International Game Theory Review 25 (2023), 2340005, Special Issue on "Computational Operations Research and Algorithmic Game Theory", S.K. Neogy, R.B. Bapat and T. Parthasarathy eds.
Some computational studies of the root distribution of Ehrhart polynomials
by Masahiro Hachimori, Yumi Yamada
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 Re(z) = -1/2.
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 Q and its dual Q. We discuss the CL-ness
and realness of Q and Q in pair, of dimensions up to 4. Throughout this paper, we
investigate these problems by computer calculation.