# Séminaires : Structures algébriques ordonnées

Logique Mathématique
F. Delon, M. Dickmann, D. Gondard
1016 Sophie Germain

Mardi de 14h00 à 15h45
Page du séminaire et programme
Abonnement à la liste de diffusion

## Séances à suivre

+ Séances antérieures

### Séances antérieures

+ Olivier Benoist Sur les mauvais points des polynômes positifs 01/06/2021 14:00 https://u-paris.zoom.us/j/87835993947?pwd=ejBSNVJBR1luWUU2S2ZjWjkzRTFMdz09 Zoom id 878 3599 3947
Un mauvais point d'un polynôme réel positif est un point en lequel un pôle apparaît dans toute représentation de celui-ci comme somme de carrés de fractions rationnelles. Nous présenterons un exemple de polynôme positif en trois variables pour lequel l'origine est un mauvais point, mais qui est néammoins une somme de carrés de séries formelles. Nous donnerons également un exemple de polynôme positif en trois variables admettant un mauvais point complexe non réel. Ces exemples répondent à des questions de Brumfiel et de Scheiderer.
+ Cordian Riener Vandermonde varieties and efficient algorithms for computing the Betti numbers of symmetric semi-algebraic sets 18/05/2021 14:00 https://u-paris.zoom.us/j/88499863030?pwd=YlhvclZzbG02UlNmZjNjVENwbXFiZz09 Zoom id 884 9986 3030
For 1 ≤ i ≤ n denote by \$p_i=\sum_{j=1}^k the power-sums polynomials. For d ∈ N and y ∈ Rd the algebraic set Vd(y) = {x ∈ Rn : p1(x) = y1, . . . , pd(x) = yd} is called a Vandermonde-variety. These algebraic sets have been studied by Arnold and Giventhal in connection to hyperbolic polynomials. In this talk we will generalise some results of Arnold and Giventhal on the Homology of Vandermonde varieties, more concretely: let H∗(Vd, Q) denote the cohomology group of a Vandermonde variety. Since Vd is invariant by the natural action of the symmetric group Sn the cohomology group H∗(Vd, Q) has the structure of an Sn module. We prove that for all λ ⊢ k, the multiplicity of the Specht-module Sλ in Hi(Vd(y), Q), is zero if length(λ) ≥ i+2d−3. This vanishing result allows us to prove similar vanishing result for arbitrary symmetric semi- algebraic sets defined by symmetric polynomials of degrees bounded by d. As a result, we obtain for each fixed l ≥ 0, an algorithm for computing the first l + 1 Betti numbers of such sets, whose complexity is polynomially bounded (for fixed d and l).
+ Daniel Plaumann Hyperbolic Polynomials 11/05/2021 14:00 https://u-paris.zoom.us/j/85413611691?pwd=NHBvY1VoenFCNU05eVVKdTVqRTJjdz09 Zoom Id 854 1361 1691
Hyperbolic polynomials are real homogeneous polynomials in several variables with a certain reality condition on their roots. They orginate in PDE theory but have more recently been studied extensively in combinatorics, convex optimization and real algebraic geometry. A lot of their theory can be thought of as generalising symmetric determinants. In this talk, I will give a gentle introdcution, survey several important results and point to some recent developments and open problems.
+ Salma Kuhlman On Rayner Structures 23/03/2021 14:00 https://u-paris.zoom.us/j/83288534029?pwd=M1dmZ3RBa01LSlA2OU5lUDVIQVUvZz09 Zoom Id 832 8853 4029
In this note, we study substructures of generalised power series fields induced by families of well-ordered subsets of the group of exponents. We relate set theoretic and algebraic properties of the families to algebraic features of the induced sets. By this, we extend the work of Rayner ('An algebraically closed field', GMJ 1968) to truncation closed substructures of generalised power series fields.
+ Vincent Bagayoko Hyperseries and surreal numbers 09/03/2021 14:00 https://u-paris.zoom.us/j/83552104627?pwd=Y1BhbmZxY1JpU1hLSFpFZnNGSDgzZz09 Zoom Id 835 5210 4627
Transseries are formal series, involving exponentials and logarithms, which provide a natural model for the model theory of Hardy fields, i.e. ordered fields of real-valued differentiable germs. Transseries are not closed under many functional equations, whose solutions still behave like germs in Hardy fields. Hyperexponential functions, which grow faster or than any finite iteration of the exponential, naturally appear in this context. Although such functions are somewhat analytically exotic, the works of Ecalle, van der Hoeven and Schmeling show that some of them are amenable to geometric or formal descriptions. Hyperseries are an extension of transseries that can act as formal counterparts to more general germs including hyperexponentials. It turns out that hyperseries can be interpreted as Conway's surreal numbers. I will show that surreal numbers fill the gaps left in transseries, and I will explain how one can exploit this connection to endow the class of surreal numbers with a structure of field of hyperseries.
+ Marie Françoise Roy Complexité du calcul de la topologie d'une courbe algébrique réelle 16/02/2021 14:00 https://us02web.zoom.us/j/85841560806?pwd=SG5KSEUwNlUvNnJFWHBiaHhUNGt5UT09 Zoom ID 858 4156 0806
We give a deterministic algorithm to compute the topology of a real algebraic curve defined by an integer bivariate polynomial of degree bounded by d and bitsize bounded by τ. . Our analysis yields the upper bound Õ(d^5 τ + d^6 ) on the bit complexity of our algorithm. Compared to existing algorithms with similar complexity, our method does not consider any change of coordinates, and gives cylindrical algebraic decomposition of the curve. We use two main ingredients: First, we derive amortized quantitative bounds on the the roots of polynomials with algebraic coefficients as well as adaptive methods for computing the roots of such polynomials that actually exploit this amortization. Our second ingredient is a novel approach for the computation of the local topology of the curve in a neighbourhood of all critical points. Joint work with Daouda Niang Diatta · Sény Diatta ·Fabrice Rouillier · Michael Sagraloff
+ Sylvy Anscombe Approximation for spaces of orderings and valuations 09/02/2021 14:00 https://us02web.zoom.us/j/84425422716?pwd=ZkdzL09ZdWtyQzROSXdwQm1QVzVRQT09 Zoom ID 844 2542 2716
(Joint work with Philip Dittmann and Arno Fehm.) By the classical Artin--Whaples approximation theorem we may simultaneously approximate finitely many different elements of a field with respect to finitely many pairwise inequivalent absolute values. Several variants and generalizations exist, for example for finitely many valuations, where one usually requires that these are independent (i.e. induce different topologies). Ribenboim proved a generalization for finitely many valuations where the condition of independence is relaxed for a natural compatibility condition, and Ershov proved a statement about simultaneously approximating finitely many different elements with respect to finitely many possibly infinite sets of pairwise independent valuations. We prove approximation theorems which generalize both of these: we work with infinite sets of valuations and orderings and we weaken the requirement of pairwise independence. On the way we'll use the notation of a locality', generalising both valuations and orderings. We will discuss the space of localities and explore some advantages and deficiencies of this approach.
+ Salma Kuhlmann Projective limit techniques for Positivstellensätze. 18/02/2020 14:00

In this talk we discuss Positivstellensätze in the general context of a unital, commutative, not necessarily finitely generated, real algebra. We focus on the dual problem, and give an introduction to (real) infinite dimensional moment problems (i.e. when measures are supported on real infinite dimensional spaces). We will focus on the following problem: when can a linear functional on a unital commutative real algebra A be represented as an integral w.r.t. a Radon measure on the real character space X(A) equipped with the Borel σ-algebra generated by the weak topology? Our main idea is to construct X(A) as a projective limit of the character spaces of all finitely generated subalgebras of A, to be able to exploit the classical finite dimensional moment theory in the infinite dimensional case. We thus obtain existence results for representing measures defined on the cylinder σ-algebra on X(A), carried by the projective limit construction. If in addition the well-known Prokhorov (ε-K) condition is fulfilled, then we can solve our problem by extending such representing measures from the cylinder to the Borel σ-algebra on X(A). These results allow us to establish e.g. infinite dimensional analogues of the classical Riesz-Haviland.

+ Francisco Miraglia Special groups and quadratic forms over rings with non zero-divisor coefficients. 11/02/2020 14:00

The purpose is to begin to generalize the theory developed in [DM] (see below) to diagonal quadratic forms with non zero-divisor coefficients in reduced unitary pre-ordered commutative rings. We present:
1. Horn-geometric axioms entailing the equivalence between ring-theoretic isometry and representation by this class of forms is faithfully coded by the corresponding concepts in an associated reduced special group.
A preordered ring, <A, T>, satisying these axioms is called NT-faithfully quadratic (N for "multiplicative set of non zero-divisors in A").
2. A natural correspondence between rings that satisfy the axioms in (1) above and faithfully quadratic rings in the sense of [DM].
3. We prove that any reduced f-ring satisfies the axioms in (1) and, in fact, any preordered f-ring, <A,T>, where T is a preorder satisfying a certain weak cancelation property, verifies the axioms mentioned in item (1). In particular, this applies to real closed rings and rings of continuous functions (bounded or not) over any completely regular topological space.
Joint work with M. Dickmann (IMJ-PRG) and Hugo Ribeiro (USP).
[DM] M. Dickmann, F. Miraglia "Faithfully Quadratic Rings(Memoirs of the AMS 1128, november 2015).



+ Jorge GUIER ACOSTA Divisibilité locale et modèle complétude en théorie des anneaux réels-clos 21/01/2020 14:00

On introduira les relations de divisibilité locale ainsi que ces propriétés élémentaires.  Par la suite on démontrera que la théorie des f-anneaux réels clos, sc-reguliers, projetables, divisible-projetables ayant la premiere propriété de convexité, sans idempotents minimaux est modèle complète en utilisant la relation de divisibilité locale ainsi que la relation radicale associée au spectre des idéaux premiers minimaux.

+ Eberhard Becker STUDY OF SUMS OF POWERS IN REAL FUNCTION FIELDS VIA TOPOLOGY 08/10/2019 14:00

In formally real fields K sums of powers with arbitrary even exponents can be
studied by invoking the compact space M = M (K) of all real places, i.e. certain
maps λ : K → P 1 := R ∪ ∞, and the natural representation K → C(M, P 1 ), a 7→ â
based on the evaluation maps â : λ 7→ λ(a). The real holomorphy H = H(K) is
defined as the subring of K consisting of all elements a admitting a finite evaluation
map â : M → R. One is further led to study the series of representations
S n (H) → C(M, S n ), n ∈ N.
In the case of a real (=formally real) function field F over R this general approach
will be substantiated by relating M and H to the family of all smooth models of
F with a compact real locus X which is a real algebraic set and a C ∞ -manifold.
The investigation of line bundles on those manifolds leads to information on the
invertible ideals of H, recent results on continuous rational maps help to study
the image of S n (H) in C(M, S n ). All these informations can be used to derive
qualitative and quantitative results on the representation of an element as a sum of
powers with a given even exponent. Purely algebraic proofs for results of this type
have not been found so far.

+ Patrick Speissegger Expansions of the real field by canonical products 11/06/2019 14:00
We consider expansions of o-minimal structures on the real field by certain
canonical Weierstrass products and/or associated functions, such as their
logarithmic derivatives. We show that there are only three possible
outcomes for the resulting structures: they are either o-minimal, or
d-minimal but not o-minimal, or they define the set of integers. Joint work with Chris Miller.
+ Salma Kuhlmann ORDERED FIELDS DENSE IN THEIR REAL CLOSURE AND DEFINABLE CONVEX VALUATIONS 14/05/2019 14:00
In this work, we undertake a systematic model and valuation theoretic
study of the class of ordered fields which are dense in their real closure. We apply this
study to determine definable henselian valuations on ordered fields, in the language of
ordered rings. In light of our results, we re-examine recent conjectures in the context of
strongly NIP ordered fields. Joint work with Lothar Sebastian Krapp and Gabriel Lehericy.
+ Alexi Block Gorman O-minimal Expansions of Groups with a Predicate for a Dense Substructure Expanding a Group 16/04/2019 14:00
This talk concerns a couple properties of the theory obtained by adding a dense/codense algebraic substructure to an o-minimal expansion of an ordered divisible abelian group. I will discuss a characterization of when the expansion of an o-minimal group by a generic subgroup has a model companion. This characterization proves to be geometric in essence, and hence is similar in spirit to criteria for the property of near-model completeness. I will discuss a few examples of an o-minimal theory with a predicate for an algebraic substructure that is not generic, but satisfies some geometric criteria that imply near-model completeness. Namely, the examples are pairs of ordered vector spaces with different base fields, and pairs of fields such that one is real closed and one is pseudo-real closed.
+ Olivier Benoist Le u-invariant du corps de fonctions d'une surface réelle 19/02/2019 14:00
Lang a conjecturé qu'une forme quadratique en au moins 5 variables sur le corps de fonctions d'une surface réelle sans point réel a un zéro non trivial. Nous expliquerons une preuve de cet énoncé, ainsi que d'une généralisation, considérée par Pfister, à des surfaces réelles arbitraires.
+ Pablo Cubides Ensembles définissables d'une courbe de Berkovich 04/12/2018 14:00
Soit K un corps valué algébriquement clos complet de rang 1. Soit X une courbe algébrique sur K et X^\mathrman son analytifié au sens de Berkovich. Nous montrerons comment associer à X^\mathrman, de façon fonctorielle, un ensemble définissable Y dans un langage naturel. On obtient ainsi une preuve alternative d’un résultat de Hrushovski-Loeser sur l’iso-définissabilité des courbes. Notre association étant explicite nous permettra de donner une description concrète des sous-ensembles définissables de Y. Il s’agit d’un travail un commun avec Jérôme Poineau.
+ Silvain Rideau Imaginaires dans les corps Henséliens II: Invariance des types au dessus de RV 13/11/2018 14:00
Dans ce second exposé, je présenterais le second ingrédient principal de
la preuve d'élimination des imaginaires pour certains corps Henséliens
(avec opérateurs) d'équicharactéristique nulle qui consiste à compléter
n'importe quel type sans quantificateur définissable en un type
invariant au dessus de RV. La preuve au dessus d'un modèle consiste à
rendre canonique la preuve d'élimination des quantificateurs de corps.
Au dessus d'ensembles algébriquement clos, des types stablement dominés
font leur apparition.
Ces travaux sont joints avec Martin Hils.
+ Silvain Rideau Imaginaires dans les corps Henséliens I : densité des types sans quantificateurs définissables 06/11/2018 14:00
Dans cet exposé nous aborderons la question de l'élimination des
imaginaires dans les corps henséliens potentiellement munis d'opérateurs
d'équicharactérsitique nulle. Dans un premier temps je présenterais le
résultat optimal que l'on peut espérer prouver, à savoir une élimination
relative aux sortes géométriques, à RV ainsi qu'à certains imaginaires
d'espaces vectoriels définissables. Dans un second temps, je
présenterais une approche pour certains corps en équicharactéristique
nulle et enfin je présenterais le premier ingrédient principal de cette
approche: la construction de types sans quantificateurs définissables,
qui est une amélioration de mes résultats précédents sur les corps
valués différentiels contractants existentiellement clos.
Ces travaux sont joints avec Martin Hils.
+ Journée thématique en l'honneur de Paulo Ribenboim 20/03/2018 10:30 Institut Henri Poincaré - Amphi Hermite
Programme

Welcome 10:30

10:45 - 11:30 Lou van den Dries: Hardy fields, transseries, and surreal numbers

11:45 - 12:30 Franziska Jahnke: Definable Henselian Valuations

Lunch 12:30 -14:30

14:30 - 15:15 Sibylla Priess-Crampe: Asymptotic approximations - a short insight into my joint work with Paulo for our book on Ultrametric Spaces

15:30 - 16:15 Paulo Ribenboim: Roots of Polynomials in Ultranormed Rings

Coffee and tea break 16:15 - 17:00

17:00 - 17:45 Daniel Bertrand: Pell equations over polynomial rings

Birthday cocktail 18:00 - 20:00

This event is supported by IMJ-PRG, Projets Logique and Théorie des nombres, and by the GDR-STN.
+ Pantelis Eleftheriou Counting rational points in tame expansions of o-minimal structures by a dense set 13/03/2018 14:15 IHP - Amphithéatre Darboux
ATTENTION : EXCEPTIONNELLEMENT, CET EXPOSÉ AURA LIEU À L’INSTITUT HENRI POINCARÉ.
We work in an expansion (M, P) of an o-minimal structure M by a dense set P, such that three tameness conditions hold. Examples include dense pairs, expansions of M by an independent set, and expansions by a multiplicative group with the Mann property. In the first part of the talk, we give a structure theorem for definable sets in (M, P), in analogy with the cell decomposition theorem known for o-minimal structures, and analyze the relevant notion of dimension (joint with Günaydın and Hieronymi). In the second part, we propose a generalized "Pila-Wilkie" statement and prove it in the three examples. Namely, if X is a definable set in (M, P) and contains many rational points, then it is dense in an infinite semialgebraic set. The proof of the statement is by reduction to the standard Pila-Wilkie theorem, using the structure theorem.
+ Immanuel Halupczok Classifying definable sets in ℤ-groups 27/02/2018 14:15 IHP - Amphithéatre Darboux
ATTENTION : EXCEPTIONNELLEMENT, CET EXPOSÉ AURA LIEU À L’INSTITUT HENRI POINCARÉ.
Definable sets in the ordered abelian group ℤ are very well understood, via Presburger arithmetic. In particular, one can easily give a complete classification of definable subsets of ℤ^n up to definable bijection: Finite sets are classified by their cardinality and infinite sets are classified by a notion of dimension. Surprisingly, things become more difficult if one works in an elementary extension of ℤ. In the talk, I will present a complete classification of definable sets in that setting. This is joint work with Raf Cluckers.
+ Mickaël Matusinski Surreal numbers as transseries... and vice versa ! 06/02/2018 14:15 IHP - Amphithéatre Darboux
ATTENTION : EXCEPTIONNELLEMENT, CET EXPOSÉ AURA LIEU À L'INSTITUT HENRI POINCARÉ.
Surreal numbers strike by the richness of their structure together with their universality: generalization of the reals and of the ordinal numbers, representation as generalized series with real coefficients, universal domain for real closed exponential fields. Several other important achievements have been obtained recently: structure of real differential field of transseries (Berarducci-Mantova), universal domain for Hardy fields (Aschenbrenner-van den Dries-van der Hoeven). Even a notion of partial composition has just been developed (Berarducci-Mantova).
In this talk, we'll get started with an intuitive presentation of surreal numbers (definitions and classical results). Then, we'll develop some aspects of the cited recent works. If time permits, we'll mention a work in progress with Berarducci, S. Kuhlmann and Mantova. We underline that this talk is also aimed to be complementary to the mini-course of A. Berarducci "Surreal models of the reals with exponentiation".
+ Gabriel Lehéricy Dérivations de type Hardy sur les corps de séries généralisées 16/01/2018 14:00
On sait grâce à Kaplansky que tout corps valué qui a même caractéristique que son corps résiduel est isomorphe à un sous-corps d'un certain corps de séries généralisées.
On peut alors se demander si un analogue du théorème de Kaplansky existe pour les corps différentiellement valués, c'est-à-dire les corps valués munis d'une dérivation `de type Hardy'' tels que les H-corps étudiés par Aschenbrenner et v.d.Dries. Cela nécéssite de pouvoir définir une dérivation de type Hardy sur les corps de séries généralisées. On aimerait également que la dérivation satisfasse une condition de linéarité forte (c'est-à-dire que la dérivation commute
avec les sommes infinies) et une règle de Leibniz forte (la dérivation commute avec certains produits infinis). Dans cet exposé, nous considérerons un corps de séries généralisées k((G)) et on donnera des conditions sur k et G pour l'existence d'une telle dérivation. On donnera également une méthode pour définir la dérivation explicitement. Il s'agit d'un travail en collaboration avec Salma Kuhlmann.
+ Tomás Ibarlucía Quand l'ergodicité est une propriété du premier ordre 07/11/2017 14:00
Je présenterai des applications nouvelles de la logique continue à la théorie ergodique, notamment pour l'étude de phénomènes de rigidité associés à des actions fortement ergodiques de groupes dénombrables. Avant ceci j'introduirai les notions nécessaires tant de théorie ergodique que de logique continue.
+ Ayhan Günaydin Topological Study of Pairs of Algebraically Closed Fields 13/06/2017 14:10
The study of an algebraically closed field K with a distinguished algebraically closed field L goes back to Keisler's work in 1964, where he proves a completeness result. Since then there had been many developments around such pairs and more general kinds of pairs of stable structures.

In this talk, we equip each K^n with a topology refining Zariski topology in a way that sets definable in the pair (K,L) are precisely the constructible sets of this topology.

We shall also mention the relations of this topology with Morley rank and another notion of dimension arising from a pregeometry.
+ Daniel Plaumann Hyperbolic polynomials and interlacers 06/06/2017 14:15
A real polynomial in one variable is real-rooted if and only if it is the characteristic polynomial of a real symmetric matrix. Hyperbolic polynomials are the multivariate generalizations of real-rooted polynomials and their relation to matrices is far more intricate. A key role is played by interlacing polynomials, whose zeros are nested in between those of a given hyperbolic polynomial. This talk will survey some known results as well as work in progress.
+ Tim Netzer On non-commutative quantifier elimination in real algebra 30/05/2017 14:15
Quantifier elimination is a strong and useful tool in classical (commutative) real algebra and geometry. Non-commutative real algebra and geometry is a recently emerging area of research, with many interesting applications in pure and applied mathematics. Quantifier elimination would be a highly desirable tool here as well. There are some results in that direction, mostly negative. We will give a survey on this question, presenting some new results, among which is also a first positive one.
+ Christopher Miller Component-closed expansions of the real line 23/05/2017 14:15
We consider expansions M of the real line (R,<) having the property that, for all sets E definable in M, each connected component of M is definable in M; we then say that M is "component closed". Some notable examples are: (a) o-minimal M; (b) M=(R,<,+,x,Z); and (c) the "component closure" of M (defined in an obvious way). I will demonstrate that, in contrast to cases (a) and (b), the question "Is M component closed?" can be difficult to answer even if the model theory of M is well understood. This is very preliminary joint work with Athipat Thamrongthanyalak.
+ Françoise DELON Dérivabilité des fonctions définissables dans un corps valué C-minimal 16/05/2017 14:15
Dans une expansion o-minimale d'un corps réel clos, une fonction définissable unaire est dérivable presque partout. La preuve est facile (il n'y a quelque chose à montrer que si la structure du corps est enrichie, puisqu'une fonction définissable dans le pur corps est algébrique par morceaux). Que se passe-t-il dans un corps valué C-minimal ? On appelle ainsi un corps valué tel que, dans n'importe laquelle de ses extensions élémentaires, toute partie définissable est combinaison booléenne de boules. Un pur corps valué C-minimal n'est rien d'autre qu'un corps valué algébriquement clos. La caractéristique peut être positive, il y a alors des fonctions nulle part dérivables, c'est le cas du Frobenius inverse. La question est ouverte en caractéristique nulle. Nous considérerons plus précisément le cas des corps valués dans Q et complets. Il s'agit d'un travail en commun avec Pablo Cubides-Kovacsics.
+ Fred WEHRUNG Espaces spectraux de groupes réticulés Abéliens 02/05/2017 14:15
Le spectre X d'un groupe réticulé Abélien G est l'ensemble de ses l-idéaux premiers, muni de la topologie dont les fermés sont les ensembles d'idéaux premiers contenant un l-idéal donné. Il est connu depuis longtemps que X est un espace spectral généralisé, c'est à dire que tout fermé irréductible est l'adhérence d'un unique singleton (on dit que X est sobre) et les ouverts quasi-compacts de X forment une base de la topologie de X, close par intersection de deux membres quelconques. Il est également connu que X est complètement normal, c'est à dire que pour tous points x, y, z de X, si x et y appartiennent à la fermeture de z, alors x appartient à la fermeture de y ou y appartient à la fermeture de x. Un exemple de Delzell et Madden montre que ces propriétés ne caractérisent pas les spectres de groupes réticulés Abéliens. Cependant, cet exemple n'a pas une base dénombrable d'ouverts. Le but de cet exposé est d'esquisser ma preuve (aussi longtemps qu'elle survit...) que tout espace spectral généralisé, complètement normal, à base dénombrable, est le spectre d'un groupe réticulé Abélien. L'étape préliminaire de la preuve est la réduction du problème à un problème de théorie des treillis, en l'occurrence la caractérisation des treillis d'idéaux principaux de groupes réticulés Abéliens (disons l-représentables). Dans le cas dénombrable, les l-représentables sont caractérisés au premier ordre, alors que dans le cas général, les l-représentables ne peuvent pas être décrits par une classe d'énoncés L_\infty,\omega.
+ Algèbres quasi-analytiques d'Ilyashenko et o-minimalité journée thématique 28/03/2017 10:00 Sophie Germain, salles 2015 et 1016
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)
+ Structures algébriques ordonnées 30/03/2016 14:00