Julien Roques

Dynamics of Rational Symplectic Mappings and Difference Galois Theory

In this paper we study the relationship between the integrability of rational symplectic maps and difference Galois theory. We present a Galoisian condition, of Morales-Ramis type, ensuring the non-integrability of a rational symplectic map in the non-commutative sense (Mishchenko-Fomenko). As a particular case, we obtain a com- plete discrete analogue of Morales-Ramis Theorems for non-integrabi- lity in the sense of Liouville.

On the nature of the generating series of walks in the quarter plane

In the present paper, we introduce a new approach, relying on the Galois theory of difference equations, to study the nature of the generating series of walks in the quarter plane. Using this approach, we are not only able to recover many of the recent results about these series, but also to go beyond them. For instance, we give for the first time hypertranscendency results, i.e., we prove that certain of these generating series do not satisfy any nontrivial nonlinear algebraic differential equation with rational function coefficients.

On generalized hypergeometric equations and mirror maps

This paper deals with generalized hypergeometric differential equations of order n ≥ 3 having maximal unipotent monodromy at 0. We show that among these equations those leading to mirror maps with integral Taylor coefficients at 0 (up to simple rescaling) have special parameters, namely R-partitioned parameters. This result yields the classification of all generalized hypergeometric differential equations of order n ≥ 3 having maximal unipotent monodromy at 0 such that the associated mirror map has the above integrality property.

Galois Groups of Difference Equations of Order Two on Elliptic Curves

This paper is concerned with difference equations on elliptic curves. We establish some general properties of the difference Galois groups of equations of order two, and give applications to the calculation of some difference Galois groups. For instance, our results combined with a result from transcendence theory due to Schneider allow us to identify a large class of discrete Lam e equations with difference Galois group GL 2(C).

An Introduction to Difference Galois Theory

This article comes from notes written for my lectures at the summer school “Abecedarian of SIDE” held at the CRM (Montreal) in June 2016. They are intended to give a short introduction to difference Galois theory, leaving aside the technicalities.