Algebraic groups as difference Galois groups of linear differential equations

Abstract

We study the inverse problem in the difference Galois theory of linear differential equations over the difference-differential field C ( x ) with derivation d d x and endomorphism f ( x ) ↦ 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 C ( x ) .