q-analogues of Ehrhart polynomials

TL;DR

This paper introduces a q-analogue framework for Ehrhart polynomials, extending classical lattice point enumeration with weighted sums involving a parameter q, and explores properties like reciprocity and evaluations at q-integers.

Contribution

It develops a novel q-analogue theory of Ehrhart polynomials, incorporating weights and reciprocity, expanding the classical combinatorial enumeration methods.

Findings

Established a q-analogue of Ehrhart series and polynomials

Proved Ehrhart reciprocity in the q-analogue setting

Analyzed evaluations at q-integers

Abstract

One considers weighted sums over points of lattice polytopes, where the weight of a point v is the monomial q^f(v) for some linear form f. One proposes a q-analogue of the classical theory of Ehrhart series and Ehrhart polynomials, including Ehrhart reciprocity and involving evaluation at the q-integers.