Explicit Construction of Polytopes whose Ehrhart Polynomials Realize any Given Sign Pattern

In Ehrhart theory, the well-known sign pattern problem asks: given a positive integer $d\ge 3$ and integers $1\le i_1<\cdots <i_k\le d-2$, does there exist a $d$-dimensional integral polytope $\mathcal{P}$ such that in its Ehrhart polynomial $i(\mathcal{P},t)$ the coefficients of $t^{i_1},\dots ,t^{i_k}$ 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 $h^{\ast }$-polynomial of a class of simplices $\mathrm{\Delta }(0,q)$. This algorithm is crucial for constructing the simplices ${\mathcal{S}}_d(m)$.