| Résume | Le problème de Dulac à paramètres concerne l’existence de bornes uniformes sur le nombre de cycles limites d’une famille de champs de vecteurs analytiques dans le plan.
Une approche au problème de Dulac à travers la o-minimalité a été explorée, sous certaines hypothèses, par Kaiser, Rolin et Speissegger en 2009.
Cette journée est dédiée à l’exposition des travaux en cours des orateurs, dans l’esprit d’une solution du problème de Dulac à paramètres grâce à des techniques o-minimales.
Programme de la journée :
10:00 -11:30 Sophie Germain, salle 2015 (2ème étage)
Orateur : Patrick Speissegger (Konstanz/McMaster)
Titre : Quasianalytic Ilyashenko algebras I
14:00 - 15:00 Sophie Germain, salle 1016 (1er étage)
Orateur : Tobias Kaiser (Passau)
Titre : A holomorphic extension theorem for log-exp-analytic functions
15:15 - 15:45 Sophie Germain, salle 1016 (1er étage)
Oratrice : Zeinab Galal (Paris Diderot)
Titre : Quasianalytic Ilyashenko algebras II
Résumés :
- Quasianalytic Ilyashenko algebras I (Patrick Speissegger)
In 1923, Dulac published a proof of the claim that every real analytic vector field on the plane has only finitely many limit cycles (now known as Dulac's Problem). In the mid-1990s, Ilyashenko completed Dulac's proof; his completion rests on the construction of a quasianalytic class of functions. Unfortunately, this class has very few known closure properties. For various reasons I will explain, we are interested in constructing a larger quasianalytic class that is also a Hardy field. This can be achieved using Ilyashenko's idea of superexact asymptotic expansion. (Joint work with Zeinab Galal and Tobias Kaiser)
- A holomorphic extension theorem for log-exp-analytic functions (Tobias Kaiser)
The germs of functions definable in the o-minimal structure R_an,exp are known to be real analytic and therefore have holomorphic extensions on some complex domains. We give a precise description of domains on the Riemann surface of the logarithm on which they have biholomorphic extensions, and we show that these extensions are maximal in a certain sense. As an application, we obtain an upper bound on the complexity of a term defining the compositional inverse of a germ f, in terms of the complexity of the term defining f itself. (Joint work with Patrick Speissegger)
- Quasianalytic Ilyashenko algebras II (Zeinab Galal)
As our goal is to prove o-minimality of the structure generated by the functions in the Ilyashenko algebra constructed earlier, we need an extension to several variables stable under certain operations (such as blow-up substitutions). As a first step towards the several variable extension, we construct a one-variable extension where the monomials are allowed to be any so-called principal monomial in R_an,exp. This can done thanks to the holomorphic extension theorem for the germs in the Hardy field H of R_an,exp. The resulting Hardy field also extends H itself. (Joint with Tobias Kaiser and Patrick Speissegger)
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