Abstract:In Ehrhart theory, the well-konwn sign pattern problem asks: given a positive integer $d\geq 3$ and integers $1 \leq i_1 < \cdots < i_k \leq d-2$, does there exist a $d$-dimensional integral polytope $\mathcal{P}$ such that in its Ehrhart polynomial $i(\mathcal{P}, t)$ the coefficients of $t^{i_1}, \ldots, t^{i_k}$ are negative, while the remaining coefficients are positive? This problem was proposed by Hibi, Higashitani, Tsuchiya, and Yoshida. In this paper, we first construct a class of simplices $\mathcal{S}_d(m)$ whose Ehrhart polynomial has leading coefficient $m$ and all remaining coefficients fixed positive constants. Then, using the Cartesian product of $\mathcal{S}_d(m)$ and the Reeve tetrahedron, we obtain the first complete solution to the sign pattern problem. Finally, in attacking the sign pattern problem, we provide a fast algorithm for computing the $h^*$-polynomial of a class of simplices $\Delta(0,q)$.
| Subjects: | Combinatorics (math.CO) |
| Cite as: | arXiv:2605.23544 [math.CO] |
| (or arXiv:2605.23544v1 [math.CO] for this version) | |
| https://doi.org/10.48550/arXiv.2605.23544 arXiv-issued DOI via DataCite (pending registration) |
From: Sihao Tao [view email]
[v1]
Fri, 22 May 2026 12:08:26 UTC (28 KB)
此内容由惯性聚合(RSS阅读器)自动聚合整理，仅供阅读参考。 原文来自 — 版权归原作者所有。