Unimodular triangulations and Ehrhart theory for two families of Hermite normal form simplices

@misc{bruckamp2026unimodular,
  title = {{Unimodular triangulations and Ehrhart theory for two families of Hermite normal form simplices}},
  author = {Justus Bruckamp and Jhon B. Caicedo and Martina Juhnke},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.28282},
  eprint = {2608.28282},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.28282},
}

TY  - UNPB
TI  - Unimodular triangulations and Ehrhart theory for two families of Hermite normal form simplices
AU  - Justus Bruckamp
AU  - Jhon B. Caicedo
AU  - Martina Juhnke
PY  - 2026
DA  - 2026/08/28
PB  - arXiv
DO  - 10.48550/arxiv.2608.28282
UR  - https://arxiv.org/abs/2608.28282
AB  - We study regular unimodular triangulations, the integer decomposition property, and Ehrhart-theoretic properties of two families of Hermite normal form simplices. We first consider the one-row case associated with the vector (N − 1, … ,N − 1 , N)∈ ℕ^d, and completely characterize when the corresponding simplices admit a regular unimodular triangulation. Our constructions are explicit and also yield closed formulas for the h^∗-polynomial and the local h^∗-polynomial. Moreover, we prove Ehrhart positivity and derive explicit dimension-dependent conditions under which the Ehrhart polynomial is not unimodal. Finally, we extend our approach to the two-row cases associated with (1, … ,1 , N)∈ℕ^d and (M−1, … ,M−1, M, 0)∈ℕ^d. In these cases, we construct regular unimodular triangulations, derive closed formulas for the h^∗-polynomial and the local h^∗-polynomial, and prove Ehrhart positivity.
ER  -