📝 SummarXiv

Abstract

We investigate 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 yields partial results on the sign pattern problem for Ehrhart polynomials. Moreover, we shown that for each $d\geq 4$, there exists a $d$-dimensional integral polytope $\mathcal{P}$ such that an arbitrary number of low-degree coefficients in the Ehrhart polynomial $i(\mathcal{P},t)$ are negative, while all remaining higher-degree coefficients are positive. Finally, we establish five embedding theorems that enables the sign pattern of a lower-dimensional integral polytope to be embedded into a highly-dimensional integral polytope in various ways. As an application, we provide an affirmative answer to the sign pattern problem for $d=7,8,9$.