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arXiv · 2608.28282

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

Justus Bruckamp·Jhon B. Caicedo·Martina Juhnke
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Abstract

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 $(N - 1, \dots ,N - 1 , N)\in \mathbb{N}^d$, and completely characterize when the corresponding simplices admit a regular unimodular triangulation. Our constructions are explicit and also yield closed formulas for the $h^\ast$-polynomial and the local $h^\ast$-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 $(1, \dots ,1 , N)\in\mathbb{N}^d$ and $(M-1, \dots ,M-1, M, 0)\in\mathbb{N}^d$. In these cases, we construct regular unimodular triangulations, derive closed formulas for the $h^\ast$-polynomial and the local $h^\ast$-polynomial, and prove Ehrhart positivity.

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Justus Bruckamp, Jhon B. Caicedo, Martina Juhnke. 2026-08-28. Unimodular triangulations and Ehrhart theory for two families of Hermite normal form simplices. https://arxiv.org/abs/2608.28282

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