Hypertranscendance et Groupes de Galois aux differences

TL;DR

This paper develops criteria for algebraic independence of derivatives of solutions to rank one difference equations using difference Galois theory, emphasizing the role of Galois groups and field extensions.

Contribution

It introduces a novel approach linking algebraic independence to difference Galois groups via iterated extensions and Tannakian categories over non-closed fields.

Findings

Criteria for algebraic independence derived from Galois group properties

Analysis of Galois group behavior under base field extensions

Reduction of the problem to linear algebra within difference Galois theory

Abstract

This paper deals with criteria of algebraic independence for the derivatives of solutions of rank one difference equations. The key idea consists in deriving from the commutativity of the differentiation and difference operators a sequence of iterated extensions of the original difference module, thereby setting the problem in the framework of difference Galois theory and finally reducing it to an exercise in linear algebra. The involved tannakian categories are neutral over non necessarily algebraically closed fields, and this leads us to study the behaviour of Galois groups under base field extensions.