Ehrhart Theory over Abelian Group Rings

Abstract

We introduce a unified framework for Ehrhart theory in which lattice point enumerators take coefficients in an Abelian group ring, encoding substantially richer algebraic data than classical counts. We prove that fundamental results of Ehrhart theory extend to this setting through a generalized Brion theorem, including rational generating functions, reciprocity phenomena, connections to volume, and vertex-cone decompositions. We further show how to derive $q$-enumerative and weighted theories from this setting, recasting several major refinements of Ehrhart theory as consequences of a single algebraic mechanism. We also show how our framework combines with equivariant Ehrhart theory.