Université de Versailles-St Quentin, Laboratoire de Mathématiques, 45 avenue des États-Unis 78035 Versailles cedex, France
e-mail: divizio[at]math.cnrs.fr          Office: bâtiment Fermat, office 3305

Organisateurs : Alin Bostan et Lucia Di Vizio

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### 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.

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.

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