Some Computational Study of the Root Distribution of Ehrhart Polynomials

Abstract

In this paper, we investigate the root distribution of the Ehrhart polynomials of lattice polytopes. When the lattice polytope is reflexive, the roots of the Ehrhart polynomial distribute symmetrically with respect to the line [Formula: see text]. A special case of this distribution is when all the roots lie on this line. Our main concern is to find out which reflexive polytopes satisfy this special condition. Such lattice polytopes are called CL-polytopes. Another special case opposite to this is when all the roots are real. Such lattice polytopes are called real polytopes. The first topic of this paper is the Ehrhart polynomials of equatorial spheres, which are related to graded posets. We discuss the CL-ness of the equatorial Ehrhart polynomials for complete graded posets and zig-zag posets. The second topic is to investigate the relation of the root distribution of the Ehrhart polynomials of a reflexive polytope [Formula: see text] and its dual [Formula: see text]. We discuss the CL-ness and realness of [Formula: see text] and [Formula: see text] in pair, of dimensions up to 4. Throughout this paper, we investigate these problems by computer calculation.