Counting Lattice Points by Means of the Residue Theorem
By Matthias Beck. 2000
59 citations recorded by the source corpus.
We use the residue theorem to derive an expression for the number of lattice points in a dilated n-dimensional tetrahedron with vertices at lattice points on each coordinate axis and the origin. This expression is known as the Ehrhart polynomial. We show that it is a polynomial in t, where t is the integral dilation parameter. We prove the Ehrhart-Macdonald reciprocity law for these tetrahedra, relating the Ehrhart polynomials of the interior and the closure of the tetrahedra. To illustrate our method, we compute the Ehrhart coefficient for codimension 2. Finally, we show how our ideas can be used to compute the Ehrhart polynomial for an arbitrary convex lattice polytope.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/b41323213468e4a1eb1a19000aa50095140f0e2c
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In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- Effective lattice point counting in rational convex polytopes (2004, background): “For example, it was seen in Beck(2000) that if Mi denotes the i th column of the defining matrixM, then”