Séminaires : Structures algébriques ordonnées

Equipe(s) Responsable(s)SalleAdresse
Logique Mathématique
F. Delon, M. Dickmann, D. Gondard
1013 Sophie Germain

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

Séances à suivre

Orateur(s)Titre Date DébutSalleAdresseDiffusion
+ Séances antérieures

Séances antérieures

Orateur(s)Titre Date DébutSalleAdresse
+ Victor Vinnikov Some local properties of functions of noncommuting variables 28/05/2024 14:00
Elements of the free associative algebra and of the free skew field (polynomials and rational functions, as defined by Amitsur and P.M. Cohn, in noncommuting variables) can be fruitfully viewed as functions on tuples of square matrices of all sizes, in the spirit of noncommutative function theory that originated in the work of J.L. Taylor in the early 1970s and developed extensively in the last two decades. This leads naturally to certain noncommutative power series around a matrix centre, called Taylor--Taylor series after both  J.L. Taylor and Brook Taylor of the calculus fame. After setting the stage, I will describe two results: one shows that the ring of these series is exactly the completion of the free algebra with respect to the (two sided) ideal of all elements vanishing at the centre (provided the centre is semisimple); the other one identifies the subring of rational series, generalizing the results of Kronecker in the single variable case and of Fliess in the case of a scalar (rather than matrix) centre.
The talk is based on joint works with Igor Klep and Jurij Volcic and with Motke Porat.
S. A. Amitsur. Rational identities and applications to algebra and geometry, J. Algebra 3:304–359, 1966.
P. M. Cohn. Skew Fields. Theory of general division rings, Encyclopedia of Mathematics and its Applications 57,  Cambridge University Press, Cambridge, 1995;
Free ideal rings and localization in general rings, New Mathematical Monographs 3, Cambridge University Press, Cambridge, 2006.
M. Fliess, Matrices de Hankel, J. de Math. Pures et Appliquees, 53, pp. 197–222, 1974.
Joseph L. Taylor. A general framework for a multi-operator functional calculus, Advances in Math., 9:183–252, 1972;
Functions of several noncommuting variables, Bull. Amer. Math. Soc., 79:1–34, 1973.
Dmitry S. Kaliuzhnyi-Verbovetskyi and Victor Vinnikov. Foundations of free noncommutative function theory, volume 199 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2014.
I. Klep, V. Vinnikov, and Ju. Volcic. Local theory of free noncommutative functions: germs, meromorphic functions and Hermite interpolation, Trans. Amer. Math. Soc. 373 (2020), 5587–5625.
M. Porat, V. Vinnikov. Realizations of non-commutative rational functions around a matrix centre, I: synthesis, minimal realizations and evaluation on stably finite algebras, J. London Math. Soc. 104 (2021), 1250–1299;
Realizations of non-commutative rational functions around a matrix centre, II: The lost-abbey conditions, Int. Equat. Oper. Theory 95 (2023), article 1.
+ Pawel Gladki Superpowersets and superpowergroups 23/04/2024 14:00

In this talk we shall introduce the notion of a new category called the category of superpowersets. Examples superpowersets are found among powersets of nonempty sets, presentable posets in the theory of quadratic forms, and fuzzy subsets of a given set, and we shall see that the category of superpowersets forms a topos. Superpowersets can be equipped with a form of binary operation yielding objects that we shall call superpowergroups. Examples of superpowergroups are naturally built from groups, hypergroups, presentable groups, fuzzy groups and fuzzy hypergroups. The question of when superpowergroups form a topos will be also addressed.

+ Mickaël Matusinski Sur la clôture algébrique des séries formelles de plusieurs variables 09/01/2024 14:00 Salle 1016 Sophie Germain Salle 1016

Soit $K$ un corps de caractéristique nulle et $x=(x_1,...,x_r)$. Nous considérons la clôture algébrique de $K[[x]]$ en tant que sous-corps du corps des "séries polyédrales rationnelles" (lui-même sous-corps des séries de Puiseux itérées), et appelons "séries de Puiseux algébroïdes" ses éléments. Nous traitons les deux problèmes suivants :

- étant donné une équation $P(x, y) = 0$ avec $P ∈ K[[x]][y]$, fournir une formule close pour les coefficients d'une série algébroïde solution $y(x)$ en fonction des coefficients de $P$ ;

- étant donné une série algébroïde $y(x)$, reconstruire algorithmiquement les coefficients d'un polynôme annulateur.

Notre stratégie s'appuie sur le traitement de ces deux problèmes à propos des "séries de Puiseux algébriques, c'est-à-dire les éléments de la clôture algébrique de $K(x)$.

Il s'agit d'un travail en commun avec Michel Hickel (U. Bordeaux)."

+ Ordered transexponential fields 21/11/2023 14:00 Salle 1016
Sebastian Krapp (Universität Konstanz, Allemagne)

(Joint work with Salma Kuhlmann)
Studying the growth properties of definable functions in o-minimal settings, Miller established the following remarkable growth dichotomy: an o-minimal expansion of an ordered field is either power bounded or admits a definable exponential function (see [2]). Going one step further in the hierarchy of growth, Miller's dichotomy result naturally led to the question whether there exist o-minimal expansions of ordered fields that are not exponentially bounded. Recent research activity in this area is therefore motivated by the search for either an o-minimal expansion of an ordered exponential field by a transexponential function that eventually exceeds any iterate of the exponential or, contrarily, for a proof that any o-minimal expansion of an ordered field is already exponentially bounded.
In this talk, I will present our axiomatic approach towards the study of ordered fields equipped with a transexponential function from [1]. Namely, denoting by e an exponential and by T a compatible transexponential, we establish a first-order theory of ordered transexponential fields in which the functional equation $T(x + 1) = e(T(x))$ holds for any positive $x$. While the archimedean models of this theory are readily described, the study of the non-archimedean models leads to a systematic examination of the induced structure on the residue field and the value group under the natural (i.e. the finest non-trivial convex) valuation. Moreover, I will illustrate construction methods for transexponentials on non-archimedean ordered exponential fields.

All relevant valuation theoretic background will be introduced.

[1] L. S. Krapp and S. Kuhlmann, ˜Ordered transexponential fields™, preprint, 2023, arXiv:2305.04607v2.
[2] C. Miller, "A Growth Dichotomy for O-minimal Expansions of Ordered Fields™, Logic: from Foundations to Applications (eds W. Hodges, M. Hyland, C. Steinhorn and J. Truss; Oxford Sci. Publ., Oxford Univ. Press, New York, 1996) 385–399.
+ Sarah Hess A Refinement of Hilbert’s 1888 Theorem 10/10/2023 14:00

The cone of all positive semidefinite (PSD) real forms in n + 1 (n ≥ 1) variables of degree 2d
(d ≥ 1) contains the subcone of all forms that are representable as finite sums of squares (SOS) of
forms of half degree. In 1888, Hilbert showed in a seminal paper that both cones coincide exactly
in the Hilbert cases n + 1 = 2 or 2d = 2 or (n + 1, 2d) = (3, 4).
In this talk, we construct a filtration of intermediate cones between the SOS and PSD cones
along a filtration of varieties containing the Veronese variety and investigate it for proper inclusions
in any non-Hilbert cases. To this end, we establish a sufficient criterion for a given intermediate
cone to coincide with the SOS cone on the one hand. On the other hand, we develop a tool
with which we are able to determine the greatest cone in the filtration containing an a priori
fixed PSD-extremal not-SOS circuit form. This allows us to introduce and prove a refinement of
Hilbert’s 1888 Theorem in three steps. First, we lay out the situation for (n + 1)-ary quartics
(n ≥ 3). Secondly, we extend our findings to (n + 1)-ary sextics (n ≥ 2) and, then, thirdly, to any
non-Hilbert case via a degree-jumping principle.

+ Rainer Sinn Sums of Squares and Convexity 03/10/2023 14:00

Recent results about sums of squares in connection with projective algebraic geometry have relied on the convex geometry of spectrahedra to cover the divide between the two subjects. I want to explain some of these recent results and try to make this bridge by convexity concrete. This is based on joint work with Greg Blekherman, Greg Smith, and Mauricio Velasco.

+ Pawel Gladki Natural homomorphisms of Witt rings of a certain cubic order 04/04/2023 14:00

Let $K$ be a number field and $\mathcal{O}_K$ its ring of integers. A famous result by Knebusch asserts that the natural homomorphism of Witt rings $W\mathcal{O}_K \rightarrow W K$ is injective. This, however, fails to be true if we replace $\mathcal{O}_K$ with an arbitrary ring $R$ whose field of fractions is equal to $K$. We shall consider one particular class of such rings here, namely orders, that is subrings $\mathcal{O}$ of $\mathcal{O}_K$ which are also $\mathbbm{Z}$-modules of rank $n = [K :\mathbbm{Q}]$. The case of orders in quadratic number fields is relatively well understood with both examples of natural homomorphisms of Witt rings being injective and not. In this talk we shall take a closer look at orders in cubic number fields. While orders in quadratic number fields are easy to describe and classify, cubic orders are considerably more difficult to handle. Nevertheless, we manage to
exhibit an example of a cubic order $\mathcal{O}$ whose Witt ring $W\mathcal{O}$ naturally embedds into the Witt ring of its field of fractions.

+ Salma Kuhlmann Generalised power series determined by linear recurrence relations 28/03/2023 14:00
In 1882, Kronecker established that a given univariate formal Laurent series over a field can be expressed as a fraction of two univariate polynomials if and only if the coefficients of the series satisfy a linear recurrence relation. 
We introduce the notion of generalised linear recurrence relations for power series with exponents in an arbitrary ordered abelian group, and generalise Kronecker's original result. In particular, we obtain criteria for determining whether a multivariate Laurent series lies in the fraction field of the corresponding polynomial ring. 
+ Françoise DELON Construction de groupe dans les structures C-minimales, suite 24/01/2023 14:00

Nous considérons le cas où les C-structures sont denses, définissablement maximales et sans bijection définissable dans leur arbre canonique entre un intervalle borné d'une branche et un intervalle non borné d'une autre branche (« non borné » signifie ici que la borne supérieure de l'intervalle est une feuille). Elles sont également géométriques et non triviales. Cette situation permet de montrer l'existence de bonnes familles de fonctions, de développer une notion de tangente et de dérivation, et enfin de définir (au sens de la théorie des modèles) un groupe infini.

+ Françoise DELON Construction de groupe dans les structures C-minimales 17/01/2023 14:00

Dans les années 80 un important courant de pensée en théorie des modèles considérait que des propriétés de pure théorie des modèles de certaines structures abstraites révélaient que ces structures étaient en fait des avatars de structures algébriques classiques. L'exemple même de cette tentative de reconstruction est la conjecture de Cherlin-Zilber : une théorie fortement minimale à géométrie non triviale interprète un groupe infini. Si sa géométrie n'est pas modulaire, elle interprète un corps infini. La conjecture s'est révélée fausse, déjà en ce qui concerne l'existence d'un groupe. Des contre-exemples ne sont apparus qu'au prix de la construction des amalgames de Fraïssé-Hrushovski. Pour qu'elle devienne exacte il a fallu introduire un soupçon de topologie, c'est ce qu'ont fait Ehud Hrushovski et Boris Zilber avec ce qu'ils ont appelé les structures de Zariski. 
Les structures o-minimales portent quant à elles une topologie définissable. Elles sont de plus géométriques, au sens où leur clôture algébrique satisfait le lemme de l'échange. Kobi Peterzil et Sergei Starchenko ont montré qu'elles satisfont (à quelques nuances près) la conjecture de Cherlin-Zilber. 
De façon analogue Fares Maalouf a montré que toute structure C-minimale géométrique modulaire et non triviale permet de définir un groupe. Fares, Patrick Simonetta et moi-même nous intéressons maintenant au cas non modulaire.

+ Hugo Mariano K-theories and free inductive graded rings in abstract quadratic forms theories 10/01/2023 14:00
(Joint work with Kaique M.A. Roberto (IME-USP))

We will expand a fundamental tool in algebraic theory of quadratic forms to the more general multivalued setting: the K-theory. We introduce and develop the K-theory of hyperbolic hyperfields that generalize simultaneously Milnor's K-theory and Special Groups K-theory, developed by Dickmann-Miraglia. We develop some properties of this generalized K-theory, that can be seen as a free inductive graded ring, a concept introduced in order to provide a solution to Marshall's Signature Conjecture by Dickmann-Miraglia. Moreover, we will show how the extended version of Arason-Pfister Hauptsatz - presented in the previous talk "Multirings and applications to algebraic theory of quadratic forms, IV" - entails some interesting properties concerning K-theory and Marshall's conjecture.
+ Hugo Mariano Multirings and applications to algebraic theory of quadratic forms, IV 15/11/2022 14:00

(Joint work with Kaique M.A. Roberto (IME-USP) and Hugo R.O. Ribeiro (IME-USP))

A superring is essentially a ring with multivalued sum and product: for instance, the set of polynomials with coefficients in a multiring has a natural structure of superring.
In the first part of the talk, we consider some constructions of superrings, explore some properties of the superring of polynomials with coefficients in a hyperfield and develop some fragments of the theory of algebraic extension for superfields, such as the addition of roots to a superfield.
The significance of these multialgebraic methods to (univalent) Commutative Algebra is indicated by applying these results to algebraic theory of quadratic forms:
(i) obtaining new relevant constructions in the category of special groups (or its equivalent category special hyperfields);
(ii) extending the validity of the Arason-Pfister Hauptsatz - a positive answer to a question posed by Milnor in a classical paper of 1970 - and established by Dickmann-Miraglia in the realm of reduced special groups (or its equivalent category real reduced hyperfields) in 2000.
In the second part of the talk, we expand a fundamental tool in algebraic theory of quadratic forms to the more general multivalued setting: the K-theory. We introduce and develop the K-theory of hyperbolic hyperfields that generalize simultaneously Milnor's K-theory and Special Groups K-theory, developed by Dickmann-Miraglia. We develop some properties of this generalized K-theory, that can be seen as a free inductive graded ring, a concept introduced in order to provide a solution of Marshall's Signature Conjecture. Moreover, the extended version of Arason-Pfister Hauptsatz entails some interesting properties concerning K-theory and Marshall's conjecture.
We finish this series of talks indicating some future developments of superrings theory and possible applications.

+ Hugo Mariano Multirings and applications to algebraic theory of quadratic forms, III 08/11/2022 14:00
(Joint work with Hugo R.O. Ribeiro (IME-USP))
In this third talk, we focus on constructions of multirings associated with real semigroups:
(i) we describe further properties of the functor Q, the reflection (= left adjoint functor) of the natural inclusion of the category of real reduced multirings into the category of pre-ordered multirings and explore some of these properties; 
(ii) by the employ of sheaf-theoretic methods, we characterize the real reduced hyperrings as certain "geometric" von Neumann regular real hyperring and describe the functor V, "geometric" von Neumann regular hull of a multiring;
(iii) we present some interesting logical-algebraic interactions between the functors Q and V that are useful to describe the Witt ring of a real semigroup (or real reduced multiring).
+ Hugo Mariano Multirings and applications to algebraic theory of quadratic forms, III 18/10/2022 14:00
(Joint work with Hugo R.O. Ribeiro (IME-USP))
In this third talk, we focus on constructions of multirings associated with real semigroups:
(i) we describe further properties of the functor Q, the reflection (= left adjoint functor) of the natural inclusion of the category of real reduced multirings into the category of pre-ordered multirings and explore some of these properties; 
(ii) by employing sheaf-theoretic methods, we characterize the real reduced hyperrings as certain "geometric" von Neumann regular real hyperrings and describe the functor V, the "geometric" von Neumann regular hull of a multiring;
(iii) we present some interesting logical-algebraic interactions between the functors Q and V that are useful to describe the Witt ring of a real semigroup (or real reduced multiring).
+ Hugo Mariano Multirings and applications to algebraic theory of quadratic forms, II 11/10/2022 14:00

(Joint work with Kaique M.A. Roberto (IME-USP) and Hugo R.O. Ribeiro (IME-USP))
In the first talk of this series, we have presented the concept of multiring and showed how it can (functorially) encode the abstract theories of quadratic forms of special groups and real semigroups.  In this second talk, we present other notions of multirings R (as hyperbolic multirings and quadratic multirings) and pairs (R,T), where R is a multiring and T is a certain multiplicative subset (as DM-multirings, DP-multirings and quadratic pairs), that seem relevant to algebraic theory of quadratic forms. 

+ Hugo Mariano Multirings and applications to algebraic theory of quadratic forms, I 04/10/2022 14:00
(Joint work with Kaique M.A. Roberto (IME-USP) and Hugo R.O. Ribeiro (IME-USP))
The concept of multiring was introduced by M. Marshall in 2006 and generalizes the Krasner's hyperrings (introduced in the 1950's), but multifields and hyperfields coincide. A multiring is essentially a ring with a multivalued sum satisfying a weak distributive law, but it can be viewed also as a first-order relational structure satisfying some $\forall\exists$ sentences.
In this first talk of a series, we start describing the basic notions, examples and main constructions in the category of multirings. 
In the sequel, we present detailed functorial encoding of abstract theories of quadratic forms (abstract ordering spaces, (pre)special groups, real semigroups, etc) into the theory of multirings (respectively real reduced hyperfields, (pre)special hyperfields, real reduced multirings, etc).
+ Jorge GUIER ACOSTA Théorie universelle d'une classe modèle-complète d'anneaux réels clos. 14/06/2022 14:00

Soit T* la théorie des f-anneaux réduits, projetables, divisible-projetables, sc-réguliers et réels clos sans idempotents minimaux (non nuls). La modèle-complétude de cette théorie a été établie auparavant dans le langage des anneaux réticulés avec une relation radicale (dans le sens de Prestel-Schwartz) déterminée par la classe des idéaux premiers minimaux, et une relation de divisibilité locale. Dans cet exposé je décrirai la théorie universelle de T* dans ce langage, augmenté par la relation de divisibilité globale. L'étude détaillé de cette théorie universelle est le pas préalable à une preuve d'élimination des quantificateurs de T* sur laquelle je travaille actuellement.

+ Victor Vinnikov Hyperbolic polynomials and their determinantal representations 12/04/2022 14:00
A homogeneous polynomial of degree d with real coefficients is called hyperbolic with respect to a point if any real line through this point intersects the corresponding hypersurface in d real points (counting multiplicities). Hyperbolic polynomials are in a sense the opposite of strictly positive polynomials: they have as many real zeroes as possible. Hyperbolic polynomials were first introduced by Garding in the study of linear hyperbolic PDEs with constant coefficient in the 1950s; he showed that a hyperbolic polynomial determines a convex cone, called a hyperbolicity cone. In recent years hyperbolic polynomials and hyperbolicity cones came to play an important in convex programming as well as combinatorics and other areas.
Much like a representation as a sum of squares certifies the positivity of a polynomial, its hyperbolicity is certified by a representation as a determinant of a matrix of linear forms, with the coefficient matrices of the linear forms satisfying some positivity conditions. I will describe some of what is known about the existence of these determinantal representations, usually "with denominators". One fruitful approach uses a Hermitian Positivstellensatz that gives a representation of a polynomial satisfying matrix positivity conditions as a weighted sum of hermitian squares.
+ Arno Fehm The existential theory of discrete equicharacteristic henselian valued fields 29/03/2022 14:00
From a model theoretic point of view, local fields of positive characteristic, i.e. fields of Laurent series over finite fields, are much less well understood than their characteristic zero counterparts - the fields of real, complex and p-adic numbers. I will discuss different approaches to axiomatize and decide at least their existential theory in various languages and under various forms of resolution of singularities.
+ Salma Kuhlmann Distinguished Subfields of Hahn Fields 22/03/2022 14:00
(Work in progress with Michele Serra and Sebastian Krapp). Let k be a field and G a totally ordered Abelian group. A Hahn field is an intermediate field K between k(G) (the fraction field of the group ring k[G], which we call the minimal Hahn field) and k((G)) (the field of generalised series, which we call the maximal Hahn field). While studying the group of valuation preserving automorphisms of K, we identified two crucial lifting properties of K which allow a fine description of that group. We still lack an understanding of the class of Hahn fields which do enjoy those properties. In the talk, we will discuss the properties, their relation to the automorphism group, and our current methods to obtain examples and counterexamples.
+ Françoise DELON Structures C-minimales denses définissablement complètes 23/11/2021 14:00 S.G. salle 1013
Comme on le voit dans l'exposé de Pablo du 9 novembre, un corps p-minimal est toujours définissablement complet. Il satisfait même une forme forte de complétude définissable. Un corps C-minimal n'a aucune raison d'être définissablement complet, mais s'il l'est alors, modulo une additionnelle sur le groupe de valuation, il satisfait une forme forte très semblable de complétude définissable. Ce résultat vaut pour n'importe quelle structure C-minimale et est le sujet de l'exposé. Il a déjà été utilisé pour montrer des résultats allant dans le sens de la trichotomie dans les structures C-minimales géométriques (selon laquelle une telle structure interprète un groupe si la géométrie est non triviale et un corps si elle est non modulaire). Rappels : un corps p-minimal est un corps p-adiquement clos enrichi dans lequel les sous-ensembles à un variable définissables sont déjà définissables dans le pur corps, et si la même propriété est vraie dans toute structure élémentairement équivalente. Une C-relation, comme on l'a rappelé Marie-Hélène dans son exposé du 21 octobre, est un affaiblissement ternaire d'une distance ultramétrique, situation où elle est donnée ainsi : C(x,y,z) ssi d(x,y) = d(x,z) > d(y,z), id est : y et z sont plus près l'un de l'autre que de x. Une structure dans un langage contenant C plus d'autres choses est C-minimale si tout sous-ensemble à une seule variable définissable est définissable sans quantificateurs dans le pur langage { C }, et idem dans toute structure élémentairement équivalente. C'est la même chose de dire qu'un tel ensemble est une combinaison booléenne de ce qui généralise les boules ouvertes et fermées (id est de cônes et cônes épais ou '' 0-levelled sets ''). Dans notre contexte les C-relations sont de plus supposées denses : pour tous x et y distincts, il existe z plus près de x que de y.
+ Max Dickmann Rings of formal power series and symmetric real semigroups 16/11/2021 14:00 S.G. salle 1016
On commencera par un bref survol de la notion de semigroupe réel (RS) et de ses relations avec les anneaux (commutatifs, unitaires) semi-réels et préordonnés. On fera une description rapide des classes de RS connues, en mettant l'accent sur celle des éventails (fans). Ensuite nous donnons une caractérisation arithmétique des objets du spectre réel des anneaux F[[G]] des séries formelles à coefficients dans un corps formellement réel (i.e., ordonnable) F et exposants dans un groupe abélien totalement ordonné G arbitraire; on obtient ainsi des renseignements sur la structure ordinale de leurs spectres réels et des leurs RS associés. On déduit que les espaces de caractères de ces RS possédent des symétries très fortes qui conduisent naturellement à une notion générale de RS symétrique. On étudie en détail ces RS ainsi que les nombreuses propriétés qui en découlent. On prouve un théorème de représentation qui montre que les RS symétriques finis sont isomorphes aux RS associés à des anneaux du type F[[G]]; pourtant, il y a des contrexemples à la validité d'un tel résultat pour les RS symétriques de cardinalité arbitraire.
+ Pablo Cubides Kovacsics Complétude définissable et applications 09/11/2021 14:00 S.G. salle 1016
La complétude définissable est une propriété qui n'est pas forcément préservée dans une expansion modérée de corps valués. Par exemple, il y a des expansions C-minimales de corps valués algébriquement clos n'ayant pas cette propriété. En revanche, je montrerai dans cet exposé que toute expansion P-minimale d'un corps p-adiquement clos K satisfait une version forte de complétude définissable : toute famille définissable d'ensembles emboîtés, fermés et bornés est d'intersection non-vide. On en déduit entre autres qu'une telle expansion est polynômialement bornée (répondant positivement à une question de R. Cluckers et I. Halupczok). Il s'agit d'un travail en commun avec Françoise Delon.
+ Silvain Rideau-Kikuchi Groupes définissables, génériques et loi birationnelle 19/10/2021 14:00 https://zoom.us/j/91443442825?pwd=VmFTTWsxcmFQSmQ5dGRjUW1tSm1sUT09 Zoom id 914 4344 2825
Un concept central dans l'étude des groupes stables est celui de généricité, qui capture une notion de « gros » sous-ensemble du groupe et qui généralise la notion de point générique de la géométrie algébrique. Et comme l'énonce un théorème de Weil pour les groupes algébriques, il s'avère qu'un groupe stable est entièrement déterminé par le comportement générique de sa loi de groupe. Dans cet exposé, je présenterai cette notion de généricité dans le cadre des théories stables et l'une des approches pour la généraliser à des théorie instables. Dans un deuxième temps, j'expliquerais comment ces résultats peuvent être utilisés pour classifier les groupes définissables dans certaines théories de corps, en particulier, pour montrer qu'ils sont essentiellement des groupes algébriques.
+ Marie-Hélène Mourgues Classification des purs C-ensembles C-minimaux et aleph-zéro-catégoriques 12/10/2021 14:00 1016 Sophie Germain
J'exposerai les résultats de classification des $C$-ensembles purs $C$-minimaux et $\aleph_0$-catégoriques obtenus dans l'article Classification of $\aleph_0$-categorical $C$-minimal pure $C$-sets. Le théorème de Ryll-Nardzewsky permet de se ramener tout d'abord à la classification des $C$-ensembles purs $C$-minimaux et $\aleph_0$-catégoriques et indiscernables, c'est cette classification que nous détaillerons principalement à l'aide d'exemples. Si j'en ai le temps, je dirai ensuite quelques mots de la classification générale qui utilise des arbres finis étiquetés par des entiers et des théories de $C$-ensembles purs $C$-minimaux et $\aleph_0$-catégoriques et indiscernables
+ 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. 


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.
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

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
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
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
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