Séminaire différentiel

Le séminaire différentiel a vu le jour en 2003 à l’IMJ sous l’impulsion de Daniel Bertrand, qui a participé à son organisation jusqu’en 2014. Depuis 2012, il a lieu à Versailles. Actuellement, il est organisé par A. Bostan (Inria Saclay) et L. Di Vizio (Univ. Versailles), à raison de quelques mardis par semestre et 2 ou 3 exposés par réunion, avec le soutien du Laboratoire de Mathématiques de Versailles, de l'équipe SpecFun de l'INRIA, du GDR EFI et du projet ANR De Rerum Natura.

À titre expérimental nous avons créé une playlist Viméo, qui s'appelle Séminaire différentiel et qui contient les enregistrements des exposés à partir de la journée du 30 mars 2021.

Dernières vidéos du séminaire :

Error

Joomla Gallery makes it better. Balbooa.com

POUR RECEVOIR LES ANNONCES :

Les annonces des exposés ci-dessous sont diffusés sur la liste la liste News du GDR EFI : pour s'inscrire (ou se désinscrire) suivre ce lien. Il existe aussi un canal Telegram (https://t.me/gdrefi) et un agenda Google "GDR EFI" (url de l'agenda, lien ical, ).

Organisateurs : Alin Bostan et Lucia Di Vizio

POUR RECEVOIR LES ANNONCES :

Programme

10h00 : accueil des participants, café et viennoiseries.

10h30-11h30 : Klara Nosan (IRIF, Université Paris-Cité), On the membership problem for hypergeometric sequences with rational parameters [SLIDES]

Abstract: We investigate the Membership Problem for hypergeometric sequences: given a hypergeometric sequence \(\langle u_n \rangle_{n = 0}^\infty\) of rational numbers and a target \(t \in \mathbb{Q}\), decide whether \(t\) occurs in the sequence. We show decidability of this problem under the assumption that in the defining recurrence \(p(n) u_{n+1} = q(n) u_{n}\), the roots of the polynomials \(p(x)\) and \(q(x)\) are all rational numbers. Our proof relies on bounds on the density of primes in arithmetic progressions. We also observe a relationship between the decidability of the Membership problem (and variants) and the Rohrlich-Lang conjecture in transcendence theo This work is in collaboration with Amaury Pouly, Mahsa Shirmohammadi and James Worrell. The full version of the paper is available at https://arxiv.org/abs/2202.07416.

12h : Déjeuner à l'Entre Nous (plan)

14h-15h : Julien Roques (ICJ, Université Lyon 1), Around Mahler equations

Abstract: In this talk, we will report on some recent results about linear Mahler equations. We will notably speak about automata, difference Galois theory, Hahn series.

15h30-16h30 : Éric Delaygue (ICJ, Université Lyon 1), On Abel's problem and Gauss congruences [SLIDES]

Abstract : A classical problem due to Abel is to determine if a homogeneous linear differential equation of order 1, with an algebraic function as coefficient, admits a non-zero algebraic solution. Given such an equation, Risch designed an algorithm that determines whether there exists a non-zero algebraic solution or not. I will prove an arithmetic characterization of Abel’s problem, in terms of Gauss congruences, in the case where the coefficient of the equation admits a Puiseux expansion with rational coefficients. I will use this criterion to completely solve the hypergeometric case and prove a prediction Golyshev made using the theory of motives. This is a joint work with T. Rivoal.

Details: Parent Category: Séminaires et groupes de travail

Organisateurs : Alin Bostan et Lucia Di Vizio

POUR RECEVOIR LES ANNONCES :

Programme

10h30-11h30 : Michel Waldschmidt (IMJ-PRG, Sorbonne Université), Interpolation de Lidstone en une et plusieurs variables

Résumé : L'interpolation classique de Lidstone (1929) est une variante du développement de Taylor: au lieu de considérer en un point toutes les dérivées, on considère en deux points les dérivées d'ordre pair. L'existence et l'unicité d'un développement pour les polynômes s'étend aux fonctions entières de type exponentiel \(<\pi\); l'existence d'un développement est encore vrai pour toute fonction entière de type exponentiel fini. Cette théorie a donné lieu à des développements et de nombreuses variantes, toujours en une variable : Gontcharoff (1930), Poristky (1932), Whittacker (1934), Schoenberg (1936), Straus (1950), Macintyre (1954), Buck (1955), Sato (1964). Le but principal de l'exposé sera de présenter d'abord la théorie classique, puis une généralisation en plusieurs variables.

11h30-12h30 : Marina Poulet (IF, Université Grenoble Alpes), Computing Galois groups of difference equations of order 3

Résumé : One important application of the difference Galois theory is the study of the (differential) transcendence of solutions of difference equations. Roughly speaking, if the difference Galois group \(G\) of a difference equation is sufficiently big then the nonzero solutions of this equation are (differentially) transcendent. More generally, the larger \(G\) is, the fewer algebraic relations there are. However, the computation of difference Galois groups is in general a difficult task, we do not have a way to do it for general difference equations. For difference equations of order \(1\) or \(2\), many things are known and we can compute Galois groups of \(q\)-difference equations, Mahler equations and other well-known types of equations. The aim of this talk is to present the main ideas used to compute difference Galois groups. In particular, we will give an extension of these results for difference equations of order \(3\) and, in some cases, of order greater than \(3\). It is a joint work with Thomas Dreyfus.

12h30-14h30 : pause déjeuner

14h30-15h30 : Henri Cohen (LFANT, INRIA, Université de Bordeaux), Modular, algebraic, and Γ-evaluations of hypergeometric series [AFFICHER LES SLIDES]

Résumé : In a first part, we explain how to obtain a conjecturally complete list of pure gamma-evaluations of the Euler-Gauss hypergeometric function 2F1(a,b;c;z), and discuss their extensions to mixed gamma-evaluations. In a second part, we give a complete list of the modular evaluations obtained as values on Hauptmoduln of the 2F1 coming from noncompact arithmetic triangle groups, and the corresponding CM algebraic evaluations. In the third and final part, we give an extensive and probably 90% complete list of the corresponding algebraic evaluations for compact arithmetic triangle groups, as well as many not corresponding to triangle groups. Joint work with Frits Beukers.

Bon ton du Séminaire différentiel en ligne : merci de vous connecter en utilisant vos nom et prénom réels et complets. À défaut de voir le visage des personnes qui suivent l'exposé, nous pensons que l'orateur doit au moins savoir qui sont les collègues connectés.

Details: Parent Category: Séminaires et groupes de travail

Organisateurs : Alin Bostan et Lucia Di Vizio

POUR RECEVOIR LES ANNONCES :

Programme

11h00-12h00 : Sara Checcoli (Institut Fourier, UGA, Grenoble)

Titre : On small height and local degrees (joint work with A. Fehm) [AFFICHER LES SLIDES]

Resumé : A field of algebraic numbers has the Northcott property (N) if it contains only finitely many elements of bounded absolute logarithmic Weil height. While for number fields property (N) follows immediately by Northcott's theorem, to decide property (N) for an infinite extension of the rationals is, in general, a difficult problem.

This property was introduced in 2001 by Bombieri and Zannier, who raised the question of whether it holds for fields with uniformly bounded local degrees. They also remarked that, for a (possibly infinite) Galois extension of the rationals whose local degrees are bounded at (at least) one prime, property (N) is implied by the divergence of a certain sum, but suggested that this phenomenon might occur only for number fields. In 2011 Widmer gave a criterion for an infinite extension of the rationals to have property (N) under some condition on the growth of the discriminants of certain finite subextensions of the field.

In this talk I will present several results obtained in this context with A. Fehm. In particular, we show the existence of infinite Galois extensions of the rationals for which the sum considered by Bombieri and Zannier is divergent and to which Widmer's criterion does not apply and we also show the existence of fields without property (N) and having (non-uniformly) bounded local degrees at all primes. This last result is a corollary of a theorem of Fili on totally \(S\)-adic numbers of small height, of which I will present an effective version.

14h30-15h30 : Veronika Pillwein (RISC, Linz, Austria)

Titre : On a sequence of polynomials generated by a Kapteyn series of the second kind (joint work with D. Dominici) [AFFICHER LES SLIDES]

Resumé : Kapteyn series are series expansions in terms of the Bessel function of the first kind. The first researcher to investigate such series in a systematic way was Willem Kapteyn (not to be confused with his brother Jacobus Cornelius Kapteyn). The topic of this talk is the explicit representation for a particular Kapteyn series of the second kind in terms of a family of polynomials. The appearing sums and sequences involve Stirling numbers and are just outside the class of holonomic functions. Still, an extension of the holonomic systems approach due to Chyzak, Kauers, and Salvy, allows to use symbolic computation to find a recurrence for the coefficients of this family of polynomials.

16h00-17h00 : Vesselin Dimitrov (University of Toronto, Canada)

Titre : The unbounded denominators conjecture for vector-valued modular forms [AFFICHER LES SLIDES]

Resumé : We will discuss some new developments arising from a solution of the unbounded denominators conjecture in the theory of noncongruence and vector-valued modular forms. One form of our result, confirming in particular certain new cases of the algebraicity conjectures of Grothendieck and Christol, is the complete determination of those integer coefficients formal power series that fulfill a linear ODE without singularities outside of 0, 1/16 and infinity, and whose local monodromy around 0 is semisimple. We also raise a few related open questions, notably what can be said whenever one lifts the constraint on semisimple local monodromy. This is a joint work with Frank Calegari and Yunqing Tang.

Séminaire différentiel

Dernières vidéos du séminaire :

Error