a real cuspidal cubic with Newton triangle { (O~'O~) ~3] ~ ~ (O) 3) 3 and with prescribed sections on edges

[(o;o)~

(3;o)] 2

C{o;o.~.,(o,.~)].

Proof. It is enough to show that anhsmmonic relation of origin and intersection points of C 3 on coordinate axes ~K and are arbitrary. This can be got by means of a suitable choice of axes ~

and

~

Suppose that

V~=

~ ~

. The q~adrangles

~ . : ~ ( 3 ~ 3d'), C3~;3j-,-D~ (3~.3;3j), ~3~.3; 3j'+3.)3, and the triangles

L = o~ . . . , ~ - / ,

(O,'OJ, (I,l,4"OJ) i'o)i~,,JJ.

form the regular subdivision of the triangle ~ Orient all arcs of the adjoining graph [- upwards or to the right. Define the following transformations of polynomials ~ ( ~ / ~ with Newton polygon ~ -

:

398

Then the desired curve with ing of curves ~ f 3 6 Bj ~ i H j

~2__

~(~j~)~

~ ~/0~

real cusps is a glu~'~0/

('kj~ ~ - 2 ~

and also suitable cuspidal cubics with Newton triangles ~ , ' O_< i ~ ~- ~ . Here conditions of theorem 1.1.4 are satisfied, because ~ = ~ for any ordinary cusp. If ~ = 3 ~ + 4 or 3 ~ ~ ~ , it is necessary to add to the above gluing some curves ~ with S(~) = ~ and Newton triangles {~0]~)) {3~ j B@-3£) 2 {'3Le3 ~ 3~.-3~-3) ~

(3¢,.o)9.

7,-¢)

4. Oonstruct,i, on of cu~v,es,, with arbi,trar y s i r ~ u ! a r i t i e s 4.1. Theorem. For any collection ~ D~; . . . ) 5 ~ ~ al singularities and for integer iS9 ~ O satisfying

of re-

where ~ is a Milnor number, there exists a real irreducible curve of degree DI with S~;...~ 5 ~ as only its singularities. 4.3. Remarks. (1) The left hand side of (4.2) cannot be increased, because a singularities number of a degree ~4 curve is at most M 4 2 / 2 [14]. The left hand sides of known estimates [3, 10] are linear functions of the curve degree. At the same time, probably, the singularities invariants in (4.2) can be decreased. (2) The getting of the type (4.2) estimate in the ~ y of the formal comparison of a curve coefficients number with a number of conditions imposed by singularities is hardly possible, because it is very difficult to prove the system of these conditions gives a curve without additional degenerations. Proof of theorem 4. I. If ~{Dg, ~ ) = ~ defines the singularity S at { 6 )] (9) then the equation ~ ( o d j ~ ) ----0 with the same {~{~)+ 43 -jet defines the same singularity at ( ~ # ' O) (see [I]). Therefore, according to 1.2.3, it is enough to construct a reduced curve of degree ~/I , satisfying (4.2), with points ~ £ of orders ~ Z d { ~ ) ~{S~)~ 3 # 4-.<~ < ~ 4.4. Lemma. For any integers K ~ >I _._ >~ ~ ~ u/ there exists a reduced curve of degree

~) with points

~

of orders

(4.~)

399 (4.6) Proof.

Put

~TC(L) = { I . - I ) ( L ~ - 2 ) / 2

,

;. ~_. 9__.

Let ~YC(~) ~ ~ < ~f-{9. 4) and points 2 ~ ) , , , ; general position in a plane. Note that ~ T U [ ~ 4.) of the space of the degree ~ curves. Since there are different irreducible curves

~ be in is a dimension

,/_.< ( of degree

~

, which pass through ~ i)---) ~, . For any is a dimension of the , since ~i-[~-j+ ¢) space of the degree ( ~ - j ) curves, there are different irreducible curves

j = 4, ---,9 - ~

,

/

of degree $ - j At last let

4_~i , which pass through ~ Z/ 4 ~< Z ~¢7%-~9-#'+ 4)- /.

be different straight lines through 2 I • Consider the reduced curve ~--- Q)~/J) point ~ Z ~ ~'V[(~,)~ Z ~ ~ , belongs to ~p/o") Since~_
.,.

.j

~'I~'(9~

~,

then

o"zd~(~,z.) ~ K..:,.cc,],)~ / ( z

,, ,-v'c(~.) ~ z __~ e t .

(4.7)

Analogously, any point ~ z~ ~ ~/'C'(~,-~.) ~ ~. < ~/'C($ - ~ -e- { ) > ~< ~-< ~ - 2 , belongs to ~¢(J)) 0 ~ < ~ , therefore

>~ K ~z )

~T~-C$ - ~ ) <_ Z

< "TC [ ~/ - ~ + {.) .

(4.8)

-4d

(4.9)

At last

-(J)

,~-,/}

K.~

Inequalities (4.7), (4.8), (4.9) imply (4.6). Now we'll give an upper bound for ~ ~

,4= a,

1, =

+

:

%-0+...÷

400

-- K,rC(~ j f

K,ri.(~_f ) -l- -.. 4- K~_cC~) -~- K, I = ~

~L = K,~r-~L) ,,

where

K z ~K.rc~ )

,,

~

C ~< ~ )

-h

Z~.?"~-...+ "~Z -l- "~f/

"Z'# ----- ~ - Z

. Since

~---~'C($_) ~ Z ~ : I T ( 3 ) ~

then 3,

Kt 3

3

=

@ i---3

z l3 + z ~ + .s

_3

..

3

.

/<

-L

#

3

---

~

The ginkovski inequality implies

m=~._

m2 ~

~ /~z~ (1+ I+ ~ / ~ - -

N~ ~ ~

~-jH(eD

~

¢TS/3.

-<

+

Since

+ Z~

(4. lO

then

~ ~ (r.~-l)~ ~ 9 ~ ~ ~A:~/2.

Therefore (4.10) implies

what is equivalent to (4.5) i Now for any degree satisfying (4.2), we have to add to the curve from lemma 4.4 a suitable set of straight lines in general position. References 1. Arnol'd V.I., Gusein-zade S.N., Varchenko A.N. Singularities of differentiable maps, vol. I. Boston, Basel, Stuttgart: Birkhauser Verlag, 1985. 2. Brusotti L. Sulla "piccola variazione" di una curva piana algebrica reali. Rend. Rom. ACo Lincei (5), 30, 375-379 (1921) 3. Gradolato N., Mezzetti E. Curves with nodes, cusps and ordinary triple points. Ann. Univ. Ferrara, sez. 7, vol. 31, 23-47

401

(1985) 4, Gudkov D,A. On the curve of 5th order with 5 cusps. Function. anal. Pril. 16, no. 3, 54-55 (1982) (Russian) 5. Hirzebruch ~. Singularities of algebraic surfaces and characteristic numbers. Contemp. Math. 58, 141-155 (1986) 6. Ivinskis K. Normale Flachen und die Miyaoka-Kobayashi Ungleichung. Diplomarbeit, Bonn, 1985. 7. Severi F. Vorlesungen uber algebraische Geometrie (~nhang F). Leipzig, Teubner, 1921. 8. Shustin E.I. Gluing of singular algebraic curves. In: Methods of Qualitative Theory, Gorky Univ. Press, Gorky, 1985, pp. 116-128 (Russian) 9. Shustin E.I. On manifolds of singular algebraic curves. Selecta Math. Soy. 10, no. I, 27-37 (1991) 10. Vainstein A.D., Shapiro B.Z. Singularities of Hyperbolic Polynomials and Boundary of Hyperbolicity Domain. Uspekhi Math. Nauk 40, no. 5, 305 (1985) (Russian) 11. Viro O.Ya. Real varieties with prescribed topological properties. Doct. thesis, Leningrad Univ., Leningrad, 1983 (Russian) 12. Viro 0.Ya. Gluing of algebraic hypersurfaces, smoothing of singularities and construction of curves. In: Proc. Leningrad Intern. Topological Conf., Leningrad, Nauka, 1983, pp. 149-197 (Russian) 13. Viro O.Ya. Gluing of plane real algebraic curves and construction of curves of degrees 6 and 7. In: Lect. Notes Math., vol. 1060, Springer, 1984, pp. 187-200. 14. Walker R. Algebraic curves. New York, Dover, 1950. 15. Zeuthen H.G. Sum les differentes formes des courbes planes du quadrieme ordre. Math. Annalen, 408-432 (1893)

Effective stratification o f regadar real algebraic varieties Nicolai N. Vorobjov st. Petersburg Dept. of Mathematical Steklov Institute of Academy of Sci. 27, Fontanka, St. Petersburg, 191011 Russia A b s t r a c t . An algorithm is proposed, producing a W h i t n e y stratification for a real algebraic v a r i e t y which is a union of t r a n s v e r s a l l y intersecting smooth varieties. The complexity of the algorithm and the estimates on the p a r a m e t e r s of the produced s t r a t a are single exponential in the n u m b e r of variables of the i n p u t polynomials. IntroductionLet us define a regular real algebraic variety to be the union k ) W i of a l_
(-~

Wje ~ ~ t h e n this i n t e r s e c t i o n is

l_<e_<S1

transversal. I n t h i s p a p e r a n a l g o r i t h m is p r e s e n t e d for c o n s t r u c t i n g in s u b e x p o n e n t i a l time (see below) a W h i t n e y stratification (see e.g. [I]) of a regular real algebraic variety given by an a r b i t r a r y formula of the kind

k]

&

1~/~ l_<j~r where polynomial flj e Z [ X 1.... ,Xn].

( f ij = 0)

(I)

L e t us a s s u m e t h a t t h e i n p u t f o r m u l a (I) s a t i s f i e s t h e following e s t i m a t e s : degx1,...~n (fij) < d, the absolute value of every (integer) coefficient a p p e a r i n g in fij does not exceed 2 M (hence bit-length g(flj) of every coefficient does not exceed M) and t+r = K for some natural d, M. T h e o r e m : There is an algorithm that produces for a n y formula of the form (I), defining a r e g u l a r real algebraic variety V c IR~ and satisfying t h e mentioned estimates, a W h i t n e y stratification of V. This stratification is r e p r e s e n t e d by a family of s t r a t a - semialgebraic sets V o = {0 0}..... Vm = {Om} c IR~ (m < n), where Oj (0 -<j < m) are quantifier-free formulas. E v e r y Oj containes (Kd) n°(1) atomic subformulas of the kind (g > 0) where polynomial g e Z [ X 1..... X~] satisfies the bounds :

403

degx t .....x, (g) < (Kd)n°(1),

e(g) < M °(1) ( K d ) n°(1) .

The r u n n i n g time of the algorithm is polynomial in M, ( K d ) ~°(1) . L e t us a d m i t t h a t the n a t u r a l idea of c o n s t r u c t i n g a s t r a t i f i c a t i o n by recursion (roughly speaking : defining firstly singular points with the help of a formula of the first-order theory of reals h a v i n g small n u m b e r of quantifiers, after t h a t defining the singular points of the l a t t e r set a n d so on) will give the a l g o r i t h m w i t h the complexity double exponential in n u m b e r of variables (more precisely - in the "depth" of the singularities). This idea can be applied to an arbitrary (not necessary regular) algebraic variety [9]. The set V admits a certain s t a n d a r d stratification. Namely, suppose t h a t the stratification for the set W (e) = k.J W j (1 < e < s) is a l r e a d y defined. T h e n l<_i<_e define the stratification for W(e+t) = L3 W j whose s t r a t a are : l
~ , J~ij . Let f = 1-I f i . It is easy to l <_j
check t h a t the variety V = {f=-0}.

I. I n f i n i t e s i m a l s . Let F be a n y formally real field, denote by P its real closure (see e.g. [4]). Introduce an element e > 0 infinitesimal relative to F (i.e. for every 0 < a e F the inequality e < a is valid). The elements o f F ( c ) are P u i s e u x series in s i.e. for a e F(e) holds : a = ~ a i £ i>O

vii #

where 0 ~ a i ~ F for all i > O, integers vo < v 1 < ...

increase a n d the n a t u r a l n u m b e r # > 1. If v o < O, t h e n a e F(e) is infinitely large, if v o > 0 t h e n a is infinitesimal relative to F. If v o > 0 t h e n the s t a n d a r d p a r t st~(a) ~ i s

defined : 0 if vo>0 stc(a) =

a o if vo=O

404

The s t a n d a r d p a r t of a vector (/31..... /3,) is defined component-wise. By stl,2(.) we shall denote the s u p e r p o s i t i o n st~,(st~2(.)) in case the several extensions are involved. The following "transfer principle" is valid : i f F 1 c F 2 are two real closed fields, F 1 is a subfield of F 2 t h e n every proposition expressed by a formula of f i r s t - o r d e r t h e o r y of real closed fields w i t h coefficients (of the atomic polynomials) in F : is true over F 1 if and only if it is true over F 2 . Let A c ~ ) n be a semialgebraic set defined by a formula with coefficients in F . T h e n by A (e) we shall denote the set defined in (F(~.)) n by the same f o r m u l a . Analogous n o t a t i o n we shall use i f several i n f i n i t e s i m a l s are involved.

2. Auxiliary propositions. I n t r o d u c e tree i n f i n i t e s i m a l s 0

<

E2 < C1 < t:0 S0 t h a t Q (i = 0,1,2) is

infinitesimal relative to the field Q(c 0..... Q_I ) and the v a r i e t y ~ = {f-e2=0} c (Q(eo,Q,e2)) n . Note t h a t :g is a smooth hypersurface. The m a i n idea of the algorithm is t h a t close to any point of p-dimension s t r a t u m Vp (of the s t a n d a r d s t r a t i f i c a t i o n of V) the h y p e r s u r f a c e ~r looks like a cylinder over ( n - p - 1 ) d i m e n s i o n m a n i f o l d h a v i n g i n f i n i t e l y large a b s o l u t e v a l u e s of principal curvatures. This property one can express in the language of first-order theory of real closed fields and t h u s "define" the s t r a t u m Vp. The role of the base of the mentioned cylinder plays a section of ~ with a ( n - p ) - p l a n e which is orthogonal to Vp at some point from Vp. The idea of studying a stratification of a zero-level set of a function via the behaviour of a close-level smooth set was introduced (in much more abstract setting)) in [10] (see also [11]). Choose x e Vp. Note t h a t the intersection M = V(e°'el'C2)r~ ~ x(eO) is p connected since Vp n ~x(5) is connected for all sufficiently small 0 < 6 • Q and according to t r a n s f e r principle. Here ~z(R) denotes a n open ball of r a d i u s R about z. Let = U

yeM

(gnU(q)).

Let L, L : : (Q(~o,sl,~2)) n be r-and m-planes correspondingly, r > m > 0. We shall say t h a t L a n d L 1 are eo-collinear if for every u n i t vector v I e L 1 there exists such a vector v • L t h a t v:=v+w w h e r e all c o m p o n e n t s of w are infinitesimals relative to Q.

405

Denote by Pz the (n-p)-plane orthogonal to M a t the point z e M a n d by ~£ c J/ a subset of points of J/ a t each of which the n o r m a l to ~ / i s n o t Eocollinear to (n-p)-plane Px" L e t us r e m a r k t h a t ~ is open in :g in the topology with the base of all open disks i n ~ . Note t h a t the set £e is n o t s e m i a l g e b r a i c (its d e f i n i t i o n i n c l u d e s t h e indication on unspecified element infinitesimal relative to Q). Nevertheless we shall consider the connected components of ~, namely, the l i n e a r l y connected components. The subset LPo c ~¢ will be called l i n e a r l y connected i f for every pair of points x~y • ~o there exists a connected semialgebraic curve belonging to

"~0 C (Q(e0,E 1,£2)) n a n d containing x, y. L e t us prove prove t h a t ~ h a s a finite n u m b e r of l i n e a r l y connected components. Recall a t first t h a t if a semialgebraic set is defined by a quantifierfree f o r m u l a of t h e first-order t h e o r y of real closed field w i t h K atomic polynomials h a v i n g n variables and degrees not exceeding d t h e n the n u m b e r of its connected components is less t h e n a value of a certain function a in K, n, (as such a function one can t a k e (K d )0(~) b u t this is not essential here). Suppose t h a t ~ has as infinite n u m b e r of (linearly) connected components. Choose a finite n u m b e r t of them. Then there exists an element 0 < a • ~ s u c h t h a t in each of the chosen components there is a point z for which the u n i t vector v 1 normal to ~ at z a n d every vector v • Px the n o r m [ v-v 11 > a. Indeed, c o n s i d e r t h e f u n c t i o n z : ~ .....

> ~

which to every p o i n t z • ~¢ p u t s in

correspondence t h e e l e m e n t 0 < fl • ~ equal to t h e m i n i m u m of the v a l u e Sto,l,2(Iv'-v ~ [) where v~ is the u n i t vector normal to r a t z a n d v" r u n s t h r o u g h all vectors v' • Px- For every connected component LP0 of L¢ the image z(~eo) = (0,~) c ~

where 1 > X • Q . Therefore for chosen c o m p o n e n t s of ~ t h e

intersection of z-images is also an interval of the kind (0,~) c ~ t h u s for a one can t a k e a n y element from (0,X). Now consider a semialgebraic set £2 of all point z from J~ such t h a t for a u n i t vector v 1 normal to ~ at z and every vector v • Px the n o r m [ v-v 1 [ > a. The set ~ can be given by a formula of the first order t h e o r y w i t h pl.k atomic polynomials of degrees less t h e n P2.d a n d h a v i n g P3.n v a r i a b l e s for some n a t u r a l n u m b e r s Pl,P2,Pa" Therefore the n u m b e r of the connected components of ~ is less t h e n a(plk,P2d,p3n). Obviously ~ c ~. T h u s the n u m b e r of the connected components of ~ is not less then t. We can choose t > a(Plk,P2d,p3n)contradiction.

406

We shall call a s u b s e t J/1 c J / p - l o n g i f p is the maximal n a t u r a l n u m b e r such t h a t for every point z from some p-dimension s u b s e t M 1 of M holds : P~ r~ J/1 ¢ ~#-It is clear t h a t J/1 is p-long iff dim(stl,2(J/1)) = p . L e m m a 1. The set ~ is not p-long. P r o o f . Suppose t h a t ~ is p-long. Choose in ~ (correspondingly - in M ) a pdimensional smooth connected semialgebraic s u b s e t ~1 (correspondingly - M 1) such t h a t for every z e M 1 the intersection P= r~ ~fl ¢ ~ consists of a unique point and this correspondence is bijective. The sets ~1 and M 1 can really be chosen. Indeed, since the n u m b e r of the c o n n e c t e d c o m p o n e n t s of ~f is finite, a m o n g t h e m t h e r e will be a p - l o n g c o m p o n e n t ~'. Choose in the set ~ ' which is open in 7z a p - d i m e n s i o n connected semialgebraic s u b s e t ~" which is a smooth manifold. According to the definition of p-long set there exists a p-dimension open subset M" of M such that for every y e M" the intersection Py c~ ~ " ~ ~. Besides, from the transfer principle follows t h a t for all pairs Y l , Y 2 • M " c M holds : Pyl ¢'~ Py2 ('~ fl.t = d). Therefore we have t h e smooth m a p of smooth manifolds 92 : CA~''

) M", if

z e ~ " c~ Py then 92(z) = y. By Sard's theorem, t h e r e exists a connected p-

dimension semialgebraic s u b s e t M 1 c M" such that for every point y e M 1 the fibre 921(y) consists of the noncritical points of the m a p 92 and t h u s - of finite n u m b e r of points. Let ~1 be such a smooth semialgebraic subset of 921(M1) t h a t 92(~.~1 ) = M 1 a n d the restriction of 92 on -~1 is injective. Therefore we h a d constructed the sets M p ~1 having the necessary properties. Choose a point x 0 • M 1 and denote by L the p-plane t a n g e n t to manifold M at x o. Since the smooth manifolds Vp can be given by a formula of first-order theory of the field Q for every point z e M the intersection Pz n L ~ ~ consists of the unique point and this correspondence 91 "L

> M is bijective.

L e t L 1 = ~0~1(M1). Denote by ~1 : L1 > Q(eo,~l,e 2) a smooth function taking the value of the euclidean distance b e t w e e n a point from L 1 and its image in M 1 in force of the smooth map 91. Denote by V2 : L1 " Q(~o,Q,~2) a smooth function which for y e L 1 is taking the value equal to the s u m of ~I(Y) and the distance between 91(Y) and 921(9](y)). Let us prove t h a t there exists a point a • 5g1 such t h a t euclidean distance form a to M is an element from Q(eo). This will contradict with the definition of J/.

407

According to Lagrange's t h e o r e m on finite i n c r e m e n t s , for every point y e L 1 there exist points $1 e L 1 and 02 e L1 such t h a t ~I(Y) = < grado~(~), (y-x o) > , ~2(Y) - ~2(Xo) = < grade2(~'2), (Y-Xo) > , therefore ~2(Y) -- ~I(Y) -- ~2(X0) = < grado2(tg2) - gradel(V1), (Y-Xo) >Choose a point y 0 e L 1 such t h a t lYo-Xol = e~ for some 0 < 7 e Q, grad=o(~ 2) is c011inear to (yo-Xo) and W2(Yo) _>~2(Xo). Then by definition of the set ~ D ~1 for c o r r e s p o n d i n g O 2 the scalar product < grade2(~g2), (Yo-Xo) > ~ a e~ w h e r e 0 < a e Q. Besides, [ g r a d o , ( ~ l ) [ < e~o for a certain 0 < fl e Q a n d t h u s < grade,(Wl) , (Yo-Xo) > Igrade,(wl)[. l yo-xo I <e~o+r . Let a = ~21(~1(yo)). Then the distance from a e A1 to M 1

v'2(Yo) - ~(Yo) > ~(a-~o) + ~(Xo) > e~. This contradicts the inclusion Stl,2(a) e M 1 which is valid by definition of the set A1 c ~q. The l e m m a is proved. Let us r e m a r k t h a t the l e m m a r e m a i n e s t r u e if we take i n s t e a d of ~ a n a r b i t r a r y s m o o t h algebraic h y p e r s u r f a c e t h e s t a n d a r d p a r t st 2 of which coinsides with V. The property to be algebraic is i m p o r t a n t : consider for example a line on a plane for V and an infinitely close sinusoid h a v i n g an infinitesimal period for the smooth hypersurface. Denote by -¢ : ~ ) S n-1 the Gauss map (see e.g. [5]) o f ~ , here

s--*= { x

Ixl = 1 }

Corollary. Fix a point x e fie°). p For almost all point y e Cep O) c~ ~=(e o) (i.e. for all points with the exception, maybe, of the points of a semialgebraic subset of the dimension less t h e n p) holds : d i m ( s t o & 2 ( J ( ~ • ~y(Q)))) < n - p - 1 . Moreover, the set Sto,l,2(J(~ n ~y(el))) belongs to the intersection of the (n-1)sphere S "-1 with (n-p)-plane, passing through 0. P r o o f . According to l e m m a I, stl,2(5~) c

fie°) p (where ~ appears from x as in

l e m m a I) is a semialgebraic subset of the dimension less t h e n p. Let the point y belong to the complement to stl,2(5~) in 1fiE°) p . Then, by definition of £P, for any point z e (Tz ¢~ ~y(~l)) the normal to ~r at this point is %-collinear to (n-p)-plane which is passing through 0 and is collinear to P=. The corollary is proved.

408

Consider the s t r a t u m Vp for a certain 0 _

j< s, for almost every point y e ~/;0) n ~x(Co) holds : dim(sto, l,2(J(~v c~ ~y(Q)))) < n - p f f l

(2)

L e m m a 2. Let for almost every point y e ~%) pj c~ ~x(Co) the equality t a k e place in (2). T h e n dim(sto&2(J(Tz c~ ~x(Q)))) = n - p - 1 .

P r o o f . According to the corollary to l e m m a I (being applied to Vp) t h e inequality holds : dim(st0j,2(J(~" c~ ~x(Q)))) = l < n - p - 1 . Assume t h a t the strict inequality takes place. Let tl= max {/, n - p 1 - 1 . . . . . n - p s 1}. T h e n there exists an element 6 : ~0 > 6 > 0 of the field Q(E o) such t h a t for all Ce 0) points y e c~ ~ ( 6 ) the following holds : dim(sto,l,2(Y(~ n ~y(el)))) < t 1 (in the opposite case on can find Yo e

Ce )

c~ ~x(6o), w h e r e 5 o is infinitesimal

relative to Q(e o) such t h a t dim(sto, l,2(J(r c~ ~yo(60)))) > t l + l therefore the last inequality is also valid if one changes So by Q a n d Yo by x). Let W = ~ c~ ~ x(6). Thus dim(sto,l,2(J(W))) = t 1 < n - p - 2 . There exists a n element 0 < v e Q(e o) infinitesimal relative to Q such t h a t

L3

~w(v) D .¢(Z).

W e (Sto,1,2(J(,~f)))(eO'eI'e2)

For each point w e (Sto,l,2(J(Ar))) (eO,el,e2) denote by Aw the fibre J-l(~w(v) c~ J(W)). Obviously W=

LJ Aw . w e (Sto,l,2(J(]~))) (eO,el,e2)

409

Since all normals to ~ at points of the set A w are eo-collinear, every point from A w is situated in an infinitesimal relative to Q distance from a certain (the same for all points) hyperplane, besides the normal to this h y p e r p l a n e is eo-collinear to every normal to ~ at the points from A w. We shall say t h a t a set P c (Q(Eo,E1,E2)) n at a point z e P has dimension r is for all sufficiently small elements 0 < a • Q(CO,El,e 2) the intersection P n ~z(a) h a s dimension r. Denote by T the s u b s e t of all points from the set Sto,l,2(J(X)) c S n-1 at which this set has dimension t < t 1 . F u r t h e r we shall use considerations very similar to those which occur in the theory of the developable surfaces. One can prove t h a t in fact every point from Aw where w • T (•0'el'e2) lies in infinitesimal relative to Q distance from a certain (the same for all points) ( n - t - 1 ) - p l a n e , herewith the orthogonal (t+l)plane is ~o-collinear to each normal to ~ at the points from A w. Indeed, consider t + l points wj • (st o 1,2(J(fl/')))(e°'cl'c2)(1 < t + l < n - p - l ) such t h a t w h e n J l ~ J2, firstly ~ h ( v ) ~ ~ w i 2 ( v ) = ~ and, secondly, t h e d i s t a n c e t wjl-wj21 i s i n f i n i t e s i m a l r e l a t i v e to Q o B e s i d e s , a s s u m e t h a t t h e h y p e r p l a n e s corresponding (see above) to the points wj are in the general position. Then the fibres A ~ i over the points wj which "do n o t d e v i a t e much" from t h e corresponding h y p e r p l a n e s and from each other, also "do not deviate much" from the intersection of the hyperplanes i.e. from ( n - t - 1 ) - p l a n e . Every fibre A w where w • T (e°'~1'c2) can not be bounded by a set of points y such t h a t

dim(sto,l,2(J(~ n ~y(el)))) > t since this would contradict the definition of T and is a ( n - t - 1 ) - l o n g set. Let us prove t h a t T belongs to the union of finite family of ( t + l ) - p l a n e s p a s s i n g t h r o u g h 0. Indeed, in the opposite case t h e r e exists a t - d i m e n s i o n s u b s e t L CSto, l,2(J(X))) (~0'~1,~2) such that none of its t-dimension s u b s e t L 1 c L lies in (t+l)-plane passing through 0. C o n s i d e r two p o i n t s w 1 a n d w 2 • L s u c h t h a t ( n - t - 1 ) - p l a n e s corresponding to fibres Awl and Aw2 are not So-collinear. E v e r y fibre Aw~ a n d Aw2iS an ( n - t - 1 ) - l o n g set and an argumentation similar to the one in the proof of l e m m a I shows t h a t t h e r e exists a point a • Aw~ (~ ~=(5) such t h a t t h e n e a r e s t to a point ]3 • Aw2 belongs to ~x(5) and the distance I a-fl [ > 0 is not infinitesimal relative to Q(So ). It follows that the intersection J - I ( L ) n ~ ( S ) is a ( n - 1 ) - l o n g set.

410

In case when there does not exist a s t r a t u m Vpjof dimension pj > n - t - 1 which is adjacent to Vp this contradicts the corollary to l e m m a 1 a n d in the opposite case the inequality (2) (applied to Vpj). Therefore we have proved that Sto, l,2(~c(JV)) ~ S n-1 belongs to the union of a finite family of some planes (of maybe different dimensions) passing through 0, note also t h a t for each pair Wl, W2 e T (%'c1'~2) the ( n - t - 1 ) - p l a n e s corresponding to Aw~, Aw2 are eo-collinear. Now let us prove t h a t this family consists of not less then two members. Moreover, there exists a connected component X 1 of X such that the same is true for W 1 Suppose the opposite : let it consist of unique t = tl-dimensional member. Since n - t - 1 > p, there exists a s t r a t u m Vpi ( l < j < s) such t h a t the intersection (~o) ¢ ~ for every w e T (%'c1'c2). Denote by B w the ( n - t - 1 ) - p l a n e stl,2(A ~) n Vpj corresponding to Aw. From the transversality condition it follows t h a t Bw is s 0(E0) n ~ ( 5 ) . C o n s i d e r two collinear to one of the connected components of _Vpj connected components V'

c

.(eo)r~ ~ x ( 5 ) of some s t r a t a Vpj, (E0) ('~ ~x(5), Y" C Ypj,,

~Eo) V'pi,, ~o) (possibly j ' =j") such that Y' ~ V", V" a: V,, ~r, (~ ~r,, ¢ $ (here the b a r V'pj,, denotes the closure in the topology with the base of all open disks). Choose a point z e Vp (~ V' r~ V" (~Qn and consider a tangent pj,-plane L ' (correspondingly pj,-plane L") to V' (correspondingly - to ~r,,) at z. Note that B

is defined over

Q ( s o) while L ' ,L "- o v e r Q- . Let us suppose t h a t B w is So-collinear to both L' and L " (we shall s a y t h a t in this case B w is eo-collinear to V' a n d V"). T h e n

Sto(B w) c L" r~ L". Therefore

min{dim(L'), dim(L")} > d i m ( L ' r~ L") > n - t - 1

and B~ is eo-collinear to ~r, n V" = ~(3) where V(3) is the intersection of ~ ( 5 ) with a connected component of a certain s t r a t u m among --Vtp~°) . R e p e a t i n g this a r g u m e n t a t i o n if necessary we can a s s u m e t h a t t h e r e does not exist a n o t h e r s t r a t u m among --Y~:.°) ( l < j < s) such t h a t for an intersection V(4) of its connected component with ~x(5) holds : V(4) ~ 3 ) , a n d V (4). The inequality

~(3) cET~4) and B~ is So-collinear to V(3)

dim(Vp) < n - t - 1 a n d the t r a n s v e r s a l i t y condition

imply t h a t there exists a connected component of some s t r a t u m a m o n g --V~°) d

whose intersection with ~x(5) is transversal to V(3). F r o m our a s s u m p t i o n it

411

follows t h a t B w is So-collinear to this intersection. T h u s t h e r e is a point S t o , l , 2 ( J ( X 1 ) ) not belonging to the ( t + l ) - p l a n e u n d e r c o n s i d e r a t i o n . This contradiction implies that there exist two points w 1, w 2 • Sto,l,2(.c(X1)) such t h a t fibres Awl, A~2 intersect transversally at a point from X. This contradicts the smoothness of X. The l e m m a is proved.

L e m m a 3. For almost every point x • Vp (with the exception of a semialgebraic subset of a dimension less then p) holds : dim(stl,2(.¢(V n ~x(Q)))) = e = n - p - 1 . P r o o f . From the transfer principle it follows t h a t it is sufficient to prove the proposition for ~/~e0)instead of Vp and sto,1,2 instead of stl, 2 . P

Proceed by induction. As a base of the induction consider the case w h e n the s t r a t u m Vp is maximal i.e. it is not incident to a n y s t r a t u m of g r e a t e r dimension. Then for almost every point x •

pe°) t h e intersection P x (~ 7r is

obviously diffeomorphic to ( n - p - 1 ) - s p h e r e , therefore e > n - p - 1 . On the other hand, according to the corollary to l e m m a I, e < n - p - 1 .

S u p p o s e t h a t the

l e m m a is proved for s t r a t a of the dimension g r e a t e r t h e n p. Consider nonmaximal s t r a t u m V (e°) and apply to it lemma 2 (the hypothesis of l e m m a 2 is P

valid by the inductive hypothesis). The lemma is proved.

3. R e p r e s e n t a t i o n s o f s o m e s e m i a l g e b r a i c sets. Consider a semialgebraic set R given by a formula of first-order theory of Q. Let the dimension of R at every point (see the proof of l e m m a 2) be the same and equal to ra. The fact that z • R is a smooth point in R (that is the fact that for a n y two pairs of distinct points in the neighbourhood of z in R, the lines passing through the first and second pair are almost collinear to the same mplan) can be expressed by a first order formula. Note t h a t in R = Vp every point is smooth ; if Vp+i is incident to Vp then the point z • Vp c (Vp+ i u Vp) = R is not smooth. It is hence clear t h a t the s u b s e t S m ( R ) of all smooth points of R is semialgebraic and can be w r i t t e n as a formula of first-order with a c o n s t a n t (i.e. not depending on R) n u m b e r of quantifier alternations. In the algorithm described below an i m p o r t a n t role is playing the set of the form :

412

Kp = stl,2( {x e//" : dim(stl,2(~¢(~ f r~ ~x(el)))) = n-p-l} ), where the external st is taken for all points of the set for which it is defined. L e t us prove t h a t Kp is a semialgebraic set which can be given b y a formula with

a constant

number

(not d e p e n d i n g

on ~ ) of q u a n t i f i e r

alternations. Indeed, the image of a semialgebraic set u n d e r the G a u s s m a p is obviously semialgebraic. The s t a n d a r d p a r t of a semialgebraic set A in the representation of which atomic polynomials belong to Q[Sl,~ 2] [X1..... X n] is, as it was noted in [3], also semialgebraic (consider ~l,S2 as n e w variables, t h e n stl,2(A) coinsides with the intersection of n-planes {Q = £2 = 0} and the closure in euclidean topology of the set A (~ {Q > 0 & E2 > 0}. The proposition dim(.) = t can be w r i t t e n in the following w a y : there exists a linear transformation of coordinates L : (X 1..... Xn) ' > (Y1 ..... Yn) with the matrix

I

I ~'2 1 "'" ... ~'n 0 1

0

0

0

0

(3)

1

such t h a t the projection of the set on the subspace of coordinates Y1 .... ,Yt containes a t-ball and for every linear m a p of the kind (3) the projection on the (t+l)-subspace does not contain a (t+l)-ball.

4. The algorithm and its m, nning ~me. As a n a u x i l i a r y p r o c e d u r e s our a l g o r i t h m

e s s e n t i a l l y involves t h e

effective algorithms for quantifier elimination in first-order t h e o r y of real closed fields proposed in [6] (see also [7]) a n d for finding all c o n n e c t e d components of a semialgebraic set from [2] or [3]. The algorithm works recursively in p. Let p = m = dim(V). The algorithm defines the set Sm({x e V/dim(V) in x equals m}) by a formula (with quantifiers) O~ ) of the first-order theory ofQ(e~,E 2) as it was explaned a t the beginning of section 3. After t h a t using several times the quantifier elimination procedure from [6] t h e a l g o r i t h m p r o d u c e s q u a n t i f i e r - f r e e f o r m u l a (~(2) w h i c h is equivalent to O ~ ). Obviously Vm = {O~)}. A s s u m e t h a t s t r a t a Vm, .... Vp+t a r e a l r e a d y c o n s t r u c t e d a n d given by formulas O m ..... ep+ 1 correspondingly. The algorithm defines the set Sm(Kp) by a formula (with quantifiers) O(p1) as it was explaned in section 3, and then, with the help of procedure from [6] produces an equivalent quantifier-free formula

413

O(p2). Using [2] the algorithm finds all connected components of {@(p2)}and after t h a t with the help of the procedure from [8] for solving systems of polynomial inequalities selects those which have empty intersections with every Vm..... Vp+1. Let the selected components be given by formulas O(p2'1)..... O(p2'sp). For every l < i <sp the algorithm writes a formula (with quantifiers) O(p3'/) defining the closure (in the topology in (Q(Q,~2)) n with the base of all open balls) of the set Op(2,/) and then, eliminating quantifiers in Op(3,/) finds an equivalent quantifierfree formula O(p4'/) . Let o(pb) =

V

O(p4J) and O(p6) be the formula defining Sm({O(pb)}).

l<-i<-sp

E l i m i n a t i n g quantifiers in O(p6) the algorithm obtaines the quantifier-free formula Op. Let us prove that Vp = {Op}. We m u s t show that if U 1..... U t are all connected components of Sm(Kp) which have empty intersections with each of the strata V m ..... Vp+1 and V~ = t.9 ~rj is the union of the closures of U 1..... U t then Vp = l <j
k.)

V i iff Uj belongs to this union.

p+ l <_i<m

The inclusion Sm(V'p) c Vp can be proved by the reverse argumentation. Let us estimate the running time of the algorithm and the parameters of the produced formulas Op. Recall that the input variety is given by the formula (I) where f ij • Z[X1 ..... Xn], degx, ..... x.(f ij) < d,e(foj) < M and the n u m b e r of the atomic subformulas is K. The image J ( ~ r~ ~x(E1)) is defined by a formula F (1) in the prenex form with one block of existential quantifiers. The polynomials occuring in the quantifier-free p a r t have total degrees less t h e n K d and bitlengths of coefficients not exceeding log(K)M. The application to F (1) of the procedure from [6] requires time M °(1) (Kd) n°(1) and the resulting quantifierfree formula F (2) has (Kd) "°(1) atomic subformulas with polynomials of the degree (Kd) ~°(1) having coefficient lengths not exceeding M °(1) (Kd) ~°(1) . The formula defining the set stl,2({F(2)}) has two quantifier alternations, the result of quantifier elimination is a quantifier-free formula F (3) with the bounds of the same kind as in formula F (2).

414

The formula which defines the value of the dimension has the n u m b e r of variables linearly depending on n and is a conjunction of two formulas in prenex forms the first of which has the prefix of existential quantifiers and the second - of universal quantifiers. As a result of quantifier elimination in both members of the conjunction a formula F(p4) will be obtained having the bounds of the indicated type. Elimnating both quantifier blocks in stl,2({l-(p4)}) the algorithm obtaines a quantifier-free formula F (5) such that Kp = {I'(p5)} again having the estimates of the same kind as in F (2). The total running time for the constructing of F(p5) will be M °(1) (Kd) nO(l) . The formula defining the set Sm({F(pS)}) has a n u m b e r of variables polynomially d e p e n d i n g on n and one block of universal quantifiers. The elimination algorithm outputs quantifier-free formula O(p2) having analogous bounds on the parameteres in the similar time. The algorithm constructs strata Vp by recursion beginning with p = m (in this case V,n = {O~)}). Assume that the formulas O m = O (2) e m - l ' " " V p +•l defining m ' strata Vm..... Vp+1 correspondingly are already obtained. The formula O(p2J) (1 < j < Sp) defining the connected component of {O(p2)} has according to [2,3]

(Kd) n°(1) atomic subformulas with polynomials of the degree (Kd) "°(I) and coefficient lengths not exceeding M °(I) (Kd) n°(1). Besides, sp < (Kd) n°(D. The r u n n i n g time of the decomposition and of the selection of the components is M °(I) (Kd) n°(1) according to [2,3] and [8] (here we use the estimates on the p a r a m e t e r s of formulas O m..... Op+1 which, as we shall see further, are of the same kind as for O(p2) and do not depend on p). The formula O(p3J) defining the closure of {O(p2J)} (1 < j < s.) has three quantifier alternations and the n u m b e r of variables linearly depending on n. The result of quantifier elimination in this formula will be the formula @(p4j) having the bounds similar to j u s t described. The formula Op(5),Op(6) obviously have the same type of bounds. Finally the formula Op defining Vp has the similar e s t i m a t e s on the p a r a m e t e r s a n d the total r u n n i n g time of the algorithm is M °(1) (Kd) n°(1). The theorem is proved. R e m a r k . The problem of deciding for a given formula of the kind (1) whether is defines a regular algebraic variety does not seem to be more simple t h a n the problem of stratification. However using quantifier elimination procedure from [6, 7] one can check whether the produced decomposition of the variety is

415

a Whitney stratification and thus make sure t h a t the algorithm had worked correctly for the given input.

Acknowledgement. I thank K. Bekka and M.-F. Roy for useful discussions and comments.

References I. Goresky M., MacPherson R. : Stratified Morse Theory. Springer-Verlag, Berlin, 1988. 2. C a n n y J., Grigor'ev D. Yu., Vorobjov N.N. Jr. : F i n d i n g connected components of a semialgebraic set in subexponential time, 1990, to appear in AAECC. 3. Heintz J., Roy M.-F., Solerno P. : Description des composantes connexes d'un ensemble semi-algdbrique en temps simplement exponentiel C.R. Acad. Sci. Paris 313,1991, p.167-170. 4. Lang S. : Algebra. Addison-Wesley, New York, 1965. 5. Thorpe J.A. : Elementary Topics in Differential Geometry. Springer-Verlag, Berlin, 1979. 6. Heintz J., Roy M.-F., Solerno P. : Sur la complexitd du principe de TarskiSeidenberg, Bull. Soc. Math. France, 118, 1990, p. 101-126. 7. Renegar J. : On the computational complexity and geometry of the first order theory of t h e reals, P a r t s I, II, III, Tech. Report 856, Cornell University Ithaca, 1989. 8. Grigor'ev D.Yu., Vorobjov N.N. Jr. : Solving s y s t e m s of polynomial inequalities in subexponential time. J. Symbolic Comp., 5, 1988, p. 37-64. 9. Mostowski T., Rannou E.: Canonical Whitney stratification of an algebraic set in C n . AAECC 1991, New Orleans. 10. Kashiwara M. : B-functions and holonomic systems. Inv. Math., 38, 1976, p. 33-53. 11. Henry J.-P., Merle M., Sabbah C. : Sur la condition de Thom stricte pour un m o r p h i s m e analytique complexe. Ann. Scient. Ec. Norm. Sup., 4 sdrie, 17, 1984, p. 227-268.

LIST OF PARTICIPANTS

ACQUISTAPACE Francesca AKBULUT Selman ALONSO Maria-Emilia ANDRADAS Carlos BECKER Eberhard BEKKA Karim BERR Ralph BIERSTONE Edward BOCHNAK Jacek BOROBIA Alberto BOUZOUBAA Taoufik BROCKER Ludwig BUCHNER Michael CANNY John CELLINI Paola COSTE Michel CYGAN Ewa CYNK Slawomir DEDIEU Jean-Pierre DEGTYAREV Alexander DE LA PUENTE Maria Jesus DIOP Mahmadou DRUZKOWSKI Ludwik EFRAT Ido FERRAROTTI Massimo FINASHIN S. FORTUNA Elisabetta FRAN~OISE Jean-Pierre GAMBOA Jose-Manuel GONDARD Danielle GONZALEZ VEGA Laureano GUARALDO Francesco GUERGUEB Ahmed HAJTO Zbigniew HAROUNA Warou HEINZ Joos HUBER Roland HUISMAN Johan ISCHEBECK Friedrich ITENBERG Ilia JAWORSKI Piotr JELONEK Zbigniew

Univ. Pisa (Italy) Michigan State Univ. (USA) Univ. Complutense Madrid (Spain) Univ. Complutense Madrid (Spain) Univ. Dortmund (Germany) Univ. Rennes 1 (France) Univ. Dortmund (Germany) Univ. Toronto (Canada) Vrije Univ. Amsterdam (Netherlands) U.N.E.D., Madrid (Spain) Univ. Rennes 1 (France) Univ. Mtinster (Germany) Univ. New Mexico, Albuquerque (USA) I.C.S.I., Berkeley (USA) Univ. Pisa (Italy) Univ. Rennes 1 (France) Uniw. Jagiellofiski, Krakbw (Poland) Uniw. Jagiellofiski, Krakbw (Poland) Univ. Paul Sabatier, Toulouse (France) Steklov Institute, Leningrad (USSR) Univ. Complutense Madrid (Spain) Univ. Rennes 1 (France) Uniw. Jagiellofiski, Krakbw (Poland) Univ. Konstanz (Germany) Univ. Pisa (Italy) Leningrad Electrotechnical Institute (USSR) Univ. Pisa (Italy) Univ. Paris 6 (France) Univ. Complutense, Madrid (Spain) Univ. Paris 6 (France) Univ. Cantabria, Santander (Spain) Univ. degli Studi, Roma (Italy) Univ. Rennes 1 (France) Uniw. Jagielloriski, Krakbw (Poland) Univ. Rennes 1 (France) Univ. Buenos Aires (Argentina) Univ. Regensburg (Germany) Vrije Univ., Amsterdam (Netherlands) Univ. Mtinster (Germany) Leningrad State Univ. (USSR) Univ. Warszaw (Poland) Uniw. Jagiellorlski, Krakbw (Poland)

417

KING Henry KNEBUSCH Manfred KORCHAGIN Anatolii KRASINSKI Tadeusz KRIEK Teresa KURDIKA Krzysztof KWIECINSKI Michal LAM Tsit Yuen LE SAUX Frederic. LIGATSIKAS Zissis LOJASIEWlCZ Stanislaw LOMBARDI Henri MACRI Patrizia MAHE Louis MARIN Alexis MARINARI Maria Grazia MARSHALL Murray MAZUROVSKII Vladimir MEGUERDITCHIAN Ivan MIKHALKIN Grigory MILMAN Pierre MIODEK Andrzej MONTANA Jose Luis MOSTOWSKI Tadeusz NATANZON Serguey NEUHAUS Ralph ORTEGA Jesus OTERO M. PARDO VASALLO Luis Miguel PARIMALA PAUGAM Anne-Marie PAWLUCKI Wieslaw PECKER Daniel PEDERSEN Paul PFISTER Albrecht POLOTOVSKII G.M. PRESTEL Alexander RAMANAKORAISINA Rodolphe RANDRIAMAHALEO Solo RANNOU Eric REZNICK Bruce RICHARDSON Donald ROBSON Robert O. ROY Marie-Fmngoise

Univ. of Maryland (U.S.A.) Univ. Regensburg (Germany) Nizhny Novgorod Univ. (USSR) Univ. Lrd~ (Poland) Univ. Buenos Aires (Argentina) Uniw. Jagiellofiski, Krakbw (Poland) Univ. Lille 1 (France) Univ. California, Berkeley (USA) Univ. Rennes 1 (France) Univ. Rennes 1 (France) Uniw. Jagiellofiski, Krakbw (Poland) Univ. Franche-Comtr, Besan~on (France) Univ. degli StuN, Roma (Italy) Univ. Rennes 1 (France) E.N.S. Lyon (France) Univ. Genova (Italy) Univ. of Saskatchewan, Saskatoon (Canada) Ivanovo Civil Engineering Institute (USSR) Univ. Rennes 1 (France) Leningrad (USSR) Univ. Toronto (Canada) Univ. Lrd2 (Poland) Univ. Cantabria, Santander (Spain) Univ. Warszaw (Poland) Moskva (USSR) Univ. Dortmund (Germany) Univ. Castilla-La-Mancha (Spain) Univ. Oxford (Great Britain) Univ. Cantabria, Santander (Spain) Tata Institute, Bombay (India) Univ. Rennes 1 (France) Uniw. Jagiellofiski, Krakrw (Poland) Univ. Paris 6 (France) Courant Institute, New York (USA) Univ. Mainz (Germany) Nizhny Novgorod Univ. (USSR) Univ. Konstanz (Germany) Univ. Rennes t (France) Univ. Fianarantsoa (Madagascar) Univ. Rennes 1 (France) Univ. Illinois (USA) Univ. Bath (Great Britain) Oregon State Univ. (USA) Univ. Rennes 1 (France)

418

RUIZ Jesus RUSEK Kamil SANDER Tomas SCHEIDERER Claus SCHMID Joachim SEPPALA Mikka SHUSTIN E.I. SILHOL Robert SIMON Odile SKIBINSKI Przemyslaw SOLERNO Pablo SPODZlEJA Stanislaw STASICA Jacek STENGLE Gilbert STES David SZAFRANIEC Zbigniew TOGNOLI Alberto TOUGERON Jean-Claude TRAVERSO Carlo TROTMAN David VAN GEEL Jan VIRO Oleg Ya. VOM HOFE Giinter VOROBJOV Nicolai N. WORMANN Thorsten ZVONILOV V.I.

Printing: Weihert-Druck GmbH, Darmstadt Binding: Buchbinderei Sch/iffer, Griinstadt

Univ. Complutense, Madrid (Spain) Uniw. Jagiellofiski, Krak6w (Poland) Univ. Dortmund (Germany) Univ. Regensburg (Germany) Univ. Konstanz (Germany) Univ. Helsinki (Finland) Samara State Univ. (USSR) Univ. Montpellier 2 (France) Univ. Rennes 1 (France) Univ. L6d2 (Poland) Univ. Buenos Aires (Argentina) Univ. L6d2 (Poland) Uniw. Jagiellofiski, Krak6w (Poland) Lehigh Univ., Bethlehem (USA) Univ. Gent (Belgium) Univ. Gdansk (Poland) Univ. Trento (Italy) Univ. Rennes 1 (France) Univ. Pisa (Italy) Univ. Provence, Marseille (France) Univ. Gent (Belgium) Steklov Institute, Leningrad (USSR) Univ. Dortmund (Germany) Steklov Institute, Leningrad (USSR) Univ. Dortmund (Germany) Syktyckar (USSR)