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Peer-reviewedCC BYJournal of Combinatorial Algebra

Harmonics and graded Ehrhart theory

  • Victor Reiner1
  • Brendon Rhoades2
  1. 1University of Minnesota, Minneapolis, USA
  2. 2University of California, San Diego, USA
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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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Cite

BibTeXRIS
BibTeX
@article{reiner2026harmonics,
  title = {{Harmonics and graded Ehrhart theory}},
  author = {Victor Reiner and Brendon Rhoades},
  journal = {Journal of Combinatorial Algebra},
  year = {2026},
  month = jun,
  doi = {10.4171/jca/128},
  url = {https://doi.org/10.4171/jca/128},
  note = {CC BY},
}
RIS
TY  - JOUR
TI  - Harmonics and graded Ehrhart theory
AU  - Victor Reiner
AU  - Brendon Rhoades
PY  - 2026
DA  - 2026/06/19
JO  - Journal of Combinatorial Algebra
DO  - 10.4171/jca/128
UR  - https://doi.org/10.4171/jca/128
AB  - 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.
SN  - 2415-6302
SN  - 2415-6310
ER  -