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Local formulas for Ehrhart coefficients from lattice tiles

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Abstract

As shown by McMullen in 1983, the coefficients of the Ehrhart polynomial of a lattice polytope can be written as a weighted sum of facial volumes. The weights in such a local formula depend only on the outer normal cones of faces, but are far from being unique. In this paper, we develop an infinite class of such local formulas. These are based on choices of fundamental domains in sublattices and obtained by polyhedral volume computations. We hereby also give a kind of geometric interpretation for the Ehrhart coefficients. Since our construction gives us a great variety of possible local formulas, these can, for instance, be chosen to fit well with a given polyhedral symmetry group. In contrast to other constructions of local formulas, ours does not rely on triangulations of rational cones into simplicial or even unimodular ones.

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Acknowledgements

We like to thank the two anonymous referees, as well as Frieder Ladisch and Erik Friese for valuable comments. Moreover, Maren H. Ring is grateful for support by a Ph.D. scholarship of Studienstiftung des Deutschen Volkes (German Academic Foundation). Both authors gratefully acknowledge support by DFG Grant SCHU 1503/6-1.

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  1. Institute for Mathematics, University of Rostock, 18051, Rostock, Germany

    Maren H. Ring & Achill Schürmann

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  1. Maren H. Ring
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  2. Achill Schürmann
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Correspondence to Achill Schürmann.

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Ring, M.H., Schürmann, A. Local formulas for Ehrhart coefficients from lattice tiles. Beitr Algebra Geom 61, 157–185 (2020). https://doi.org/10.1007/s13366-019-00457-8

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