Michel Coste

Real algebraic geometry

1. Ordered Fields, Real Closed Fields.- 2. Semi-algebraic Sets.- 3. Real Algebraic Varieties.- 4. Real Algebra.- 5. The Tarski-Seidenberg Principle as a Transfer Tool.- 6. Hilberts 17th Problem. Quadratic Forms.- 7. Real Spectrum.- 8. Nash Functions.- 9. Stratifications.- 10. Real Places.- 11. Topology of Real Algebraic Varieties.- 12. Algebraic Vector Bundles.- 13. Polynomial or Regular Mappings with Values in Spheres.- 14. Algebraic Models of C? Manifolds.- 15. Witt Rings in Real Algebraic Geometry.- Index of Notation.

Thom's lemma, the coding of real algebraic numbers and the computation of the topology of semi-algebraic sets

Thoms lemma, a very simple and basic result in real algebraic geometry, and explained in section I, has a lot of interesting computational consequences. We shall outline two of these. The first one is the fact that a real root @x of a polynomial P of degree n with real coefficients may be distinguished from the other real roots of P by the signs of the derivatives P^i of P at @x, i = 1, ..., n - 1. This offers a new possibility for the coding of real algebraic numbers and for computation with these numbers (see section 2). The second is based on a generalisation of Thoms lemma to the case of several variables. It gives, after a linear change of coordinates, a cylindric algebraic decomposition of a semi- algebraic set where the incidence relation between the cells is easily obtained (see section 3).

Dynamical method in algebra: effective Nullstellensätze

We give a general method for producing various effective Null and Positivstellensatze, and getting new Positivstellensatze in algebraically closed valued fields and ordered groups. These various effective Nullstellensatze produce algebraic identities certifying that some geometric conditions cannot be simultaneously satisfied. We produce also constructive versions of abstract classical results of algebra based on Zorns lemma in several cases where such constructive version did not exist. For example, the fact that a real field can be totally ordered, or the fact that a field can be embedded in an algebraically closed field. Our results are based on the concepts we develop of dynamical proofs and simultaneous collapse.

Approximation in compact Nash manifolds

Let Ω⊂Rn be a compact Nash manifold; A,B the rings of Nash, analytic global functions on Ω. The main result of this paper is the following: Theorem 1. Let Ω,Ω′ be a pair of Nash submanifolds of some Rn ,Rq and let us suppose Ω is compact. Let F1,⋯,Fq:Ω×Ω′→R be Nash functions. Then every analytic solution y=f(x) of the system F1(x,y)=⋯=Fq(x,y)=0 can be approximated, in the Whitney topology, by the global Nash solutions y=g(x). The main tool used to prove the above results is this version of Nerons desingularisation theorem: Any homomorphism of A-algebras C→B, with C finitely generated over A, factorizes through a finitely generated A-algebra D such that A→D is regular. Using Theorem 1 the authors are able to solve several interesting problems that have been open for many years. For example they prove: (I) Every analytic factorization of a global Nash function, defined over Ω, is equivalent to a Nash factorization. (II) Every semialgebraic subset of Ω which is a global analytic subset is also a global Nash subset. (III) Every prime ideal of A generates a prime ideal in B. (IV) Every coherent ideal subsheaf of the sheaf N(Ω) of Nash functions on Ω is generated by its global sections. The case where Ω is noncompact is only partially studied in this paper. In the reviewers opinion this article makes crucial progress in the theory of global Nash functions.

On the link of a stratum in a real algebraic set

THE AIM of this paper is to exhibit some combinatorial topological properties of real algebraic sets. These properties are in the line of Sullivan’s condition on the local Euler characteristic of a real algebraic (or analytic) set. Let us recall this condition, in a way that will introduce the main result of this paper. Let X c R” be a real algebraic set, x a point of X. Let B(x, E) (resp. S(x, E)) denote the closed ball (resp. the sphere) with center x and radius E > 0. The local conic structure theorem says that, for E small enough, fi(x, c) n X is homeomorphic to the cone with vertex x and base S(x, E) n X, via a homeomorphism which preserves the distance to x. Hence the topological type of the space S(x, E) n X is independent of E, for E small enough; it is called rhe link of x in X, and dcnotcd by Ik(x, X). When X is a curve, then Ik(x, X) has two points for each real branch of X passing through x, and so Ik(x, X) consists of an even number of points. Sullivan’s condition says that this evenness is found in any dimension.

Nash functions on noncompact Nash manifolds

Abstract Several conjectures concerning Nash functions (including the conjecture that globally irreducible Nash sets are globally analytically irreducible) were proved in Coste et al. (1995) for compact affine Nash manifolds. We prove these conjectures for all affine Nash manifolds.

Generalized Budan-Fourier theorem and virtual roots

In this Note we give a proof of a generalized version of the classical Budan-Fourier theorem, interpreting sign variations in the derivatives in terms of virtual roots.

A Simple Proof That Generic 3-RPR Manipulators Have Two Aspects

Background: Avoiding singularities in the workspace of a parallel robot is an important issue. The case of 3-RPR planar robots is an important subject of theoretical studies. Method of approach: We study the singularities of planar 3-RPR robots by using a new parameterization of the singular locus in a modi ed workspace. Results: This approach enables us to give a simple proof of a recent result of M. Husty: the complement of the singular locus in the workspace of a generic 3-RPR manipulator has two connected components (called aspects). Conclusion: The parameterization introduced in this paper, due to its simple geometric properties, proves to be useful for the study of the singularities of 3-RPR robots.

Atypical values at infinity of a polynomial function on the real plane: an erratum, and an algorithmic criterion

Abstract We correct a previously published theorem by proving a criterion to decide whether the fibration given by a real polynomial function f : R 2 → R is locally trivial at infinity. The algorithmical nature of this criterion provides a test that can be applied to any real polynomial f.

Topology of Real Algebraic Varieties

In the first section, we prove some combinatorial topological properties of real algebraic sets; the simplest and most important of these properties is the fact that, for every semi-algebraic triangulation of a bounded algebraic set of dimension d and every (d - 1)-simplex σ of such a triangulation, the number of d-simplices of the triangulation having σ as a face is even. In the second section, we use this property and an appropriate stratification to prove that, for every point a of an algebraic set V, the local Euler-Poincare characteristic χ(V, V \ a) is odd; this result gives a necessary combinatorial condition for a polyhedron to be homeomorphic to a real algebraic set. In Section 3 we define the fundamental ℤ/2-homology class of a real algebraic variety. This leads to the concept of algebraic homology groups of a real algebraic variety, consisting of the homology classes represented by algebraic subsets. These groups, which are basic invariants, will be used in Chap. 12 and 13. We construct examples of nonsingular algebraic sets whose homology is not totally algebraic. In Section 4, we use the Borel-Moore fundamental classes to prove that an injective regular mapping from a nonsingular irreducible algebraic set to itself is surjective. The analogous result in complex algebraic geometry (without the assumption of nonsingularity) is well known, but the methods of proof are completely different. Section 5 contains an upper bound for the sum of the Betti numbers of an algebraic set. Section 6 is devoted to algebraic curves in the real projective plane. We prove Harnack’s theorem concerning the maximum number of connected components of a nonsingular curve of given degree and some results concerning the first part of Hilbert’s 16th problem (without proving the crucial Rokhlin congruence, Theorem 11.6.4).