Finiteness Properties of Affine 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.

Almost-simple affine difference algebraic groups

“…The following finiteness theorem is the combination of Theorem 4.1 and Corollaries 4.2 and 4.3 in [Wib20].…”

“…In this section we introduce some notation that will be used throughout the text. We also recall the required constructions and results from [Wib20]. The reader familiar with [Wib20] may safely skip this section.…”

“…We also recall the required constructions and results from [Wib20]. The reader familiar with [Wib20] may safely skip this section.…”

“…Because of the present infantile state of the theory of difference algebraic groups, the purpose of this article is also to lay the groundwork for a further comprehensive study of affine difference algebraic groups. 1 To this end we build on [Wib20], where some basic finiteness properties of affine difference algebraic groups have been established and numerical invariants for affine difference algebraic groups, such as the σ-dimension σ-dim(G) and the limit degree ld(G), have been introduced. On our path to Theorems A and B above we encounter several basic results and constructions that we deem fundamental for the further development of the theory of affine difference algebraic groups:…”

“…We conclude this introduction with an outline of the article: In Section 1 we go through the details of the definition of affine difference algebraic groups and we recall the necessary results from [Wib20]. Sections 2 to 6 roughly correspond to the five bullet points in the above list.…”