Three Ehrhart quasi-polynomials

Baldoni, Velleda; Berline, Nicole; De Loera, Jesús A.; Köppe, Matthias; Vergne, Michèle

doi:10.5802/alco.46

Baldoni, Velleda ¹; Berline, Nicole ²; De Loera, Jesús A.³; Köppe, Matthias ³; Vergne, Michèle ⁴

¹ Dipartimento di Matematica Università degli studi di Roma “Tor Vergata” Via della ricerca scientifica 1 I-00133 Italy
² École Polytechnique Centre de Mathématiques Laurent Schwartz 91128 Palaiseau Cedex France
³ Department of Mathematics University of California Davis One Shields Avenue Davis, CA, 95616 USA
⁴ Université Paris 7 Diderot Institut Mathématique de Jussieu Sophie Germain, case 75205 Paris Cedex 13 France

Algebraic Combinatorics, Volume 2 (2019) no. 3, pp. 379-416.

Abstract

Let $𝔭 (b) \subset ℝ^{d}$ be a semi-rational parametric polytope, where $b = (b_{j}) \in ℝ^{N}$ is a real multi-parameter. We study intermediate sums of polynomial functions $h (x)$ on $𝔭 (b)$ ,

S^{L} (𝔭 (b), h) = \sum_{y} \int_{𝔭 (b) \cap (y + L)} h (x) d x,

where we integrate over the intersections of $𝔭 (b)$ with the subspaces parallel to a fixed rational subspace $L$ through all lattice points, and sum the integrals. The purely discrete sum is of course a particular case ( $L = 0$ ), so $S^{0} (𝔭 (b), 1)$ counts the integer points in the parametric polytopes.

The chambers are the open conical subsets of $ℝ^{N}$ such that the shape of $𝔭 (b)$ does not change when $b$ runs over a chamber. We first prove that on every chamber of $ℝ^{N}$ , $S^{L} (𝔭 (b), h)$ is given by a quasi-polynomial function of $b \in ℝ^{N}$ . A key point of our paper is an analysis of the interplay between two notions of degree on quasi-polynomials: the usual polynomial degree and a filtration, called the local degree.

Then, for a fixed $k \leq d$ , we consider a particular linear combination of such intermediate weighted sums, which was introduced by Barvinok in order to compute efficiently the $k + 1$ highest coefficients of the Ehrhart quasi-polynomial which gives the number of points of a dilated rational polytope. Thus, for each chamber, we obtain a quasi-polynomial function of $b$ , which we call Barvinok’s patched quasi-polynomial (at codimension level $k$ ).

Finally, for each chamber, we introduce a new quasi-polynomial function of $b$ , the cone-by-cone patched quasi-polynomial (at codimension level $k$ ), defined in a refined way by linear combinations of intermediate generating functions for the cones at vertices of $𝔭 (b)$ .

We prove that both patched quasi-polynomials agree with the discrete weighted sum $b \mapsto S^{{0}} (𝔭 (b), h)$ in the terms corresponding to the $k + 1$ highest polynomial degrees.

Received: 2018-03-27
Revised: 2018-10-25
Accepted: 2018-11-01
Published online: 2019-06-06

MR Zbl

DOI: 10.5802/alco.46

Classification: 05A15, 52C07, 68R05, 68U05, 52B20
Keywords: Ehrhart polynomials, generating functions, Barvinok’s algorithm, parametric polytopes

Author's affiliations:

Baldoni, Velleda ¹; Berline, Nicole ²; De Loera, Jesús A. ³; Köppe, Matthias ³; Vergne, Michèle ⁴

¹ Dipartimento di Matematica Università degli studi di Roma “Tor Vergata” Via della ricerca scientifica 1 I-00133 Italy
² École Polytechnique Centre de Mathématiques Laurent Schwartz 91128 Palaiseau Cedex France
³ Department of Mathematics University of California Davis One Shields Avenue Davis, CA, 95616 USA
⁴ Université Paris 7 Diderot Institut Mathématique de Jussieu Sophie Germain, case 75205 Paris Cedex 13 France

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Baldoni, Velleda and Berline, Nicole and De Loera, Jes\'us A. and K\"oppe, Matthias and Vergne, Mich\`ele},
     title = {Three {Ehrhart} quasi-polynomials},
     journal = {Algebraic Combinatorics},
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Baldoni, Velleda; Berline, Nicole; De Loera, Jesús A.; Köppe, Matthias; Vergne, Michèle. Three Ehrhart quasi-polynomials. Algebraic Combinatorics, Volume 2 (2019) no. 3, pp. 379-416. doi : 10.5802/alco.46. https://alco.centre-mersenne.org/articles/10.5802/alco.46/

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