MultivariateApart: Generalized partial fractions

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. Program summaryProgram title:MultivariateApartCPC Library link to program files:https://doi.org/10.17632/zbt9tfpkgv.1Developer's repository link:https://gitlab.msu.edu/vmante/multivariateapartLicensing provisions: GPLv3Programming language: Mathematica (Wolfram Language)Nature of problem: Partial fraction decomposition is widely used in particle physics to bring rational functions into a unique form. Treating the multivariate case by applying the univariate method iteratively risks the introduction of spurious singularities. Here, we formulate an algorithm that handles non-linear multivariate denominators, avoids the introduction of spurious denominators, and aims at providing good performance for typical applications in particle physics. It can be used to obtain unique results also when decomposing individual terms of a sum separately, independent of the details of the input form. A variant of the approach allows to reach a Leĭnartas' decomposition as the output form.Solution method: We reformulate the problem of calculating a partial fraction decomposition into the problem of reducing a polynomial with respect to a specific ideal. The polynomial reduction is based on a Gröbner basis calculation and guarantees a unique result for the partial fraction decomposition. We provide a complete implementation as a Mathematica package. Optionally, the included interfaces to Form and Singular can be used to speed up the computation.Additional comments including restrictions and unusual features: Our algorithm does not introduce new singularities that were not presented in the input. If, however, the input contains spurious singularities, our package can be used in such a way that the cancellation of these singularities is not guaranteed. By default, the main MultivariateApart function of our package avoids this problem by first putting terms in a sum over a common denominator, and canceling factors in the result.