LinApart3: efficient algorithm for multivariate partial fraction decomposition with linear denominators

Abstract

We present LinApart3, an efficient multivariate partial fraction decomposition algorithm for rational functions with linear denominators. Our decomposition algorithm guarantees that each term contains at most as many distinct denominators from the original set as partial fraction variables, introduces no spurious singularities, is independent of variable ordering, and is insensitive to the presence of spectator variables. While general multivariate approaches based on Gröbner bases or Leinartas' method handle arbitrary polynomial denominators, they suffer from intermediate expression swell. LinApart3 replaces polynomial-ideal computations with linear algebra and residue extraction by exploiting the geometry of the hyperplane arrangement defined by the denominators, circumventing this issue just as LinApart did in the univariate case. Because the individual basis contributions are independent, the algorithm is moreover naturally parallelizable. To showcase the utility of our algorithm we implemented the algorithm both in Wolfram Mathematica and FORM.