$h^{\ast }$-Polynomials With Roots on the Unit Circle

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