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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Chen Lattice Points, Dedekind Sums, and Ehrhart Polynomials of Lattice Polyhedra. Discrete Comput Geom 28, 175–199 (2002). https://doi.org/10.1007/s00454-002-2759-7
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00454-002-2759-7