Hal will be stopped for maintenance from friday on june 10 at 4pm until monday june 13 at 9am. More information
Skip to Main content Skip to Navigation
Conference papers

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 $X$.
Complete list of metadata

Cited literature [16 references]  Display  Hide  Download

Contributor : Coordination Episciences Iam Connect in order to contact the contributor
Submitted on : Thursday, October 1, 2015 - 9:28:56 AM
Last modification on : Tuesday, March 7, 2017 - 3:26:28 PM
Long-term archiving on: : Saturday, January 2, 2016 - 10:51:47 AM


Publisher files allowed on an open archive




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, ⟨10.46298/dmtcs.2379⟩. ⟨hal-01207590⟩



Record views


Files downloads