Roots of Ehrhart Polynomials and Symmetric Δ-vectors
نویسنده
چکیده
Abstract. The conjecture on roots of Ehrhart polynomials, stated by Matsui et al. [15, Conjecture 4.10], says that all roots α of the Ehrhart polynomial of a Gorenstein Fano polytope of dimension d satisfy − d 2 ≤ Re(α) ≤ d 2 − 1. In this paper, we observe the behaviors of roots of SSNN polynomials which are a wider class of the polynomials containing all the Ehrhart polynomials of Gorenstein Fano polytopes. As a result, we verify that this conjecture is true when the roots are real numbers or when d ≤ 5.
منابع مشابه
Ehrhart polynomials of convex polytopes with small volumes
Let P ⊂ R be an integral convex polytope of dimension d and δ(P) = (δ0, δ1, . . . , δd) be its δ-vector. By using the known inequalities on δ-vectors, we classify the possible δ-vectors of convex polytopes of dimension d with P
متن کاملEhrhart Polynomials of Integral Simplices with Prime Volumes
For an integral convex polytope P ⊂ R of dimension d, we call δ(P) = (δ0, δ1, . . . , δd) the δ-vector of P and vol(P) = ∑d i=0 δi its normalized volume. In this paper, we will establish the new equalities and inequalities on δ-vectors for integral simplices whose normalized volumes are prime. Moreover, by using those, we will classify all the possible δ-vectors of integral simplices with norma...
متن کاملRoots of Ehrhart Polynomials Arising from Graphs
Several polytopes arise from finite graphs. For edge and symmetric edge polytopes, in particular, exhaustive computation of the Ehrhart polynomials not merely supports the conjecture of Beck et al. that all roots α of Ehrhart polynomials of polytopes of dimension D satisfy −D ≤ Re(α) ≤ D − 1, but also reveals some interesting phenomena for each type of polytope. Here we present two new conjectu...
متن کاملNotes on the Roots of Ehrhart Polynomials
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 , 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 po...
متن کاملEhrhart f*-Coefficients of Polytopal Complexes are Non-negative Integers
The Ehrhart polynomial LP of an integral polytope P counts the number of integer points in integral dilates of P . Ehrhart polynomials of polytopes are often described in terms of their Ehrhart h∗-vector (aka Ehrhart δ-vector), which is the vector of coefficients of LP with respect to a certain binomial basis and which coincides with the h-vector of a regular unimodular triangulation of P (if o...
متن کامل