## 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 language | English |
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Article number | #38 |

Journal | Seminaire Lotharingien de Combinatoire |

Issue number | 85 |

State | Published - 2021 |

### Bibliographical note

Funding Information:*andres.vindas@uky.edu. 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.

## Keywords

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

## ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics