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:
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
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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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