$h^*$-Polynomials With Roots on the Unit Circle
Abstract
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.
- Publication:
-
arXiv e-prints
- Pub Date:
- June 2018
- DOI:
- 10.48550/arXiv.1807.00105
- arXiv:
- arXiv:1807.00105
- Bibcode:
- 2018arXiv180700105B
- Keywords:
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- Mathematics - Combinatorics;
- Mathematics - Number Theory;
- 52B20;
- 05A15;
- 26C10
- E-Print:
- minor clarifications added to version 2