h*-Polynomials with Roots on the Unit Circle

Abstract

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

Keywords:

• Ehrhart positivity
• unit circle rooted
• h*-polynomial
• lattice simplex

2010 Mathematics Subject Classification:

• Primary: 52B20
• 05A15
• 26C10

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The first author was partially supported by grant H98230-16-1-0045 from the U.S. National Security Agency. The second author was partially supported by a grant from the Simons Foundation #426756. This material is also based in part upon work supported by the National Science Foundation under Grant No. DMS-1440140 while both authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2017 semester.