Ehrhart polynomials of polytopes and spectrum at infinity of Laurent polynomials
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Ehrhart polynomials of polytopes and spectrum at infinity of Laurent polynomials Antoine Douai1 Received: 5 June 2019 / Accepted: 19 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Gathering different results from singularity theory, geometry and combinatorics, we show that the spectrum at infinity of a tame Laurent polynomial counts (weighted) lattice points in polytopes. We deduce an effective algorithm in order to compute the Ehrhart polynomial of a simplex containing the origin as an interior point. Keywords Toric varieties · Polytopes · Ehrhart theory · Spectrum of polytopes · Spectrum of regular functions Mathematics Subject Classification 52B20 · 05A15 · 32S40 · 14J33
1 Introduction The spectrum at infinity of a Laurent polynomial f , defined by Sabbah [13], is a sequence β1 , . . . , βμ of nonnegative rational numbers which is related to various concepts in singularity theory (monodromy, Hodge and Kashiwara-Malgrange filtrations, Brieskorn lattices...) and it can be difficult to handle. Fortunately, if f is convenient and nondegenerate with respect to its Newton polytope in the sense of Kouchnirenko [10], μ and this is generically the case, the generating function Spec f (z) := i=1 z βi has a very concrete description; it is equal to the Hilbert-Poincaré series of the Jacobian ring of f graded by the Newton filtration [7]. As noticed in [5,6], it follows from Kouchnirenko’s work that
Spec f (z) = (1 − z)n
z ν(v)
(1)
v∈N
B 1
Antoine Douai [email protected] CNRS, LJAD, Université Côte d’Azur, Nice, France
123
Journal of Algebraic Combinatorics
where ν is the Newton filtration of the Newton polytope P of f and N := Zn . Because the right-hand side depends only on P, we will also call it the Newton spectrum of P and we will denote it by Spec P (z). With this terminology, the spectrum at infinity of a Laurent polynomial is equal to the Newton spectrum of its Newton polytope. All this is recorded in Sect. 2, where we also recall the basic definitions. Once we have this description, it follows from the work of Stapledon [14] that the spectrum at infinity of a convenient and nondegenerate Laurent polynomial counts “weighted” lattice points in its Newton polytope P and its integer dilates. More precisely, the δ-vector δ P (z) = δ0 + δ1 z + · · · + δn z n of a lattice polytope P in Rn containing the origin as an interior point, hence its Ehrhart μ polynomial, can be obtained effortlessly from its Newton spectrum Spec P (z) = i=1 z βi : the coefficient δi is equal to the number of βk ’s such that βk ∈]i − 1, i]. This is explained in Sect. 4.1. Thereby, singularity theory meets Ehrhart theory by the means of the δ-vector; the spectrum at infinity of a convenient and nondegenerate Laurent polynomial determines the δvector of its Newton polytope P. And we show in Proposition 5.1 that both coincide if and only if P is reflexive. This is particularly fruitful when P is a reduced simplex in Rn in the sense of [4] (the definition of a reduced simplex a
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