# Algebraic independence and linear difference equations

### Boris Adamczewski

Univ Lyon, Université Claude Bernard Lyon 1, CNRS, France### Thomas Dreyfus

Université de Strasbourg, France### Charlotte Hardouin

Université Paul Sabatier, Toulouse, France### Michael Wibmer

Graz University of Technology, Austria

## Abstract

We consider pairs of automorphisms $(ϕ,σ)$ acting on fields of Laurent or Puiseux series: pairs of shift operators $(ϕ:x↦x+h_{1},σ:x↦x+h_{2})$, of $q$-difference operators $(ϕ:x↦q_{1}x$, $σ:x↦q_{2}x)$, and of Mahler operators $(ϕ:x↦x_{p_{1}},σ:x↦x_{p_{2}})$. Given a solution $f$ to a linear $ϕ$-equation and a solution $g$ to an algebraic $σ$-equation, both transcendental, we show that $f$ and $g$ are algebraically independent over the field of rational functions, assuming that the corresponding parameters are sufficiently independent. 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 question and is based on a suitable Galois theory: the $σ$-Galois theory of linear $ϕ$-equations.

## Cite this article

Boris Adamczewski, Thomas Dreyfus, Charlotte Hardouin, Michael Wibmer, Algebraic independence and linear difference equations. J. Eur. Math. Soc. (2023), published online first

DOI 10.4171/JEMS/1316