The Sign Pattern Problem for Ehrhart Polynomials
Feihu Liu, Sihao Tao, Guoce Xin
TL;DR
This paper advances the understanding of sign patterns in Ehrhart polynomials of integral polytopes by (i) deriving explicit Ehrhart polynomials for pyramids over Reeve tetrahedra and their Cartesian products with hypercubes, (ii) constructing high-dimensional polytopes whose low-degree coefficients are negative while higher-degree coefficients are positive, and (iii) introducing five embedding theorems that systematically transfer sign patterns across dimensions. These embedding theorems enable partial resolutions of the sign-pattern problem, reducing the conjectural verification to a Fibonacci-based finite set of cases and yielding complete results for dimensions 7–9. The work combines explicit polytope constructions with generating-function techniques and Newton-type analyses to establish both specific examples and general transfer principles. Collectively, the results deepen the structural understanding of Ehrhart positivity and lay groundwork for a more general resolution of sign-pattern phenomena in high dimensions.
Abstract
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 $d = 7, 8, 9$.