Roots of Ehrhart polynomials of Gorenstein Fano polytopes

Abstract

Given arbitrary integers $k$ and $d$ with $0 \leq 2k \leq d$, we construct a Gorenstein Fano polytope \Pc \subset \RR^d of dimension $d$ such that (i) its Ehrhart polynomial i(\Pc, n) possesses $d$ distinct roots; (ii) i(\Pc, n) possesses exactly $2k$ imaginary roots; (iii) i(\Pc, n) possesses exactly $d - 2k$ real roots; (iv) the real part of each of the imaginary roots is equal to $- 1 / 2$; (v) all of the real roots belong to the open interval $(-1, 0)$.Comment: 6 page