MultivariateApart: Generalized Partial Fractions
Matthias Heller, Andreas von Manteuffel
TL;DR
The paper introduces MultivariateApart, a generalized multivariate partial fraction decomposition method that avoids spurious denominators by using polynomial reductions anchored in Gröbner-basis theory and a block monomial ordering. Implemented in a Mathematica package with interfaces to Singular and Form, the approach delivers unique reduced forms even for sums of rational functions and scales to complex Feynman-amplitude calculations. It provides practical workflows, performance optimizations, and a path toward rational reconstruction from finite-field samples, enabling efficient code generation for numerical evaluation. The demonstrated applications to one- and two-loop amplitudes show substantial reductions in expression size and improved computational performance, underscoring the method’s utility for high-energy physics computations and symbolic manipulation pipelines.
Abstract
We present a package to perform partial fraction decompositions of multivariate rational functions. The algorithm allows to systematically avoid spurious denominator factors and is capable of producing unique results also when being applied to terms of a sum separately. The package is designed to work in Mathematica, but also provides interfaces to the Form and Singular computer algebra systems.