Ehrhart Theory over Abelian Group Rings

Summary · qwen2.5:32b

The article introduces a unified framework for Ehrhart theory using Abelian group rings to encode richer algebraic data than traditional counts, proving that key results like the Brion theorem extend to this setting. This framework enables the derivation of $q$-enumerative and weighted theories as consequences of a single mechanism, enhancing the algebraic versatility of lattice point enumeration.

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Learn about Ehrhart Theory over Abelian Group Rings in this new study, extending fundamental results and connecting to volume and vertex-cone decompositions.

Excerpt

arXiv:2511.10373v2 Announce Type: replace 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.