Partial fraction decompositions on hyperplane arrangements

TL;DR

This paper develops criteria and algorithms for partial fraction decompositions of multivariable rational functions with poles on hyperplane arrangements, aiding calculations in scattering amplitudes and Feynman integrals.

Contribution

It introduces a novel algebraic approach using primary decomposition to determine and compute desirable partial fraction decompositions in multiple variables.

Findings

Criteria for PFD existence based on ideal decompositions

An algorithm for practical PFD computation

Successful application to Feynman integral examples

Abstract

We study partial fraction decompositions (PFDs) in several variables using tools from commutative algebra. We give criteria for when a rational function with poles on a hyperplane arrangement has a desirable PFD. Our criteria are obtained by examining the primary decomposition of ideals coming from hyperplane arrangements. We then present an algorithm for finding a PFD that satisfies properties desired for simplifying the calculation of scattering amplitudes. We demonstrate the effectiveness of this algorithm by computing practical examples coming from Feynman integrals.