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h*-POLYNOMIALS OF ZONOTOPES
San Francisco State Univ, Dept Math, San Francisco, CA 94132 USA..
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
San Francisco State Univ, Dept Math, San Francisco, CA 94132 USA..
2019 (English)In: Transactions of the American Mathematical Society, ISSN 0002-9947, E-ISSN 1088-6850, Vol. 371, no 3, p. 2021-2042Article in journal (Refereed) Published
Abstract [en]

The Ehrhart polynomial of a lattice polytope P encodes information about the number of integer lattice points in positive integral dilates of P. The h*-polynomial of P is the numerator polynomial of the generating function of its Ehrhart polynomial. A zonotope is any projection of a higher dimensional cube. We give a combinatorial description of the h*-polynomial of a lattice zonotope in terms of refined descent statistics of permutations and prove that the h*-polynomial of every lattice zonotope has only real roots and therefore unimodal coefficients. Furthermore, we present a closed formula for the h*-polynomial of a zonotope in matroidal terms which is analogous to a result by Stanley (1991) on the Ehrhart polynomial. Our results hold not only for h*-polynomials but carry over to general combinatorial positive valuations. Moreover, we give a complete description of the convex hull of all h*-polynomials of zonotopes in a given dimension: it is a simplicial cone spanned by refined Eulerian polynomials.

Place, publisher, year, edition, pages
AMER MATHEMATICAL SOC , 2019. Vol. 371, no 3, p. 2021-2042
Keywords [en]
Ehrhart polynomials, h*-polynomials, zonotopes, unimodality, real-rooted polynomials, combinatorial positive valuations
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-241306DOI: 10.1090/tran/7384ISI: 000454641300018OAI: oai:DiVA.org:kth-241306DiVA, id: diva2:1282741
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QC 20190125

Available from: 2019-01-25 Created: 2019-01-25 Last updated: 2019-01-25Bibliographically approved

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Jochemko, Katharina

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CiteExportLink to record
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Citation style
  • apa
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  • vancouver
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  • en-GB
  • en-US
  • fi-FI
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  • nn-NB
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  • Other locale
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Output format
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