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Transactions of the American Mathematical Society

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$ h^\ast$-polynomials of zonotopes


Authors: Matthias Beck, Katharina Jochemko and Emily McCullough
Journal: Trans. Amer. Math. Soc. 371 (2019), 2021-2042
MSC (2010): Primary 05A05, 05A15, 26C10, 52B20, 52B40, 52B45
DOI: https://doi.org/10.1090/tran/7384
Published electronically: October 11, 2018
MathSciNet review: 3894043
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Abstract: 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^\ast $-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^\ast $-polynomial of a lattice zonotope in terms of refined descent statistics of permutations and prove that the $ h^\ast $-polynomial of every lattice zonotope has only real roots and therefore unimodal coefficients. Furthermore, we present a closed formula for the $ h^\ast $-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^\ast $-polynomials but carry over to general combinatorial positive valuations. Moreover, we give a complete description of the convex hull of all $ h^\ast $-polynomials of zonotopes in a given dimension: it is a simplicial cone spanned by refined Eulerian polynomials.


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Additional Information

Matthias Beck
Affiliation: Department of Mathematics, San Francisco State University, San Francisco, California 94132-4163
Email: mattbeck@sfsu.edu

Katharina Jochemko
Affiliation: Department of Mathematics, Royal Institute of Technology (KTH), SE 1004 Stockholm, Sweden
Email: jochemko@kth.se

Emily McCullough
Affiliation: Department of Mathematics, San Francisco State University, San Francisco, California 94132-4163
Email: emac@mail.sfsu.edu

DOI: https://doi.org/10.1090/tran/7384
Keywords: Ehrhart polynomials, $h^\ast$-polynomials, zonotopes, unimodality, real-rooted polynomials, combinatorial positive valuations
Received by editor(s): January 9, 2017
Received by editor(s) in revised form: August 22, 2017
Published electronically: October 11, 2018
Additional Notes: The first author was partially supported by the U.S. National Science Foundation (DMS-1162638).
The second author was partially supported by a Hilda Geiringer Scholarship of the Berlin Mathematical School (BMS) and wants to thank the BMS especially for funding her research stay at San Francisco State University in Fall 2013, during which this project evolved.
The third author was partially supported by the U.S. National Science Foundation (DGE-0841164).
Article copyright: © Copyright 2018 American Mathematical Society