The Noether-Lefschetz theorem and sums of 4 squares in the rational function field
The nonreduced order spectrum of a commutative ring.
The Nullstellensatz for real coherent analytic surfaces.
The number of conics tangent to five given conics: the real case.
It is a classical result, first established by de Jonquières (1859), that generically the number of conics tangent to 5 given conics in the complex projective plane is 3264. We show here the existence of configurations of 5 real conics such that the number of real conics tangent to them is 3264.
The number of connected components of certain real algebraic curves.
The principle of moduli flexibility for real algebraic manifolds
Given a real closed field R, we define a real algebraic manifold as an irreducible nonsingular algebraic subset of some Rⁿ. This paper deals with deformations of real algebraic manifolds. The main purpose is to prove rigorously the reasonableness of the following principle, which is in sharp contrast with the compact complex case: "The algebraic structure of every real algebraic manifold of positive dimension can be deformed by an arbitrarily large number of effective parameters".
The Real Jacobian Conjecture for polynomials of degree 3
We show that every local polynomial diffeomorphism (f,g) of the real plane such that deg f ≤ 3, deg g ≤ 3 is a global diffeomorphism.
The special Schubert calculus is real.
The theorem of the complement for a quasi subanalytic set
Let X ⊂ (ℝⁿ,0) be a germ of a set at the origin. We suppose X is described by a subalgebra, Cₙ(M), of the algebra of germs of functions at the origin (see 2.1). This algebra is quasianalytic. We show that the germ X has almost all the properties of germs of semianalytic sets. Moreover, we study the projections of such germs and prove a version of Gabrielov’s theorem.
The tropical Grassmannian.
The underlying real algebraic structure of complex elliptic curves.
Theoreme de Pfister pour les varietes et anneaux de Witt reduits.
Théorème de préparation pour les fonctions logarithmico-exponentielles
Nous donnons une preuve géométrique du théorème d’élimination des quantificateurs pour les fonctions logarithmico-exponentielles prouvé initialement par van den Dries, Macintyre et Marker. Notre démonstration n’utilise pas de Théorie des Modèles. Elle repose sur un théorème de préparation pour les fonctions sous-analytiques.
Théorèmes de finitude pour les variétés pfaffiennes
On introduit, dans ce travail, une hypothèse sur le spiralement d’une feuille d’un feuilletage analytique réel de codimension un (hypersurface pfaffienne). On en tire des résultats très généraux de finitude du type de Khovanskii. Des exemples précis montrent la généralité de ces hypersurfaces pfaffiennes. Une description complété des bouts de telles variétés en dimension trois est donnée.
Thom's lemma in real geometry
Topological invariants of analytic sets associated with Noetherian families
Let be a compact semianalytic set and let be a collection of real analytic functions defined in some neighbourhood of . Let be the germ at of the set . Then there exist analytic functions defined in a neighbourhood of such that , for all .
Topological Invariants of Real Algebraic Curves
Topological methods in representation theory.
Topological types of fewnomials
Topologische Typen reeller kubischer Flächen.