Three Ehrhart quasi-polynomials
Algebraic Combinatorics, Volume 2 (2019) no. 3, pp. 379-416.

Let 𝔭(b) d be a semi-rational parametric polytope, where b=(b j ) N is a real multi-parameter. We study intermediate sums of polynomial functions h(x) on 𝔭(b),

S L (𝔭(b),h)= y 𝔭(b)(y+L) h(x)dx,

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 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 N , S L (𝔭(b),h) is given by a quasi-polynomial function of b 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 kd, 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 bS {0} (𝔭(b),h) in the terms corresponding to the k+1 highest polynomial degrees.

Published online:
DOI: 10.5802/alco.46
Classification: 05A15,  52C07,  68R05,  68U05,  52B20
Keywords: Ehrhart polynomials, generating functions, Barvinok’s algorithm, parametric polytopes
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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     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.

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