h^*-Polynomials With Roots on the Unit Circle

2018/06/30 by Braun, Benjamin, Liu, Fu · 1 citation
#05A15 #26C10 #52B20 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1807.00105

Abstract

For an n-dimensional lattice simplex Δ_(1,q) with vertices given by the standard basis vectors and -q where q has positive entries, we investigate when the Ehrhart h^*-polynomial for Δ_(1,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 Δ_(1,q) for which q has only two or three distinct entries, providing both theoretical results and conjectures/questions motivated by experimental evidence.

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• Smith Normal Forms of Graphical Hermite Simplices

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