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 \(\varphi :V\rightarrow {\mathbb {C}}\) be a homogeneous polynomial function of degree d. For \(q\in {\mathbb {Z}}_{>0}\) 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 \(G_{\varphi }(q,y) \in {\mathbb {Q}}[q] [y]\) 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.
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Notes
This (standard) definition of the f polynomial is dual to the definition of the h-polynomial (1). The f-polynomial favors simplicial polytopes, in that Dehn–Sommerville holds with no g-polynomial corrections. The h-polynomial, on the other hand, favors simple polytopes.
We remark that the factor \((y+1)^{\deg {\varphi }}\) is not really needed for \(G_{\varphi }\), at least as far as the results in this section are concerned. This factor appears naturally when one considers the Todd operator formula, so it is reasonable to include it here.
This condition is the same as the toric variety \(X_{P}\) determined by P being nonsingular [8].
With this convention the Bernoulli numbers are \(B_{0} = 1\), \(B_{1}=\frac{1}{2}\), \(B_{2}=\frac{1}{6}\), \(B_{4}=-\frac{1}{30}\), ..., and \(B_{2k-1}=0\) for \(k > 1\). Note that for many authors \(B_{1}=-\frac{1}{2}\).
We thank an anonymous referee for pointing this out to us.
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Acknowledgements
We thank David Cox and Michèle Vergne for helpful comments. Two of us (MB and EM) thank the Max Planck Institute for Mathematics for its hospitality, where some of these results were initially worked out. We thank Toru Ohmoto, who informed us of [9] after this work was completed. Finally, we thank the anonymous referees for their comments.
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MB was partially supported by the NSF through Grant DMS-1162638. PG was partially supported by the NSF through Grants DMS-1101640 and DMS-1501832.
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Beck, M., Gunnells, P.E. & Materov, E. Weighted Lattice Point Sums in Lattice Polytopes, Unifying Dehn–Sommerville and Ehrhart–Macdonald. Discrete Comput Geom 65, 365–384 (2021). https://doi.org/10.1007/s00454-020-00175-2
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DOI: https://doi.org/10.1007/s00454-020-00175-2