Séminaire Lotharingien de Combinatoire, 86B.44 (2022), 12 pp.
Matthias Beck, Sophia Elia and Sophie Rehberg
Rational Ehrhart Theory
Abstract.
The Ehrhart quasipolynomial of a rational polytope P encodes fundamental arithmetic
data of P, namely, the number of integer lattice points in positive integral dilates of 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), Baldoni-Berline-K\"oppe-Vergne (2013),
and Stapledon (2017).
We introduce a generating-function Ansatz for rational Ehrhart
quasipolynomials, which unifies several known results in classical and rational Ehrhart
theory.
In particular, we define γ-rational Gorenstein polytopes, which extend the classical notion to the rational setting and encompass the generalized reflexive polytopes studied by Fiset-Kasprzyk
(2008) and Kasprzyk-Nill (2012).
Received: November 25, 2021.
Accepted: March 4, 2022.
Final version: April 1, 2022.
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