Roots of Ehrhart polynomials of Gorenstein Fano polytopes

Abstract

Let P ⊂ R N be an integral convex polytope of dimension d and ∂P its boundary. (An integral convex polytope is a convex polytope all of whose vertices have integer coordinates.) Given integers n = 1, 2, . . ., we write i(P, n) for the number of integer points belonging to nP, where nP = {nα : α ∈ P}. In other words,

Gorenstein graphic matroids

“…[5,Section 8.3] and references therein. Furthermore, relations of the Gorenstein property to combinatorics are also an important research topic [4,11,12,13,14,20].…”

On algebraic properties of matroid polytopes

“…It is already in use on its own combinatorial sake -see e.g. [3,4,6,17,29,30,31,33]. However, the notion of a Gorenstein polytope originates from algebra.…”

Roots of Ehrhart polynomials and symmetric $δ$-vectors

“…It is known [4,Proposition 1.8] that if all roots α ∈ C of i(P, n) of an integral convex polytope P of dimension d satisfy Re(α) = − 1 2 , then P is unimodularly equivalent to a Gorenstein Fano polytope whose volume is at most 2 d . In [11], Gorenstein Fano polytopes whose Ehrhart polynomials have a reasonable root distribution are studied. In [8], the roots of the Ehrhart polynomials of smooth Fano polytopes with small dimensions are completely determined.…”