## 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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Pages (from-to) | 50-71 |

Number of pages | 22 |

Journal | Discrete and Computational Geometry |

Volume | 68 |

Issue number | 1 |

DOIs | |

State | Published - 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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