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 v1, . . . , vk+1 in Z. We discuss an algorithm for solving the latter problem, also for a fixed d. Problem 1.1. Consider a simplex P in Z with integer vertices v1, . . . , vk+1 in Z. Determine how many integer points lie in P . An important tool in the algorithm is the exponential sum. Let {c, x} be the standard inner product on c = (c1, . . . , cd) and x = (x1, . . . , xd). The formal sum ∑ P∩Zd exp {〈c, x〉} is attained by substituting ai = exp (ci) into the Laurent series ∑ x∈P∩Zd a . Definition 1.2. Let P ⊂ R be a polyhedron and v be a vertex of P . Suppose that P = {x ∈ R : Ax ≤ b} and A′x = b′ is the subsystem held at equality at v. Then the supporting or tangent cone cone(P, v) of P at v is defined by cone(P, v) := {x ∈ R : A′x ≤ b′}. The algorithm uses the following theorem of Brion ([4], [5]). Theorem 1.3. Let P be an integral polytope. Then ∑ x∈P∩Zd exp {〈c, x〉} = ∑ v∈Vert (P ) exp {〈c, v〉} ∑ x∈cone(P,v)∩Zd exp{〈c, x〉}  Substituting c = 0 seems to give the number of integer points in P , but c = 0 is a singular point on the right hand side. Instead, the algorithm calculates the constant term of the Taylor series about t = 0 of the functions (1) exp {t〈c, v〉} ∑ x∈cone(P,v)∩Zd exp{t〈c, x〉}. Define LinK to be the linear span of a cone K. We define a cone to be simple if it can be generated by linearly independent vectors. A simple rational cone is unimodular with primitive generators {u1, . . . , uk} if u1, . . . , uk is a basis of the lattice Date: April 5, 2008. 1 2 ANDREW CRITES, MICHAEL GOFF, MATT KORSON, LEE PATROLIA, LUKE WOLCOTT Z ∩ LinK. There is an explicit formula for (1) when cone(P, v) is a unimodular cone. The following is proven in [2]. Theorem 1.4. If the dimension d is fixed, then given a rational polyhedral cone K ⊂ R, there exists a polynomial time algorithm that computes unimodular cones Ki and numbers i ∈ {−1, 1} such that [K] = ∑