Continous, piecewise-polynomial functions which solve Hilbert's 17th problem.
CR singular immersions of complex projective spaces.
Cubic differential forms and the group law on the Jacobian of a real algebraic curve
In an earlier paper [6], we gave an explicit geometric description of the group law on the neutral component of the set of real points of the Jacobian of a smooth quartic curve. Here, we generalize this description to curves of higher genus. We get a description of the group law on the neutral component of the set of real points of the Jacobian of a smooth curve in terms of cubic differential forms. When applied to canonical curves, one gets an explicit geometric description of this group law by...
Cycles on algebraic models of smooth manifolds
Every compact smooth manifold is diffeomorphic to a nonsingular real algebraic set, called an algebraic model of . We study modulo 2 homology classes represented by algebraic subsets of , as runs through the class of all algebraic models of . Our main result concerns the case where is a spin manifold.
Déchirures de variétés de dimension trois et la conjecture de Nash de rationalité en dimension trois.
Description of algebraically constructible functions.
Discriminant Sets of Families of Hyperbolic Polynomials of Degree 4 and 5
∗ Research partially supported by INTAS grant 97-1644A real polynomial of one real variable is hyperbolic (resp. strictly hyperbolic) if it has only real roots (resp. if its roots are real and distinct). We prove that there are 116 possible non-degenerate configurations between the roots of a degree 5 strictly hyperbolic polynomial and of its derivatives (i.e. configurations without equalities between roots). The standard Rolle theorem allows 286 such configurations. To obtain the result we study...
Embedding of real varieties and their subvarieties into Grassmannians.
Given a compact affine nonsingular real algebraic variety X and a nonsingular subvariety Z C X belonging to a large class of subvarieties, we show how to embed X in a suitable Grassmannian so that Z becomes the transverse intersection of the zeros of a section of the tautological bundle on the Grassmannian.
Enumeration of real conics and maximal configurations
We use floor decompositions of tropical curves to prove that any enumerative problem concerning conics passing through projective-linear subspaces in is maximal. That is, there exist generic configurations of real linear spaces such that all complex conics passing through these constraints are actually real.
Equivalence of analytic and rational functions
We give a criterion for a real-analytic function defined on a compact nonsingular real algebraic set to be analytically equivalent to a rational function.
Exposants de Łojasiewicz pour les fonctions semi-algébriques
We prove the rationality of the Łojasiewicz exponent for semialgebraic functions without compactness hypothesis. In the parametric situation, we show that the parameter space can be divided into a finite number of semialgebraic sets on each of which the Łojasiewicz exponent is constant.
Extending algebraic actions.
There is a well-known procedure -induction- for extending an action of a subgroup H of a Lie group G on a topological space X to an action of G on an associated space. Induction can also extend a smooth action of a subgroup H of a Lie group G on a manifold M to a smooth action of G on an associated manifold. In this paper elementary methods are used to show that induction also works in the category of (nonsingular) real algebraic varieties and regular or entire maps if G is a compact abelian Lie...
Generalized polar varieties and an efficient real elimination
Let be a closed algebraic subvariety of the -dimensional projective space over the complex or real numbers and suppose that is non-empty and equidimensional. In this paper we generalize the classic notion of polar variety of associated with a given linear subvariety of the ambient space of . As particular instances of this new notion of generalized polar variety we reobtain the classic ones and two new types of polar varieties, called dual and (in case that is affine) conic. We show that...
Géométrie réelle des dessins d’enfant
À tout dessin d’enfant est associé un revêtement ramifié de la droite projective complexe , non ramifié en dehors de 0, 1 et l’infini. Cet article a pour but de décrire la structure algébrique de l’image réciproque de la droite projective réelle par ce revêtement, en termes de la combinatoire du dessin d’enfant. Sont rappelées en annexe les propriétés de la restriction de Weil et des dessins d’enfants utilisées.
Gonality and Clifford index for real algebraic curves.
Let X be a smooth connected projective curve of genus g defined over R. Here we give bounds for the real gonality of X in terms of the complex gonality of X.
Graded quaternion symbol equivalence of function fields
We present criteria for a pair of maps to constitute a quaternion-symbol equivalence (or a Hilbert-symbol equivalence if we deal with global function fields) expressed in terms of vanishing of the Clifford invariant. In principle, we prove that a local condition of a quaternion-symbol equivalence can be transcribed from the Brauer group to the Brauer-Wall group.
Hilbert's 17-th problem for sums of 2n-th powers.
Homology classes of real algebraic sets
There is a large research program focused on comparison between algebraic and topological categories, whose origins go back to 1952 and the celebrated work of J. Nash on real algebraic manifolds. The present paper is a contribution to this program. It investigates the homology and cohomology classes represented by real algebraic sets. In particular, such classes are studied on algebraic models of smooth manifolds.
Introduction à la géométrie algébrique réelle
Lie ideals and Jordan triple derivations in rings