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Ehrhart tensor polynomials
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
2018 (English)In: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 539, p. 72-93Article in journal (Refereed) Published
Abstract [en]

The notion of Ehrhart tensor polynomials, a natural generalization of the Ehrhart polynomial of a lattice polytope, was recently introduced by Ludwig and Silverstein. We initiate a study of their coefficients. In the vector and matrix cases, we give Pick-type formulas in terms of triangulations of a lattice polygon. As our main tool, we introduce hr-tensor polynomials, extending the notion of the Ehrhart h⁎-polynomial, and, for matrices, investigate their coefficients for positive semidefiniteness. In contrast to the usual h⁎-polynomial, the coefficients are in general not monotone with respect to inclusion. Nevertheless, we are able to prove positive semidefiniteness in dimension two. Based on computational results, we conjecture positive semidefiniteness of the coefficients in higher dimensions. Furthermore, we generalize Hibi's palindromic theorem for reflexive polytopes to hr-tensor polynomials and discuss possible future research directions.

Place, publisher, year, edition, pages
Elsevier, 2018. Vol. 539, p. 72-93
Keywords [en]
Ehrhart tensor polynomial, hr-tensor polynomial, Half-open polytopes, Pick's formula, Positive semidefinite coefficients
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-218112DOI: 10.1016/j.laa.2017.10.021ISI: 000424177500006Scopus ID: 2-s2.0-85033474584OAI: oai:DiVA.org:kth-218112DiVA, id: diva2:1160070
Funder
Knut and Alice Wallenberg Foundation
Note

QC 20171124

Available from: 2017-11-24 Created: 2017-11-24 Last updated: 2018-02-23Bibliographically approved

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Citation style
  • apa
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  • vancouver
  • Other style
More styles
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  • de-DE
  • en-GB
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  • nn-NB
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  • Other locale
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  • asciidoc
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