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Journal of Algebraic Combinatorics

, Volume 43, Issue 2, pp 277–324 | Cite as

Splines, lattice points, and arithmetic matroids

  • Matthias Lenz
Article

Abstract

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 Ardila–Postnikov and Holtz–Ron in the context of zonotopal algebra and power ideals.

Keywords

Lattice polytope Vector partition function Todd operator Brion–Vergne formula Arithmetic matroid Zonotopal algebra 

Mathematics Subject Classification

Primary: 05B35 19L10 52B20 Secondary: 13B25 14M25 16S32 41A15 47F05 52B40 52C35 

Notes

Acknowledgments

The author would like to thank Lars Kastner and Zhiqiang Xu for helpful conversations.

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Mathematical InstituteUniversity of OxfordOxfordUK

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