Decompositions of Ehrhart h -Polynomials for Rational Polytopes

Matthias Beck, Benjamin Braun, Andrés R. Vindas-Meléndez

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

Abstract

The Ehrhart quasipolynomial of a rational polytope P encodes the number of integer lattice points in dilates of P, and the h-polynomial of P is the numerator of the accompanying generating function. We provide two decomposition formulas for the h-polynomial of a rational polytope. The first decomposition generalizes a theorem of Betke and McMullen for lattice polytopes. We use our rational Betke–McMullen formula to provide a novel proof of Stanley’s Monotonicity Theorem for the h-polynomial of a rational polytope. The second decomposition generalizes a result of Stapledon, which we use to provide rational extensions of the Stanley and Hibi inequalities satisfied by the coefficients of the h-polynomial for lattice polytopes. Lastly, we apply our results to rational polytopes containing the origin whose duals are lattice polytopes.

Original languageEnglish
Pages (from-to)50-71
Number of pages22
JournalDiscrete and Computational Geometry
Volume68
Issue number1
DOIs
StatePublished - Jul 2022

Bibliographical note

Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

Keywords

  • Decomposition
  • Ehrhart theory
  • Quasipolynomial
  • Rational polytope

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics

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