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PreprintCC BY 4.0arXiv

Ehrhart positivity of smooth polytopes with few facets

  • Feihu Liu
  • Jinlong Tang
  • Sihao Tao
  • Zihao Zhang
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Abstract

For each dimension d, we determine the minimum number of facets of a smooth d-dimensional lattice polytope whose Ehrhart polynomial has at least one negative coefficient. This minimum is six in dimension d=3, seven in dimension d=4, and d+2 in every dimension d≥ 5. The lower bounds follow from the classification of smooth lattice polytopes with at most d+2 facets, together with explicit formulas for Ehrhart polynomials in dimensions at most $4$. We also prove that a lattice polytope of dimension at least $3$ with a smooth vertex cannot be $2$-integral with respect to any ordered lattice basis.

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Cite

BibTeXRIS
BibTeX
@misc{liu2026ehrhart,
  title = {{Ehrhart positivity of smooth polytopes with few facets}},
  author = {Feihu Liu and Jinlong Tang and Sihao Tao and Zihao Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.27819},
  eprint = {2610.27819},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.27819},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Ehrhart positivity of smooth polytopes with few facets
AU  - Feihu Liu
AU  - Jinlong Tang
AU  - Sihao Tao
AU  - Zihao Zhang
PY  - 2026
DA  - 2026/10/20
PB  - arXiv
DO  - 10.48550/arxiv.2610.27819
UR  - https://arxiv.org/abs/2610.27819
AB  - For each dimension d, we determine the minimum number of facets of a smooth d-dimensional lattice polytope whose Ehrhart polynomial has at least one negative coefficient. This minimum is six in dimension d=3, seven in dimension d=4, and d+2 in every dimension d≥ 5. The lower bounds follow from the classification of smooth lattice polytopes with at most d+2 facets, together with explicit formulas for Ehrhart polynomials in dimensions at most $4$. We also prove that a lattice polytope of dimension at least $3$ with a smooth vertex cannot be $2$-integral with respect to any ordered lattice basis.
ER  -