Ehrhart polynomials with negative coefficients

It is shown that, for each \(d \geq 4\), there exists an integral convex polytope \(\mathcal{P}\) of dimension \(d\) such that each of the coefficients of \(n, n^{2}, \ldots, n^{d-2}\) of its Ehrhart polynomial \(i(\mathcal{P},n)\) is negative. Moreover, it is also shown that for each \(d \geq 3\) a...

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Bibliographic Details
Published in:arXiv.org 2016-05
Main Authors: Hibi, Takayuki, Higashitani, Akihiro, Tsuchiya, Akiyoshi, Yoshida, Koutarou
Format: Article
Language:English
Subjects:
Coefficients
Integrals
Polynomials
Polytopes
Online Access:Get full text
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Summary:It is shown that, for each \(d \geq 4\), there exists an integral convex polytope \(\mathcal{P}\) of dimension \(d\) such that each of the coefficients of \(n, n^{2}, \ldots, n^{d-2}\) of its Ehrhart polynomial \(i(\mathcal{P},n)\) is negative. Moreover, it is also shown that for each \(d \geq 3\) and \(1 \leq k \leq d-2\), there exists an integral convex polytope \(\mathcal{P}\) of dimension \(d\) such that the coefficient of \(n^k\) of the Ehrhart polynomial \(i(\mathcal{P},n)\) of \(\mathcal{P}\) is negative and all its remaining coefficients are positive. Finally, we consider all the possible sign patterns of the coefficients of the Ehrhart polynomials of low dimensional integral convex polytopes.
ISSN:2331-8422