Explicit Construction of Polytopes whose Ehrhart Polynomials Realize any Given Sign Pattern
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914598781714432 |
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| author | Liu, Feihu Tao, Sihao Xin, Guoce |
| author_facet | Liu, Feihu Tao, Sihao Xin, Guoce |
| contents | In Ehrhart theory, the well-known 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 all 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 other 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, while attacking the sign pattern problem, we discovered a fast algorithm for computing the $h^*$-polynomial of a class of simplices $Δ(0,q)$. This algorithm is crucial for constructing the simplices $\mathcal{S}_d(m)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_23544 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Explicit Construction of Polytopes whose Ehrhart Polynomials Realize any Given Sign Pattern Liu, Feihu Tao, Sihao Xin, Guoce Combinatorics In Ehrhart theory, the well-known 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 all 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 other 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, while attacking the sign pattern problem, we discovered a fast algorithm for computing the $h^*$-polynomial of a class of simplices $Δ(0,q)$. This algorithm is crucial for constructing the simplices $\mathcal{S}_d(m)$. |
| title | Explicit Construction of Polytopes whose Ehrhart Polynomials Realize any Given Sign Pattern |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2605.23544 |