Roots of Ehrhart Polynomials of Smooth Fano Polytopes

Abstract

V. Golyshev conjectured that for any smooth polytope P of dimension at most five, the roots $z\in\C$ of the Ehrhart polynomial for P have real part equal to -1/2. An elementary proof is given, and in each dimension the roots are described explicitly. We also present examples which demonstrate that this result cannot be extended to dimension six.Comment: 10 page

Ehrhart Polynomial Roots of Reflexive Polytopes

“…He conjectured that every d-dimensional smooth Fano polytope with d ≤ 5 satisfies the canonical line hypothesis. This was proved in [13], along with an example of a smooth Fano polytope in dimension 6 failing to satisfy (CL).…”

Fano polytopes

“…This is very e cient, and has been used to classify all smooth polytopes up to dimension eight. Applications include new examples of Einstein-Kähler manifolds [43], and the study of Riemannian polytopes [21]. Minkowski summands of reflexive polytopes define complete intersections (CYCIs) in Gorenstein toric Fano varieties.…”

Roots of Ehrhart polynomials of Gorenstein Fano polytopes

“…Recently, many research papers on convex polytopes, including [2], [3], [4], [5], [8], [9] and [18], discuss roots of Ehrhart polynomials. One of the fascinating topics is the study on roots of Ehrhart polynomials of Gorenstein Fano polytopes.…”

“…In a recent work [8], the roots of the Ehrhart polynomials of smooth Fano polytopes with small dimensions are completely determined.…”

“…(a) It is disproved in [8] that all of the roots α of the Hilbert polynomial of any Fano variety satisfy −1 < (α) < 0, the so-called canonical strip hypothesis, which is stated in [7]. On Theorem 0.1, however, all of the roots of Ehrhart polynomials of our Gorenstein Fano polytopes satisfy this condition.…”