Skip to content
DraconiticResearch
Search Sources API
182,544 studies31 sourcesUpdated 36 min ago
PreprintarXiv licencearXiv

Equivariant Ehrhart series for rational simplices and rational cross-polytopes

  • Ryo Uchiumi
Open at sourcePDFDOICite

Abstract

Equivariant Ehrhart theory was introduced by Stapledon (2011) and has since been extensively studied. A central question is whether the numerator in a certain rational function expression for the equivariant Ehrhart series is both a polynomial and effective. In this paper, we propose a modification of the denominator in the rational function expression for the equivariant Ehrhart series of rational polytopes. For rational simplices and rational cross-polytopes whose vertices have coordinates that are reciprocals of integers, we show that the resulting numerator is a polynomial and is effective. As an application of our results, we compute the equivariant series of the characteristic quasi-polynomials of root systems.

Related

  1. The equivariant Ehrhart theory of polytopes with order-two symmetries: Part II
    Oliver Clarke
    arXivPreprintarXiv licenceFull textAlgebra & number theoryCombinatorics & discrete mathematicsarXiv
  2. Exact Ehrhart Series of Birkhoff Polytopes via Constant Terms and Finite-Field Evaluation
    Xinru Jiang, Guoce Xin, Chen Zhang +1
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
  3. Unimodular triangulations and Ehrhart theory for two families of Hermite normal form simplices
    Justus Bruckamp, Jhon B. Caicedo, Martina Juhnke
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
  4. Sylvester simplices: Triangulations and Ehrhart-theoretic aspects
    Jhon B. Caicedo, Federico Castillo, Martina Juhnke +1
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
  5. Ehrhart polynomials of cyclic polytopes as averages of zonotope Ehrhart polynomials
    Masato Konoike
    arXivPreprintCC BY 4.0Full textCombinatorics & discrete mathematicsarXiv

Cite

BibTeXRIS
BibTeX
@misc{uchiumi2026equivariant,
  title = {{Equivariant Ehrhart series for rational simplices and rational cross-polytopes}},
  author = {Ryo Uchiumi},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.41529},
  eprint = {2610.41529},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.41529},
}
RIS
TY  - UNPB
TI  - Equivariant Ehrhart series for rational simplices and rational cross-polytopes
AU  - Ryo Uchiumi
PY  - 2026
DA  - 2026/10/30
PB  - arXiv
DO  - 10.48550/arxiv.2610.41529
UR  - https://arxiv.org/abs/2610.41529
AB  - Equivariant Ehrhart theory was introduced by Stapledon (2011) and has since been extensively studied. A central question is whether the numerator in a certain rational function expression for the equivariant Ehrhart series is both a polynomial and effective. In this paper, we propose a modification of the denominator in the rational function expression for the equivariant Ehrhart series of rational polytopes. For rational simplices and rational cross-polytopes whose vertices have coordinates that are reciprocals of integers, we show that the resulting numerator is a polynomial and is effective. As an application of our results, we compute the equivariant series of the characteristic quasi-polynomials of root systems.
ER  -