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Intermediate sums on polyhedra. II: Bidegree and Poisson formula. (English) Zbl 1347.05005

Summary: We continue our study of intermediate sums over polyhedra, interpolating between integrals and discrete sums, which were introduced by A. Barvinok [Math. Comput. 75, No. 255, 1449–1466 (2006; Zbl 1093.52009)]. By well-known decompositions, it is sufficient to consider the case of affine cones \(s+\mathfrak{c}\), where \(s\) is an arbitrary real vertex and \(\mathfrak{c}\) is a rational polyhedral cone. For a given rational subspace \(L\), we define the intermediate generating functions \(S^L(s+\mathfrak{c})(\xi)\) by integrating an exponential function over all lattice slices of the affine cone \(s+\mathfrak{c}\) parallel to the subspace \(L\) and summing up the integrals. We expose the bidegree structure in parameters \(s\) and \(\xi\), which was implicitly used in the algorithms in [V. Baldoni et al., Found. Comput. Math. 12, No. 4, 435–469 (2012; Zbl 1255.05006); Mathematika 59, No. 1, 1–22 (2013; Zbl 1260.05006)]. The bidegree structure is key to a new proof for the Baldoni-Berline-Vergne approximation theorem for discrete generating functions [V. Baldoni et al., Contemp. Math. 452, 15–33 (2008; Zbl 1162.52008)], using the Fourier analysis with respect to the parameter \(s\) and a continuity argument. Our study also enables a forthcoming paper, in which we study intermediate sums over multi-parameter families of polytopes.

MSC:

05A15 Exact enumeration problems, generating functions
52C07 Lattices and convex bodies in \(n\) dimensions (aspects of discrete geometry)
68R05 Combinatorics in computer science
68U05 Computer graphics; computational geometry (digital and algorithmic aspects)
52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
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References:

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