Calculating Heegaard–Floer homology by counting lattice points in tetrahedra

Counting Lattice Points in Polyhedra

We present Barvinok’s 1994 and 1999 algorithms for counting lattice points in polyhedra. 1. The 1994 algorithm In [2], Barvinok presents an algorithm that, for a fixed dimension d, calculates the number of integer points in a rational polyhedron. It is shown in [6] and [7] that the question can be reduced to counting the number of integer points in a k-dimensional simplex with integer vertices ...

Heronian Tetrahedra Are Lattice Tetrahedra

Extending a similar result about triangles, we show that each Heronian tetrahedron may be positioned with integer coordinates. More generally, we show the following: if an integral distance set in R can be positioned with rational coordinates, then it can in fact be positioned with integer coordinates. The proof, which uses the arithmetic of quaternions, is tantamount to an algorithm.

Universal Counting of Lattice Points in Polytopes

Given a lattice polytope P (with underlying lattice L), the universal counting function UP (L ) = |P ∩ L| is defined on all lattices L containing L. Motivated by questions concerning lattice polytopes and the Ehrhart polynomial, we study the equation UP = UQ. Mathematics Subject Classification: 52B20, 52A27, 11P21 Partially supported by Hungarian Science Foundation Grant T 016391, and by the Fr...

Counting Lattice Points in Admissible Adelic Sets

Covering Lattice Points by Subspaces and Counting Point-Hyperplane Incidences

Let d and k be integers with 1 ≤ k ≤ d − 1. Let Λ be a d-dimensional lattice and let K be a d-dimensional compact convex body symmetric about the origin. We provide estimates for the minimum number of k-dimensional linear subspaces needed to cover all points in Λ ∩ K. In particular, our results imply that the minimum number of k-dimensional linear subspaces needed to cover the d-dimensional n ×...