Goulwen Fichou

Motivic invariants of arc-symmetric sets and blow-Nash equivalence

We define invariants of the blow-Nash equivalence of real analytic function germs, in a similar way that the motivic zeta functions of Denef & Loeser. As a key ingredient, we extend the virtual Betti numbers, which were known for real algebraic sets, as a generalized Euler characteristics for projective constructible arc-symmetrics sets. Actually we prove more: the virtual Betti numbers are not only algebraic invariant, but also Nash-invariant of arc-symmetric sets. Our zeta functions enable to sketch the blow-Nash equivalence classes of Brieskorn polynomials of two variables.

Grothendieck ring of semialgebraic formulas and motivic real Milnor fibers

— We define a Grothendieck ring for basic real semialgebraic formulas, that is for systems of real algebraic equations and inequalities. In this ring the class of a formula takes into consideration the algebraic nature of the set of points satisfying this formula and contains as a ring the usual Grothendieck ring of real algebraic formulas. We give a realization of our ring that allows to express a class as a Z[ 1 2 ]linear combination of classes of real algebraic formulas, so this realization gives rise to a notion of virtual Poincaré polynomial for basic semialgebraic formulas. We then define zeta functions with coefficients in our ring, built on semialgebraic formulas in arc spaces. We show that they are rational and relate them to the topology of real Milnor fibres.

Motivic invariants of real polynomial functions and their Newton polyhedrons

We propose a computation of real motivic zeta functions for real polynomial functions, using Newton polyhedron. As a consequence we show that the weights are blow-Nash invariants of convenient weighted homogeneous polynomials in three variables.

Analytic equivalence of normal crossing functions on a real analytic manifold

By Hironaka Desingularization Theorem, any real analytic function has only normal crossing singularities after a suitable modification. We focus on the analytic equivalence of such functions with only normal crossing singularities. We prove that for such functions

On almost Blow-analytic equivalence

C^{\infty}

Group law on the neutral component of the Jacobian of a real hyperelliptic curve having many real components

right equivalence implies analytic equivalence. We prove moreover that the cardinality of the set of equivalence classes is zero or countable. We apply these results to study the cardinality of the set of equivalence classes for almost blow-analytic equivalence.

Real Milnor Fibres and Puiseux Series

Approximation of real analytic functions by Nash functions is a classical topic in real geometry. In this paper, we focus on the Nash approximation of an analytic desingularization of a Nash function germ obtained by a sequence of blowings-up along smooth analytic centers. We apply the result to prove that Nash function germs that are analytically equivalent after analytic desingularizations are Nash equivalent after Nash desingularizations. Results are based on a precise Euclidean description of a sequence of blowings-up combined with Neron Desingularization.

Artin approximation compatible with a change of variables

Abstract Let C be a real algebraic curve of genus g with at least g real components B 1 ,…, B g . We give an embedding of C into a blow up in one point of the projective plane. It allows us to describe geometrically the neutral real component Pic o ( C ) o of the Jacobian of C thanks to an isomorphism with the product B 1 ×⋯× B g . This induces an explicit geometric description of Pic o ( C ) o in the projective plane, where the group law is given by intersection with curves of genus g .

Continuous functions on the plane regular after one blowing-up

Given a real polynomial function and a point in its zero locus, we defined a set consisting of algebraic real Puiseux series naturally attached to these data. We prove that this set determines the topology and the geometry of the real Milnor fibre of the function at this point. To achieve this goal, we balance between the tameness properties of this set of Puiseux series, considered as a real algebraic object over the field of algebraic Puiseux series, and its behaviour as an infinite dimensional object over the real numbers.

Fonctions R\'egulues

We propose a version of the classical Artin approximation which allows to perturb the variables of the approximated solution. Namely, it is possible to approximate a formal solution of a Nash equation by a Nash solution in a compatible way with a given Nash change of variables. This results is closely related to the so-called nested Artin approximation and becomes false in the analytic setting. We provide local and global version of this approximation in real and complex geometry together with an application to the Right-Left equivalence of Nash maps.