Dimension Polynomials for Affine Partial Difference Algebraic Groups

TL;DR

This paper develops the theory of affine partial difference algebraic groups with finitely many commuting difference operators, establishing finite generation of their defining ideals and the existence of a dimension polynomial.

Contribution

It introduces the concept of dimension polynomials for partial difference algebraic groups and proves their existence, advancing the algebraic theory of such groups.

Findings

Defining ideals of difference algebraic groups are finitely generated.

Existence of a dimension polynomial for any partial difference algebraic group.

Foundation for further algebraic and geometric analysis of difference algebraic groups.

Abstract

We develop the theory of difference algebraic groups in the case where we have finitely many pairwise commuting difference operators. We show that the defining ideal of a difference algebraic group is finitely generated as a difference ideal, and this result allows us to prove the existence of a dimension polynomial for any partial difference algebraic group.