We obtain two new algorithms for partial fraction decompositions; the first is over algebraically closed fields, and the second is over general fields. These algorithms takes $O(M^2)$ time, where $M$ is the degree of the denominator of the rational function. The new algorithms use less storage space, and are suitable for parallel programming. We also discuss full partial fraction decompositions.

The theory of iterated Laurent series and a new algorithm for partial fraction decompositions are combined to obtain a fast algorithm for MacMahon's Omega calculus, which (partially) avoids the "run-time explosion" problem when eliminating several variables.Expand

The paper describes a method of representing an n-dimensional array of rational expressions as an (n + 1)-dimensional array of scalars and shows how this representation readily supports the… Expand

An integer greatest common divisor (GCD) algorithm due to Schönhage is generalized to hold in all euclidean domains which possess a fast multiplication algorithm and a new faster algorithm for multivariate polynomial GCD's can be derived.Expand

Three methods for evaluating sums of the form $\sum_\zeta R(\zeta)$, where $R$ is a rational function and the sum is over all $n$th roots of unity $zeta$ (often with $\zeta =1$ excluded), are discussed.Expand

We develop a theory of the field of double Laurent series, iterated Laurent series, and Malcev-Neumann series that applies to most constant term evaluation problems. These include (i) MacMahon's… Expand