Rational Ehrhart Theory
Abstract
The Ehrhart quasipolynomial of a rational polytope $\mathsf{P}$ encodes fundamental arithmetic data of $\mathsf{P}$, namely, the number of integer lattice points in positive integral dilates of $\mathsf{P}$. Ehrhart quasipolynomials were introduced in the 1960s, satisfy several fundamental structural results and have applications in many areas of mathematics and beyond. The enumerative theory of lattice points in rational (equivalently, real) dilates of rational polytopes is much younger, starting with work by Linke (2011), BaldoniBerlineKoeppeVergne (2013), and Stapledon (2017). We introduce a generatingfunction ansatz for rational Ehrhart quasipolynomials, which unifies several known results in classical and rational Ehrhart theory. In particular, we define $\gamma$rational Gorenstein polytopes, which extend the classical notion to the rational setting and encompass the generalized reflexive polytopes studied by FisetKasprzyk (2008) and KasprzykNill (2012).
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 arXiv:
 arXiv:2110.10204
 Bibcode:
 2021arXiv211010204B
 Keywords:

 Mathematics  Combinatorics;
 52C07 (Primary) 05A15;
 11H06;
 52B20 (Secondary)
 EPrint:
 21 pages, 3 figures