Algebraic independence of solutions of linear difference equations

Abstract: This work is a collaboration with B. Adamczewski (ICJ, France), T. Dreyfus (IRMA, France) and M. Wibmer (Graz University of Technology, Austria).

In this talk, we will consider pairs of automorphisms $(\phi,\sigma)$ acting on fields of Laurent or Puiseux series: pairs of shift operators, of $q$-difference operators and of Mahler operators. Assuming that the operators $\phi$ and $\sigma$ are "independent", we show that their solutions are also "independent" in the sense that a solution $f$ to a linear $\phi$-equation and a solution $g$ to a linear $\sigma$-equation are algebraically independent over the field of rational functions unless one of them is a rational function. As a consequence, we settle a conjecture about Mahler functions put forward by Loxton and van der Poorten in 1987. We also give an application to the algebraic independence of $q$-hypergeometric functions. Our approach provides a general strategy to study this kind of questions and is based on a suitable Galois theory: the $\sigma$-Galois theory of linear $\phi$-equations developed by Ovchinnikov and Wibmer.

dynamical systems

Audience: researchers in the topic

BIRS workshop: Algebraic Dynamics and its Connections to Difference and Differential Equations

Series comments: The field of algebraic dynamics has emerged over the past two decades at the confluence of algebraic geometry, discrete dynamical systems, and diophantine geometry. In recent work, striking connections have been observed between algebraic dynamics and much older theories of difference and differential equations. This meeting brings together mathematicians with expertise in such diverse fields as ring theory, complex dynamics, differential and difference algebra, combinatorics and algebraic geometry. New work towards the dynamical Mordell-Lang and dense orbit conjectures as well as theorems on hypertranscendence and functional independence proven by connecting difference Galois theory, algebraic dynamics and other algebraic approaches to the study offunctional equations will be presented at this meeting.