[Paper Review] Theorems of Brion, Lawrence, and Varchenko on rational generating functions for cones

Matthias Beck, Christian Haase+1 more|arXiv (Cornell University)|Jun 23, 2005

Advanced Combinatorial Mathematics9 references4 citations

TL;DR

This paper presents and illustrates Brion's and Lawrence–Varchenko's theorems on rational generating functions for rational cones, showing that the sum of generating functions over vertex cones of a rational polytope equals the generating function of its integer points. The key contribution is a deep structural insight into how rational functions can encode finite point sets via cone decompositions, enabling efficient computation of Ehrhart quasipolynomials and integer point counting in polynomial time.

ABSTRACT

We discuss and give elementary proofs of results of Brion and of Lawrence-Varchenko on the lattice-point enumerator generating functions for polytopes and cones. This largely expository note contains a new proof of Brion's Formula using irrational decompositions, and a generalization of the Lawrence-Varchenko formula.

Motivation & Objective

• To explain and illustrate the surprising collapse of infinite rational generating functions into finite polynomials via Brion’s and Lawrence–Varchenko formulas.
• To demonstrate how rational generating functions for vertex cones of a rational polytope reconstruct the generating function of its integer points.
• To establish a foundation for efficient computation of integer point counts in rational polytopes using signed decompositions of cones.
• To connect these theorems to Barvinok’s algorithm for short signed decompositions of rational cones into unimodular cones.
• To show that these results enable polynomial-time computation of Ehrhart quasipolynomials and related generating functions.

Proposed method

• Use generating functions to encode integer points in cones and polytopes, with variables raised to lattice point exponents.
• Apply Brion’s formula: the sum of rational generating functions over vertex cones of a rational polytope equals the generating function of the integer points in the polytope.
• Apply the Lawrence–Varchenko formula: a signed sum of generating functions over cones defined by edge directions relative to a generic direction vector yields the generating function of the polytope’s integer points.
• Utilize polar duality to relate the Lawrence–Varchenko formula to the dual of the polytope’s normal fan and to derive general Brion-type formulas.
• Employ Barvinok’s algorithm to decompose rational cones into unimodular cones with signs, enabling efficient computation of generating functions.
• Leverage the fact that unimodular cones have simple generating functions of the form $ \frac{x^p}{(1 - x^{w_1}) \cdots (1 - x^{w_d})} $, where $ p $ is the unique integer point in the fundamental parallelepiped.

Experimental results

Research questions

• RQ1 How can the generating function of integer points in a rational polytope be reconstructed from the generating functions of its vertex cones?
• RQ2 Why do sums of rational generating functions over vertex cones collapse to a polynomial encoding only the integer points in the polytope?
• RQ3 What is the role of a generic direction vector in defining the cones used in the Lawrence–Varchenko formula?
• RQ4 Can rational generating functions for cones be decomposed efficiently into simpler unimodular cone functions?
• RQ5 How does Barvinok’s short signed decomposition algorithm enable polynomial-time computation of Ehrhart quasipolynomials?

Key findings

• Brion’s formula states that the sum of rational generating functions over the vertex cones of a rational polytope equals the generating function of the integer points in the polytope.
• The Lawrence–Varchenko formula provides a signed sum of generating functions over cones defined by edge directions relative to a generic direction, which also reconstructs the polytope’s integer point generating function.
• Barvinok’s algorithm computes a signed decomposition of any rational cone into unimodular cones in polynomial time, with the number of terms bounded by a polynomial in the input size.
• The generating function of a unimodular cone has a simple closed form: $ \sigma_{\mathcal{K}}(x) = \frac{x^p}{(1 - x^{w_1}) \cdots (1 - x^{w_d})} $, where $ p $ is the unique integer point in the fundamental parallelepiped.
• These results imply that the Ehrhart quasipolynomial $ L_{\mathcal{P}}(t) = \#(t\mathcal{P} \cap \mathbb{Z}^d) $ can be computed in polynomial time via rational generating functions.
• The method has been implemented in software packages such as barvinok and LattE, with performance improvements from irrational decomposition techniques.