Harmonics and graded Ehrhart theory

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.