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Splines, lattice points, and (arithmetic) matroids

Abstract : Let $X$ be a $(d \times N)$-matrix. We consider the variable polytope $\Pi_X(u) = \left\{ w \geq 0 : Xw = u \right\}$. It is known that the function $T_X$ that assigns to a parameter $u \in \mathbb{R}^N$ the volume of the polytope $\Pi_X(u)$ is piecewise polynomial. Formulas of Khovanskii-Pukhlikov and Brion-Vergne imply that the number of lattice points in $\Pi_X(u)$ can be obtained by applying a certain differential operator to the function $T_X$. In this extended abstract we slightly improve the formulas of Khovanskii-Pukhlikov and Brion-Vergne and we study the space of differential operators that are relevant for $T_X$ (ıe operators that do not annihilate $T_X$) and the space of nice differential operators (ıe 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 $X$.
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Matthias Lenz. Splines, lattice points, and (arithmetic) matroids. 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014), 2014, Chicago, United States. pp.49-60. ⟨hal-01207590⟩

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