Moduli spaces over manifolds with involutions.
Morphisms, line bundles and moduli spaces in real algebraic geometry
Multi-Harnack smoothings of real plane branches
Let be an integral convex polygon. G. Mikhalkin introduced the notion ofHarnack curves, a class of real algebraic curves, defined by polynomials supported on and contained in the corresponding toric surface. He proved their existence, viaViro’s patchworkingmethod, and that the topological type of their real parts is unique (and determined by ). This paper is concerned with the description of the analogous statement in the case of a smoothing of a real plane branch . We introduce the class...
Multivariate Descartes' rule.
Multivariate moment problems : geometry and indeterminateness
The most accurate determinateness criteria for the multivariate moment problem require the density of polynomials in a weighted Lebesgue space of a generic representing measure. We propose a relaxation of such a criterion to the approximation of a single function, and based on this condition we analyze the impact of the geometry of the support on the uniqueness of the representing measure. In particular we show that a multivariate moment sequence is determinate if its support has dimension one and...
Multivariate polynomial inequalities viapluripotential theory and subanalytic geometry methods
We give a state-of-the-art survey of investigations concerning multivariate polynomial inequalities. A satisfactory theory of such inequalities has been developed due to applications of both the Gabrielov-Hironaka-Łojasiewicz subanalytic geometry and pluripotential methods based on the complex Monge-Ampère operator. Such an approach permits one to study various inequalities for polynomials restricted not only to nice (nonpluripolar) compact subsets of ℝⁿ or ℂⁿ but also their versions for pieces...
Nash cohomology of smooth manifolds
A Nash cohomology class on a compact Nash manifold is a mod 2 cohomology class whose Poincaré dual homology class can be represented by a Nash subset. We find a canonical way to define Nash cohomology classes on an arbitrary compact smooth manifold M. Then the Nash cohomology ring of M is compared to the ring of algebraic cohomology classes on algebraic models of M. This is related to three conjectures concerning algebraic cohomology classes.
Nash functions on noncompact Nash manifolds
Nash Manifolds
Nash triviality in families of Nash manifolds.
Nash triviality in families of Nash mappings
We study triviality of Nash families of proper Nash submersions or, in a more general setting, the triviality of pairs of proper Nash submersions. We work with Nash manifolds and mappings defined over an arbitrary real closed field . To substitute the integration of vector fields, we study the fibers of such families on points of the real spectrum and we construct models of proper Nash submersions over smaller real closed fields. Finally we obtain results on finiteness of topological types in...
Nash Wings and Real Prime Divisors.
New inequalities between elementary symmetric polynomials.
Noethérianité de certaines algèbres de fonctions analytiques et applications
Let be a real-analytic submanifold and H(M) the algebra of real analytic functions on M. If K ⊂ M is a compact subset we consider ; is a multiplicative subset of . Let be the localization of H(M) with respect to . In this paper we prove, first, that is a regular ring (hence noetherian) and use this result in two situations: 1) For each open subset , we denote by O(Ω) the subalgebra of H(Ω) defined as follows: f ∈ O(Ω) if and only if for all x ∈ Ω, the germ of f at x, , is algebraic...
Non-free Projective Modules Over IH[X, Y] and Stable Bundles Over IP2(...).
Non-simple vector bundles on curves.
Note sur la règle des signes de Descartes
Notes on the algebra and geometry of polynomial representations.
Numerical evidence for a conjecture in real algebraic geometry.
O-minimal version of Whitney's extension theorem
This is a generalized and improved version of our earlier article [Studia Math. 124 (1997)] on the Whitney extension theorem for subanalytic -Whitney fields (with p finite). In this new version we consider Whitney fields definable in an arbitrary o-minimal structure on any real closed field R and obtain an extension which is a -function definable in the same o-minimal structure. The Whitney fields that we consider are defined on any locally closed definable subset of Rⁿ. In such a way, a local...