Decompositions of Ehrhart h*-polynomials for rational polytopes

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

Research output: Contribution to journalArticlepeer-review


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
Article number#38
JournalSeminaire Lotharingien de Combinatoire
Issue number85
StatePublished - 2021

Bibliographical note

Funding Information:
* Andrés R. Vindas-Meléndez was partially supported by National Science Foundation Graduate Research Fellowship DGE-1247392.

Publisher Copyright:
© 2021, Seminaire Lotharingien de Combinatoire. All Rights Reserved.


  • Ehrhart quasipolynomial
  • Ehrhart series
  • generating function
  • h,-polynomial

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics


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