Lattice Points, Dedekind Sums, and Ehrhart Polynomials of Lattice Polyhedra |
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Authors: | Chen |
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Affiliation: | (1) Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong mabfchen@ust.hk, HK |
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Abstract: | Abstract. Let σ be a simplex of R N with vertices in the integral lattice Z N . The number of lattice points of mσ (={mα : α ∈ σ}) is a polynomial function L(σ,m) of m ≥ 0 . In this paper we present: (i) a formula for the coefficients of the polynomial L(σ,t) in terms of the elementary symmetric functions; (ii) a hyperbolic cotangent expression for the generating functions of the sequence L(σ,m) , m ≥ 0 ; (iii) an explicit formula for the coefficients of the polynomial L(σ,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≤ j≤ 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. |
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