{\sigma}-Galois theory of linear difference equations

TL;DR

This paper develops a Galois theory for linear difference equations with an endomorphism, enabling the analysis of solutions satisfying {\sigma}-polynomial equations and exploring their algebraic properties.

Contribution

It introduces a novel Galois theory framework for difference equations with endomorphisms, extending classical methods to new algebraic and difference contexts.

Findings

Provides criteria to determine if solutions satisfy {\sigma}-polynomial equations

Characterizes solutions using the developed Galois theory

Applies theory to isomonodromic difference equations and meromorphic functions

Abstract

We develop a Galois theory for systems of linear difference equations with an action of an endomorphism {\sigma}. This provides a technique to test whether solutions of such systems satisfy {\sigma}-polynomial equations and, if yes, then characterize those. We also show how to apply our work to study isomonodromic difference equations and difference algebraic properties of meromorphic functions.