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Ehrhart Series, Unimodality, and Integrally Closed Reflexive Polytopes

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Abstract

An interesting open problem in Ehrhart theory is to classify those lattice polytopes having a unimodal h*-vector. Although various sufficient conditions have been found, necessary conditions remain a challenge. In this paper, we consider integrally closed reflexive simplices and discuss an operation that preserves reflexivity, integral closure, and unimodality of the h*-vector, providing one explanation for why unimodality occurs in this setting. We also discuss the failure of proving unimodality in this setting using weak Lefschetz elements.

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Authors and Affiliations

  1. Department of Mathematics, University of Kentucky, Lexington, KY, 40506-0027, USA

    Benjamin Braun

  2. Department of Mathematics, Michigan State University, 619 Red Cedar Rd., East Lansing, MI, 48824, USA

    Robert Davis

Authors
  1. Benjamin Braun
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  2. Robert Davis
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Correspondence to Robert Davis.

Additional information

The first author is partially supported by the National Security Agency through award H98230-13-1-0240. The second author is partially supported by a 2013-2014 Fulbright U.S. Student Fellowship. The authors thank Benjamin Nill for useful insights that contributed to the proof of Theorem 3.3, and Akihiro Higashitani for his thoughtful comments.

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Braun, B., Davis, R. Ehrhart Series, Unimodality, and Integrally Closed Reflexive Polytopes. Ann. Comb. 20, 705–717 (2016). https://doi.org/10.1007/s00026-016-0337-6

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  • DOI: https://doi.org/10.1007/s00026-016-0337-6

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