AI 中文总结
该研究针对两类Hermite标准形单形，构造正则单模三角剖分，推导$h^\ast$-多项式等闭式公式，证明Ehrhart正性，还得到单行情形Ehrhart多项式非单峰的维度条件。
AI 中文摘要
我们研究两类Hermite标准形单形的正则单模三角剖分、整数分解性质及Ehrhart理论性质。首先考虑与向量$(N - 1, \dots,N - 1, N)\in \mathbb{N}^d$相关的单行情形，完全刻画对应单形何时存在正则单模三角剖分，构造是显式的，还给出了$h^\ast$-多项式和局部$h^\ast$-多项式的闭式公式；此外，证明了Ehrhart正性，并推导了Ehrhart多项式非单峰的显式依赖维度的条件。最后，将方法扩展到与$(1, \dots,1, N)\in\mathbb{N}^d$和$(M-1, \dots,M-1, M, 0)\in\mathbb{N}^d$相关的两行情形，在这些情形中构造了正则单模三角剖分，推导了$h^\ast$-多项式和局部$h^\ast$-多项式的闭式公式，并证明了Ehrhart正性。
英文摘要
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.
Comments24 pages, 3 figures
