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.
|Number of pages||22|
|Journal||Discrete and Computational Geometry|
|State||Published - Jul 2022|
Bibliographical noteFunding 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.
© 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.
- Ehrhart theory
- Rational polytope
ASJC Scopus subject areas
- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics