On the pythagoras numbers of real analytic surfaces
On Witt rings of function fields of real analytic surfaces and curves.
Let V be a paracompact connected real analytic manifold of dimension 1 or 2, i.e. a smooth curve or surface. We consider it as a subset of some complex analytic manifold VC of the same dimension. Moreover by a prime divisor of V we shall mean the irreducible germ along V of a codimension one subvariety of VC which is an invariant of the complex conjugation. This notion is independent of the choice of the complexification VC. In the one-dimensional case prime divisors are just points, in the two-dimensional...
Parusinski's “Key Lemma” via algebraic geometry.
Quantifier elimination, valuation property and preparation theorem in quasianalytic geometry via transformation to normal crossings
This paper investigates the geometry of the expansion of the real field ℝ by restricted quasianalytic functions. The main purpose is to establish quantifier elimination, description of definable functions by terms, the valuation property and preparation theorem (in the sense of Parusiński-Lion-Rolin). To this end, we study non-standard models of the universal diagram T of in the language ℒ augmented by the names of rational powers. Our approach makes no appeal to the Weierstrass preparation...
Quasianalytic perturbation of multi-parameter hyperbolic polynomials and symmetric matrices
This paper investigates hyperbolic polynomials with quasianalytic coefficients. Our main purpose is to prove factorization theorems for such polynomials, and next to generalize the results of K. Kurdyka and L. Paunescu about perturbation of analytic families of symmetric matrices to the quasianalytic setting.
Rational points on a subanalytic surface
Let be a compact subanalytic surface. This paper shows that, in a suitable sense, there are very few rational points of that do not lie on some connected semialgebraic curve contained in .
Rectilinearization of functions definable by a Weierstrass system and its applications
This paper presents several theorems on the rectilinearization of functions definable by a convergent Weierstrass system, as well as their applications to decomposition into special cubes and quantifier elimination.
Reduction of the singularities of foliations and applications
Rigid subanalytic sets
Semialgebraic and semianalytic sets
Semi-monotone sets
A coordinate cone in is an intersection of some coordinate hyperplanes and open coordinate half-spaces. A semi-monotone set is an open bounded subset of , definable in an o-minimal structure over the reals, such that its intersection with any translation of any coordinate cone is connected. This notion can be viewed as a generalization of convexity. Semi-monotone sets have a number of interesting geometric and combinatorial properties. The main result of the paper is that every semi-monotone...
Separating families for semi-algebraic sets.
Separation of global semianalytic sets
Given global semianalytic sets A and B, we define a minimal analytic set N such that Ā∖N and B̅∖N can be separated by an analytic function. Our statement is very similar to the one proved by Bröcker for semialgebraic sets.
Siciak's extremal function in complex and real analysis
The Siciak extremal function establishes an important link between polynomial approximation in several variables and pluripotential theory. This yields its numerous applications in complex and real analysis. Some of them can be found on a rich list drawn up by Klimek in his well-known monograph "Pluripotential Theory". The purpose of this paper is to supplement it by applications in constructive function theory.
Signatures and flatness.
Singular open book structures from real mappings
We define open book structures with singular bindings. Starting with an extension of Milnor’s results on local fibrations for germs with nonisolated singularity, we find classes of genuine real analytic mappings which yield such open book structures.
Smooth and analytic solutions for analytic linear systems.
We give some approximation theorems in the Whitney topology for a general class of analytic fiber bundles. This leads to a classification theorem which generalizes the classical ones.
Structure locale et globale des feuilletages de Rolle, un théorème de fibration
Un feuilletage de codimension un sur une variété orientable est de Rolle s’il vérifie la propriété suivante : une courbe transverse à coupe au plus une fois chaque feuille. Soit une fonction tapissante sur , i.e. propre et possédant un nombre fini de valeurs critiques. Nous montrons que si l’ensemble des singularités de la restriction de aux feuilles de vérifie certaines propriétés de finitude, alors la restriction de au complémentaire d’un nombre fini de feuilles possède une structure...
Supplement to the paper "Quasianalytic perturbation of multi-parameter hyperbolic polynomials and symmetric matrices" (Ann. Polon. Math. 101 (2011), 275-291)
In IMUJ Preprint 2009/05 we investigated the quasianalytic perturbation of hyperbolic polynomials and symmetric matrices by applying our quasianalytic version of the Abhyankar-Jung theorem from IMUJ Preprint 2009/02, whose proof relied on a theorem by Luengo on ν-quasiordinary polynomials. But those papers of ours were suspended after we had become aware that Luengo's paper contained an essential gap. This gave rise to our subsequent article on quasianalytic perturbation theory, which developed,...
Sur les substitution linéaires