Finiteness properties of affine difference algebraic groups

TL;DR

This paper proves that certain groups defined by algebraic difference equations have finiteness properties, including finite generation of defining equations and difference ideals, advancing the understanding of difference algebraic groups.

Contribution

It establishes that subgroups of the general linear group defined by infinitely many difference equations are finitely generated, and that the difference ideal of solutions to linear differential equations is finitely generated.

Findings

Subgroups of GL_n defined by infinitely many difference equations are finitely generated.

The difference ideal of solutions to linear differential equations is finitely generated.

Provides foundational results on the structure of difference algebraic groups.

Abstract

We establish several finiteness properties of groups defined by algebraic difference equations. One of our main results is that a subgroup of the general linear group defined by possibly infinitely many algebraic difference equations in the matrix entries can indeed be defined by finitely many such equations. As an application, we show that the difference ideal of all difference algebraic relations among the solutions of a linear differential equation is finitely generated.