Theorems of Brion, Lawrence, and Varchenko on rational generating functions for cones

TL;DR

This paper provides elementary proofs and generalizations of key theorems by Brion, Lawrence, and Varchenko concerning rational generating functions for cones and polytopes, including a new proof of Brion's Formula.

Contribution

It offers simplified proofs and extends the Lawrence-Varchenko formula, enhancing understanding of lattice-point enumeration in polytopes and cones.

Findings

Elementary proof of Brion's Formula using irrational decompositions

Generalization of the Lawrence-Varchenko formula

Clarification of lattice-point generating functions for cones

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.