Dimension Polynomials for Affine Partial Difference Algebraic Groups

TL;DR

This work develops a finite-generation theory for affine $\sigma$-algebraic groups with finitely many commuting endomorphisms and proves the existence of a dimension polynomial for partial difference algebraic groups. By introducing generalised difference algebraic groups and analyzing Zariski closures, kernels, and projections, the authors extend finiteness and dimensionality results from the ordinary to the partial setting. They establish that defining $\sigma$-Hopf ideals are finitely $\sigma$-generated, construct quotients by normal subgroups, and prove the existence and invariance of dimension polynomials, yielding robust invariants: $\sigma$-dimension, $\sigma$-type, and typical $\sigma$-dimension. The paper further connects these concepts to classical difference-module theory and difference field extensions, showing that the dimension-theoretic insights transfer across frameworks and underpin a Galois-like theory for difference algebraic groups with multi-parameter shifts.

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.

Figures (13)

• Figure 1: Universal Property for $A\hookrightarrow [\sigma]_k A$
• Figure 2: Explaining Visualisation Diagrams
• Figure 3: Visualising Base Change when $n=2$
• Figure 4: $\pi[i]$, $\sigma_1[i]$ and $\sigma_2[i]$ acting on $\mathcal{G}[i]$ when $n=2$
• Figure 5: Commutative Diagram for a Generalised $\sigma$/̄Algebraic Group $(G[i])_{i\in\mathbb{N}}$

Theorems & Definitions (141)

• Theorem 1: Corollary \ref{['cor: hopf ideal fin gen']}
• Theorem 2: Corollary \ref{['cor: partial dim poly zariskis']}
• Example 1.1
• Definition 2.1
• Lemma 2.2
• proof
• Definition 2.3
• Definition 2.4
• Proposition 2.5
• proof