Interlacing Ehrhart polynomials of reflexive polytopes

Abstract

It was observed by Bump et al. that Ehrhart polynomials in a special family exhibit properties shared by the Riemann zeta function. The construction was generalized by Matsui et al. to a larger family of reflexive polytopes coming from graphs. We prove several conjectures confirming when such polynomials have zeros on a certain line in the complex plane. Our main new method is to prove a stronger property called interlacing.