Splines, lattice points, and arithmetic matroids

Lenz, Matthias

Journal of Algebraic Combinatorics, Tome 43 (2016) no. 2, pp. 277-324.

Résumé

Let $X$ be a $(d\times N)$-matrix. We consider the variable polytope $\varPi_X(u) = \{ w \ge 0 : X w = u \}$. It is known that the function $T_X$ that assigns to a parameter $u \in \mathbb {R}^d$ the volume of the polytope $\varPi _X(u)$ is piecewise polynomial. The Brion-Vergne formula implies that the number of lattice points in $\varPi_X(u)$ can be obtained by applying a certain differential operator to the function $T_X$. In this article, we slightly improve the Brion-Vergne formula and we study two spaces of differential operators that arise in this context: the space of relevant differential operators (i.e. operators that do not annihilate $T_X$) and the space of nice differential operators (i.e. operators that leave $T_X$ continuous). These two spaces are finite-dimensional homogeneous vector spaces, and their Hilbert series are evaluations of the Tutte polynomial of the arithmetic matroid defined by the matrix $X$. They are closely related to the $\mathscr {P}$-spaces studied by F. Ardila and A. Postnikov [Trans. Am. Math. Soc. 362, No. 8, 4357--4384 (2010); corrigendum ibid. 367, No. 5, 3759--3762 (2015; Zbl 1226.05019)] and O. Holtz and A. Ron [Adv. Math. 227, No. 2, 847--894 (2011; Zbl 1223.13010)] in the context of zonotopal algebra and power ideals.

Détail

Classification : 05B35, 19L10, 52B20, 13B25, 14M25, 16S32, 41A15, 47F05, 52B40, 52C35
Keywords: lattice polytope, vector partition function, Todd operator, Brion-Vergne formula, arithmetic matroid, zonotopal algebra