Three Ehrhart quasi-polynomials

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

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

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

$$S^L(𝔭\left(b\right),h)=\sum _y{\int }_{𝔭\left(b\right)\cap (y+L)}h\left(x\right)\mathrm{d}x,$$

where we integrate over the intersections of $𝔭\left(b\right)$ 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 $S^0(𝔭\left(b\right),1)$ counts the integer points in the parametric polytopes.

The chambers are the open conical subsets of ${ℝ}^N$ such that the shape of $𝔭\left(b\right)$ does not change when $b$ runs over a chamber. We first prove that on every chamber of ${ℝ}^N$, $S^L(𝔭\left(b\right),h)$ 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 $k\le d$, 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 $𝔭\left(b\right)$.

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

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

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