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Lattice points, Dedekind sums, and Ehrhart polynomials of lattice polyhedra

Authors Chen, BF View this author's profile
Issue Date 2002
Source Discrete & Computational Geometry , v. 28, (2), 2002, SEP, p. 175-199
Summary Let a be a simplex of R-N with vertices in the integral lattice Z(N). The number of lattice points of msigma (= {malpha: alpha is an element of sigma}) is a polynomial function L(sigma, m) of m greater than or equal to 0. In this paper we present: (i) a formula for the coefficients of the polynomial L(sigma, t) in terms of the elementary symmetric functions; (ii) a hyperbolic cotangent expression for the generating functions of the sequence L(sigma, m), m greater than or equal to 0; (iii) an explicit formula for the coefficients of the polynomial L (sigma, t) in terms of torsion. As an application of (i), the coefficient for the lattice n-simplex of R-n with the vertices (0,..., 0, a(j), 0,..., 0) (1 less than or equal to j less than or equal to n) plus the origin is explicitly expressed in terms of Dedekind sums; and when n = 2, it reduces to the reciprocity law about Dedekind sums. The whole exposition is elementary and self-contained.
ISSN 0179-5376
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Language English
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