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Splines, lattice points, and arithmetic matroids

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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.

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Notes

  1. For an example, consider Example 10.6 and in particular (99). There is only one hyperplane in \(\mathbb {Z}/2\mathbb {Z}\oplus \mathbb {Z}\). It corresponds to \(H= {{\mathrm{{{span}}}}}((1,0))\) in \(\mathbb {R}^2\) and representatives for the two points that it contains are \(\lambda _1=(1,0)\) and \(\lambda _2=(0,0)\). The normal vector is \(\eta = (0,1)\).

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Acknowledgments

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

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  1. Mathematical Institute, University of Oxford, Andrew Wiles Building, Oxford, OX2 6GG, UK

    Matthias Lenz

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Correspondence to Matthias Lenz.

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The author was supported by a Junior Research Fellowship of Merton College (University of Oxford).

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Lenz, M. Splines, lattice points, and arithmetic matroids. J Algebr Comb 43, 277–324 (2016). https://doi.org/10.1007/s10801-015-0621-2

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