Harmonics and graded Ehrhart theory

TL;DR

This paper introduces a $q$-deformation of Ehrhart series using harmonic spaces and Macaulay inverse systems, proposing conjectures and constructing a bigraded algebra to study lattice point enumeration in polytopes.

Contribution

It develops a novel $q$-deformation framework for Ehrhart series and constructs a related bigraded algebra, advancing the algebraic understanding of lattice point counting.

Findings

Conjecture that the $q$-Ehrhart series is a rational function.

Construction of a bigraded algebra with Hilbert series matching the $q$-Ehrhart series.

New results on Macaulay inverse systems for Minkowski sums.

Abstract

The Ehrhart polynomial and Ehrhart series count lattice points in integer dilations of a lattice polytope. We introduce and study a $q$-deformation of the Ehrhart series, based on the notions of harmonic spaces and Macaulay's inverse systems for coordinate rings of finite point configurations. We conjecture that this $q$-Ehrhart series is a rational function, and introduce and study a bigraded algebra whose Hilbert series matches the $q$-Ehrhart series. Defining this algebra requires a new result on Macaulay inverse systems for Minkowski sums of point configurations.