Ehrhart polynomials and stringy Betti numbers

By M. Mustaţă, Sam Payne. 2005

54 citations recorded by the source corpus.

We study the connection between stringy Betti numbers of Gorenstein toric varieties and the generating functions of the Ehrhart polynomials of certain polyhedral regions. We use this point of view to give counterexamples to Hibi's conjecture on the unimodality of δ-vectors of reflexive polytopes.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/6cfc661bf23bbff4a179ae8bb12dd046c4917ee0

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

Citations in the corpus

Unimodality, log-concavity, real-rootedness and beyond (2015, background): “ial, h P(x), in the numerator of (3.7) are nonnegative, and Hibi [59] conjectured that h P(x) is unimodal whenever it is symmetric. Hibi [59] proved the conjecture for n5. However Payne and Mustat˘a [69,74] found counterexamples to Hibi’s conjecture for each n6. Let us mention a weaker conjecture th”
Inequalities and Ehrhart $\delta$-Vectors (2008, unknown): “.18). We see from Lemma 2.3 that δP(t) = tdδP(t−1) if and only if δP (t) = ¯δP (t) = a(t). Payne and Musta¸ta gave examples of reﬂexive polytopes where the coeﬃcients of δP(t) = a(t) are not unimodal [15]. Further examples are given by Payne for all d ≥ 6 [16]. Remark 2.8. It follows from (18) that ”
Ehrhart polynomials and successive minima (2005, background): “mple geometric interpretation which we present in detail in (2.1). All other coeﬃcients Gi(P), 1 ≤ i ≤ n−2, have no such direct geometric meanings, except for special classes of polytopes (cf., e.g., [31, 34, 16, 8, 6, 29, 26, 32]). Since its discovery the Ehrhart polynomial and its coeﬃcients play ”
The s-eulerian polynomials have only real roots (2012, unknown): “nvex polytope with integer vertices need not be symmetric or unimodal. Although there has been much progress in the direction of characterizing those polytopes whose h∗-vector is unimodal (see, e.g., [2, 8, 25, 34]) this is still an open question. However, we can use Theorem 1.1 to answer the questi”
Unimodality Questions for Integrally Closed Lattice Polytopes (2011, background): “Hibi showed that the δ -vector of a reflexive polytope of dimension d is unimodal if d 5 and conjectured that this would hold in general [15], but Mustaţă and Payne gave counterexamples for dimension 6 and higher [19, 20].”
Existence of unimodular triangulations — positive results (2014, background): “its dilation cP for every positive integer c. With the canonical triangulations at our disposal, we can now discuss the example advertised at the beginning of the section. This example is inspired by [91] and appears in [40]. Example 4.13. Let h and k be positive integers and set d := hk −1. In Rd+1”