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PreprintCC BY 4.0arXiv

Two families of Hermite normal form simplices

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

We study Ehrhart coefficients, the integer decomposition property, and unimodular triangulations for two families of Hermite normal form simplices considered by Bruckamp, Caicedo, and Juhnke. The first family consists of simplices of the form Sa=conv(0,e1,…,ed−1,a), where a=(a1,…,ad−1,N), and the second consists of the simplices Td,N=conv(0,e1,…,ed−2,(d−2,…,d−2,d−1,0),(1,…,1,N)). For the family Td,N, we prove unimodality of the Ehrhart coefficients in every dimension and determine the log-concave and real-rooted cases. For the subfamily Sa with a=(N−q,…,N−q,N), we characterize both the integer decomposition property and the existence of a regular unimodular triangulation via a congruence condition on a negative continued fraction. We also give an arithmetic test for the integer decomposition property and finite triangulation tests for Sa with arbitrary a. These results address three open questions of Bruckamp, Caicedo, and Juhnke.

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Cite

BibTeXRIS
BibTeX
@misc{liu2026families,
  title = {{Two families of Hermite normal form simplices}},
  author = {Feihu Liu and Jinlong Tang and Sihao Tao and Zihao Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.36818},
  eprint = {2609.36818},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.36818},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Two families of Hermite normal form simplices
AU  - Feihu Liu
AU  - Jinlong Tang
AU  - Sihao Tao
AU  - Zihao Zhang
PY  - 2026
DA  - 2026/10/16
PB  - arXiv
DO  - 10.48550/arxiv.2609.36818
UR  - https://arxiv.org/abs/2609.36818
AB  - We study Ehrhart coefficients, the integer decomposition property, and unimodular triangulations for two families of Hermite normal form simplices considered by Bruckamp, Caicedo, and Juhnke. The first family consists of simplices of the form S_a=conv(0,e_1,…,e_d−1,a), where a=(a_1,…,a_d−1,N), and the second consists of the simplices T_d,N=conv(0,e_1,…,e_d−2,(d−2,…,d−2,d−1,0),(1,…,1,N)). For the family T_d,N, we prove unimodality of the Ehrhart coefficients in every dimension and determine the log-concave and real-rooted cases. For the subfamily S_a with a=(N−q,…,N−q,N), we characterize both the integer decomposition property and the existence of a regular unimodular triangulation via a congruence condition on a negative continued fraction. We also give an arithmetic test for the integer decomposition property and finite triangulation tests for S_a with arbitrary a. These results address three open questions of Bruckamp, Caicedo, and Juhnke.
ER  -