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Weighted lattice point sums in lattice polytopes, unifying Dehn-Sommerville and Ehrhart-Macdonald

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Beck,  Matthias
Max Planck Institute for Mathematics, Max Planck Society;

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Materov,  Evgeny
Max Planck Institute for Mathematics, Max Planck Society;

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Fulltext (public)

arXiv:1805.01504.pdf
(Preprint), 275KB

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Citation

Beck, M., Gunnells, P. E., & Materov, E. (2021). Weighted lattice point sums in lattice polytopes, unifying Dehn-Sommerville and Ehrhart-Macdonald. Discrete & Computational Geometry, 65(2), 365-384. doi:10.1007/s00454-020-00175-2.


Cite as: http://hdl.handle.net/21.11116/0000-0007-F156-8
Abstract
Let $V$ be a real vector space of dimension $n$ and let $M\subset V$ be a lattice. Let $P\subset V$ be an $n$-dimensional polytope with vertices in $M$, and let $\varphi\colon V\rightarrow \CC $ be a homogeneous polynomial function of degree $d$ (i.e., an element of $\Sym^{d} (V^{*})$). For $q\in \ZZ_{>0}$ and any face $F$ of $P$, let $D_{\varphi ,F} (q)$ be the sum of $\varphi$ over the lattice points in the dilate $qF$. We define a generating function $G_{\varphi}(q,y) \in \QQ [q] [y]$ packaging together the various $D_{\varphi ,F} (q)$, and show that it satisfies a functional equation that simultaneously generalizes Ehrhart--Macdonald reciprocity and the Dehn--Sommerville relations. When $P$ is a simple lattice polytope (i.e., each vertex meets $n$ edges), we show how $G_{\varphi}$ can be computed using an analogue of Brion--Vergne's Euler--Maclaurin summation formula.