Algebraic groups as difference Galois groups of linear differential equations

TL;DR

This paper proves that every linear algebraic group can be realized as a difference Galois group of some linear differential equation over the field of rational functions, advancing the inverse problem in difference Galois theory.

Contribution

It establishes that all linear algebraic groups are realizable as difference Galois groups over (x), solving the inverse problem in this setting.

Findings

Every linear algebraic group occurs as a difference Galois group.

Advances the understanding of the inverse problem in difference Galois theory.

Provides a construction for realizing algebraic groups as Galois groups.

Abstract

We study the inverse problem in the difference Galois theory of linear differential equations over the difference-differential field $\mathbb{C}(x)$ with derivation $\frac{d}{dx}$ and endomorphism $f(x)\mapsto f(x+1)$. Our main result is that every linear algebraic group, considered as a difference algebraic group, occurs as the difference Galois group of some linear differential equation over $\mathbb{C}(x)$.