## 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

Funding Information:This work was partially supported by NSF Graduate Research Fellowship DGE-1247392 (ARVM) and NSF Award DMS-1953785 (BB). ARVM thanks the Discrete Geometry group of the Mathematics Institute at FU Berlin for providing a wonderful working environment while part of this work was done. The authors would like to thank Steven Klee, José Samper, Liam Solus, and three anonymous referees for fruitful correspondence.

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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