Difference Galois theory of linear differential equations

TL;DR

This paper develops a Galois theory for linear differential equations with an endomorphism, focusing on difference algebraic relations among solutions and defining Galois groups as linear difference algebraic groups.

Contribution

It introduces a new Galois theory for differential equations incorporating endomorphisms, expanding the algebraic framework to difference algebraic groups.

Findings

Defines Galois groups as linear difference algebraic groups

Provides a framework for studying difference algebraic relations among solutions

Extends classical Galois theory to include endomorphism actions

Abstract

We develop a Galois theory for linear differential equations equipped with the action of an endomorphism. This theory is aimed at studying the difference algebraic relations among the solutions of a linear differential equation. The Galois groups here are linear difference algebraic groups, i.e., matrix groups defined by algebraic difference equations.