$h^*$-Polynomials With Roots on the Unit Circle

For an $n$-dimensional lattice simplex $\Delta_{(1,\mathbf{q})}$ with vertices given by the standard basis vectors and $-\mathbf{q}$ where $\mathbf{q}$ has positive entries, we investigate when the Ehrhart $h^*$-polynomial for $\Delta_{(1,\mathbf{q})}$ factors as a product of geometric series in powers of $z$. Our motivation is a theorem of Rodriguez-Villegas implying that when the $h^*$-polynomial of a lattice polytope $P$ has all roots on the unit circle, then the Ehrhart polynomial of $P$ has positive coefficients. We focus on those $\Delta_{(1,\mathbf{q})}$ for which $\mathbf{q}$ has only two or three distinct entries, providing both theoretical results and conjectures/questions motivated by experimental evidence.