Université de Versailles-St Quentin, Laboratoire de Mathématiques, 45 avenue des États-Unis 78035 Versailles cedex, France
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Séminaire différentiel - Archives

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 Programme :

10h30-11h30  Quentin Gazda (École polytechnique, Palaiseau) : Some thoughts on Zagier's Conjecture: From Functions to Numbers [SLIDES]

Abstract. Zagier's conjecture is a certain formulation of the following slogan: linear relations among polylogarithms evaluated at algebraic numbers arise from relations between K-theory symbols. By identifying the different attributes of this conjecture, one can state and prove a similar version in arithmetic of function fields. Classical polylogarithms are then replaced by those of Carlitz. The proof, very different from the techniques developed so far, uses ingredients from the theory of difference equations. It involves deformations of Carlitz polylogarithms where a new variable \(t\) appears. This results from a joint work with A. Maurischat. While we currently lack the technology to reproduce this argument in number theory, it is amusing to speculate on a hypothetical transcription. \(q\)-deformations of polylogarithms then replace these « \(t\)-deformations." With T. Bouis, we recently encountered a \(q-Li_1\) in the syntomic Chern class introduced by Bhatt-Lurie. This is encouraging...! I will mention these works in a second part of the presentation.

11h30-12h30  Jehanne Dousse (Université de Genève) : q-difference equations and computer algebra for partition identities [SLIDES]

Abstract. A partition of a positive integer \(n\) is a non-increasing sequence of positive integers whose sum is \(n\). A partition identity is a theorem stating that for all \(n\), the number of partitions of \(n\) satisfying some conditions equals the number of partitions of \(n\) satisfying some other conditions. In this talk, we will show how \(q\)-difference equations, recurrences and computer algebra can be used to prove such identities.

Repas

15h-16h  Maxim Kontsevich (IHES, Bures-sur-Yvette) : p-Determinants and monodromy of differential operators

Abstract. We prove that \(p\)-determinants of a certain class of differential operators can be lifted to power series over \(\mathbb{Q}\). We compute these power series in terms of monodromy of the corresponding differential operators.

 

 

Prochaine journée du séminaire différentiel : 30/11/2023-01/12/2023, 15:00-17:00, à l'Institut Henri Poincaré

Orateurs : M. Mezzarobba, M. Mishna, T. Rivoal et B. Salvy 

Programme :

  • Nov. 30th, 15:00-16:00, Values of E-functions are not Liouville numbers, Tanguy Rivoal, Institut Fourier, Grenoble, France
  • Nov. 30th, 16:00-17:00, Rounding error analysis of linear recurrences using generating series, Marc Mezzarobba, LIX, Palaiseau, France [Slides]
  • Dec. 1st, 15:00-16:00, Computation of sums and integrals by reduction-based creative telescoping, Bruno Salvy, Inria, ENS Lyon, France [Slides]
  • Dec. 1st, 16:00-17:00, Combinatorics and Transcendence: Applications of Inhomogeneous order 1 iterative functional equations, Marni Mishna, Simon Fraser University, Burnaby, Canada [Slides]

 


 

 Organisateurs : Alin Bostan et Lucia Di Vizio

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'agendalien ical).

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.

 

 Organisateurs : Alin Bostan et Lucia Di Vizio

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'agendalien ical).

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. 

 

 

 

Organisateurs : Alin Bostan et Lucia Di Vizio

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'agendalien ical).

Programme

11h00-12h00Sara 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. 

14h30-15h30Veronika 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.

16h00-17h00Vesselin 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. 

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. 

 

 

 

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