Notes on the Roots of Ehrhart Polynomials

Abstract

Abstract. We determine lattice polytopes of smallest volume with a given number of interior lattice points. We show that the Ehrhart polynomials of those with one interior lattice point have largest roots with norm of order n 2 , where n is the dimension. This improves on the previously best known bound n and complements a recent result of Braun [8] where it is shown that the norm of a root of a Ehrhart polynomial is at most of order n 2 .For the class of 0-symmetric lattice polytopes we present a conjecture on the smallest volume for a given number of interior lattice points and prove the conjecture for crosspolytopes.We further give a characterisation of the roots of the Ehrhart polyomials in the 3-dimensional case and we classify for n ≤ 4 all lattice polytopes whose roots of their Ehrhart polynomials have all real part -1/2. These polytopes belong to the class of reflexive polytopes.

Interlacing Ehrhart polynomials of reflexive polytopes

“…Furthermore, the roots of Ehrhart polynomials are also an object of intensive studies [6,9,12,20]. There are many graphs for which these roots have a remarkable property: They lie on a line R = {z : Re(z) = − 1 2 }.…”

Roots of Ehrhart polynomials arising from graphs

“…The roots of an Ehrhart polynomial should also reflect properties of a polytope that are hard to elicit just from the coefficients. Among the many papers on the topic, including [4], [5], [6], [12] and [23], Beck et al [3] conjecture that: Compared with the norm bound, which is O(D 2 ) in general [5], the strip in the conjecture puts a tight restriction on the distribution of roots for any Ehrhart polynomial.…”

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.…”