The Sign Pattern Problem for Ehrhart Polynomials

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 $d = 7, 8, 9$.