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-Koeppe-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).

Boundary $h^\ast$-polynomials of rational polytopes

“…To this end, we first recall rational Ehrhart series, which were introduced only recently [5]. Suppose the full-dimensional rational polytope P Ă R d is given by the irredundant halfspace description P " x P R d : A x ď b ( , where A P Z nˆd and b P Z n such that the greatest common divisor of b j and the entries in the jth row of A equals 1, for every j P t1, .…”

“…We define the codenominator r of P to be the least common multiple of the nonzero entries of b. It turns out [5] Possibly more important than this theorem are the consequences one can derive from it, and [5] (re-)proved several previously-known and novel results in rational Ehrhart theory. The latter include the facts that rh mpP ; zq is palindromic if 0 P P ˝and that extracting the terms with integer exponents from rh mpP ; zq returns h P pzq, which results in yet another proof of the Betke-McMullen version of Theorem 1.2 (the ℓ " 1 case).…”

Machine Learning the Dimension of a Polytope

“…Each of these polynomials is of degree with leading coefficient vol( ). When < we say that exhibits quasi-period collapse [7,8,16,23,26]. The Ehrhart series of can be expressed as a rational function:…”

“…As in the case of lattice polytopes, the -vector carries combinatorial information about [6,7,20,21,25]. Given ( + 1) terms of the Ehrhart series, one can recover the -vector.…”