Description of the Connected Components of a Semialgebraic Set in Single Exponential Time.
Diagonal series of rational functions
Some representations of Nash functions on continua in ℂ as integrals of rational functions of two complex variables are presented. As a simple consequence we get close relations between Nash functions and diagonal series of rational functions.
Directional properties of sets definable in o-minimal structures
In a previous paper by Koike and Paunescu, it was introduced the notion of direction set for a subset of a Euclidean space, and it was shown that the dimension of the common direction set of two subanalytic subsets, called the directional dimension, is preserved by a bi-Lipschitz homeomorphism, provided that their images are also subanalytic. In this paper we give a generalisation of the above result to sets definable in an o-minimal structure on an arbitrary real closed field. More precisely, we...
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...
Distance géodésique sur un sous-analytique.
Pour un ensemble sous-analytique, connexe fermé, la distance géodésique est atteinte et est uniformément équivalente, avec des constantes arbitrairement proches de 1, à une distance sous-analytique.
Division of Distributions by Locally Definable Quasianalytic Functions
We demonstrate that the Łojasiewicz theorem on the division of distributions by analytic functions carries over to the case of division by quasianalytic functions locally definable in an arbitrary polynomially bounded, o-minimal structure which admits smooth cell decomposition. Hence, in particular, the principal ideal generated by a locally definable quasianalytic function is closed in the Fréchet space of smooth functions.
Divisors in global analytic sets
We prove that any divisor of a global analytic set has a generic equation, that is, there is an analytic function vanishing on with multiplicity one along each irreducible component of . We also prove that there are functions with arbitrary multiplicities along . The main result states that if is pure dimensional, is locally principal, is not connected and represents the zero class in then the divisor is globally principal.
Divisors on real curves.
Effectiveness - non effectiveness in semialgebraic and PL geometry.
Ein Semialgebraischer Beweis der Topologischen Form des Hauptsatzes von zariski.
Eine Bemerkung zu den höheren Pythagoraszahlen reeller Körper.
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.
Embedding Semi-Algebraic Spaces.
Ensembles semi-algébriques symétriques par arcs.
Entropy, homology and semialgebraic geometry
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.
Equidistribution of Frobenius classes and the volumes of tubes
Équisingularité réelle II : invariants locaux et conditions de régularité
On définit, pour un germe d’ensemble sous-analytique, deux nouvelles suites finies d’invariants numériques. La première a pour termes les localisations des courbures de Lipschitz-Killing classiques, la seconde est l’équivalent réel des caractéristiques évanescentes complexes introduites par M. Kashiwara. On montre que chaque terme d’une de ces suites est combinaison linéaire des termes de l’autre, puis on relie ces invariants à la géométrie des discriminants des projections du germe sur des plans...
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.
Equivariant virtual Betti numbers
We define a generalised Euler characteristic for arc-symmetric sets endowed with a group action. It coincides with the Poincaré series in equivariant homology for compact nonsingular sets, but is different in general. We put emphasis on the particular case of , and give an application to the study of the singularities of Nash function germs via an analog of the motivic zeta function of Denef and Loeser.