Galois theory of difference equations with periodic parameters

TL;DR

This paper develops a Galois theory for linear difference equations with periodic parameters and introduces linear difference algebraic groups, enabling the testing of solutions against polynomial q'-difference equations.

Contribution

It introduces a new Galois theory framework for difference equations with periodic parameters and applies it to analyze solutions of q-difference equations.

Findings

Established a Galois theory for systems with periodic parameters

Constructed linear difference algebraic groups

Provided a method to test solutions against polynomial q'-difference equations

Abstract

We develop a Galois theory for systems of linear difference equations with periodic parameters, for which we also introduce linear difference algebraic groups. We then apply this to constructively test if solutions of linear q-difference equations, with complex q, not a root of unity, satisfy any polynomial q'-difference equations with q' being a root of unity.