Difference Dimension Quasi-polynomials

TL;DR

This paper studies Hilbert-type functions related to algebraic difference equations with weighted translations, showing they are quasi-polynomials linked to Ehrhart polynomials, thus generalizing difference dimension invariants.

Contribution

It introduces a framework for difference Hilbert functions with weighted translations, demonstrating their quasi-polynomial nature and connecting them to Ehrhart theory, extending existing difference dimension results.

Findings

Hilbert-type functions are quasi-polynomials.

These functions can be expressed as sums of Ehrhart quasi-polynomials.

Generalizations of difference dimension polynomials to weighted operators.

Abstract

We consider Hilbert-type functions associated with difference (not necessarily inversive) field extensions and systems of algebraic difference equations in the case when the translations are assigned some integer weights. We will show that such functions are quasi-polynomials, which can be represented as alternative sums of Ehrhart quasi-polynomials associated with rational conic polytopes. In particular, we obtain generalizations of main theorems on difference dimension polynomials and their invariants to the case of weighted basic difference operators.