Computing amoebas.
Computing the dimension of a semi-algebraic set.
Connected components of global semianalytic subsets of 2-dimensional analytic manifolds.
Constructible functions on 2-dimensional analytic manifolds.
We present a characterization of sums of signs of global analytic functions on a real analytic manifold M of dimension two. Unlike the algebraic case, obstructions at infinity are not relevant: a function is a sum of signs on M if and only if this is true on each compact subset of M. This characterization gives a necessary and sufficient condition for an analytically constructible function, i.e. a linear combination with integer coefficients of Euler characteristic of fibers of proper analytic morphisms,...
Constructing blow-analytic isomorphisms
In this paper we construct non-trivial examples of blow-analytic isomorphisms and we obtain, via toric modifications, an inverse function theorem in this category. We also show that any analytic curve in , can be deformed via a rational blow- analytic isomorphism of , to a smooth analytic arc.
Construction d'hypersurfaces réelles
Constructions de places réelles et géométrie semi-algébrique
Continous, piecewise-polynomial functions which solve Hilbert's 17th problem.
Correction to the paper "Signatures and flatness".
Counterexamples to representing homology classes by real algebraic subvarieties up to homeomorphism
Courbures intrinsèques dans les catégories analytico-géométriques
Deux types de courbures sont associés à un sous-ensemble compact et définissable d'une variété riemannienne analytique réelle. Si la variété est de courbure constante, il y a des relations linéaires entre ces mesures. Comme application, nous démontrons une formule cinématique, définissons des densités locales, et nous étudions les volumes des simplexes réguliers.
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.
Decomposition into special cubes and its applications to quasi-subanalytic geometry
The main purpose of this paper is to present a natural method of decomposition into special cubes and to demonstrate how it makes it possible to efficiently achieve many well-known fundamental results from quasianalytic geometry as, for instance, Gabrielov's complement theorem, o-minimality or quasianalytic cell decomposition.
Definable stratification satisfying the Whitney property with exponent 1
We prove that for a finite collection of sets definable in an o-minimal structure there exists a compatible definable stratification such that for any stratum the fibers of its projection onto satisfy the Whitney property with exponent 1.
Defining orbit spaces by inequalities
Density of Morse functions on sets definable in o-minimal structures
We present a tameness property of sets definable in o-minimal structures by showing that Morse functions on a definable closed set form a dense and open subset in the space of definable functions endowed with the Whitney topology.
Description of algebraically constructible functions.