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arXiv · 2407.06511

Harmonics and graded Ehrhart theory

Victor Reiner·Brendon Rhoades
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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.

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Victor Reiner, Brendon Rhoades. 2024-09-24. Harmonics and graded Ehrhart theory. https://arxiv.org/abs/2407.06511

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