The Sign Pattern Problem for Ehrhart Polynomials
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912733800169472 |
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| author | Liu, Feihu Tao, Sihao Xin, Guoce |
| author_facet | Liu, Feihu Tao, Sihao Xin, Guoce |
| contents | We investigate the sign patterns of coefficients in the Ehrhart polynomial of the Cartesian product between the $r$-th pyramid over the Reeve tetrahedron and the hypercube $[0, n]^n$. This investigation yields partial results on the sign pattern problem for Ehrhart polynomials. Moreover, we show that for each dimension $d \geq 4$, there exists a $d$-dimensional integral polytope $\mathcal{P}$ such that arbitrarily many of the low-degree coefficients in the Ehrhart polynomial $i(\mathcal{P}, t)$ are negative, while all higher-degree coefficients are positive. Finally, we establish five embedding theorems that enable the sign pattern of a lower-dimensional integral polytope to be embedded into a higher-dimensional integral polytope in various ways. As an application, we completely resolve the Ehrhart coefficient sign pattern problem for dimensions $d = 7, 8, 9$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_17714 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Sign Pattern Problem for Ehrhart Polynomials Liu, Feihu Tao, Sihao Xin, Guoce Combinatorics We investigate the sign patterns of coefficients in the Ehrhart polynomial of the Cartesian product between the $r$-th pyramid over the Reeve tetrahedron and the hypercube $[0, n]^n$. This investigation yields partial results on the sign pattern problem for Ehrhart polynomials. Moreover, we show that for each dimension $d \geq 4$, there exists a $d$-dimensional integral polytope $\mathcal{P}$ such that arbitrarily many of the low-degree coefficients in the Ehrhart polynomial $i(\mathcal{P}, t)$ are negative, while all higher-degree coefficients are positive. Finally, we establish five embedding theorems that enable the sign pattern of a lower-dimensional integral polytope to be embedded into a higher-dimensional integral polytope in various ways. As an application, we completely resolve the Ehrhart coefficient sign pattern problem for dimensions $d = 7, 8, 9$. |
| title | The Sign Pattern Problem for Ehrhart Polynomials |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2509.17714 |