Multivariate partial fraction expansion

Abstract

The Derive computer-algebra program has Expand as one of the menu choices: The user is prompted for successively less main expansion variables, which can be all of the variables or any proper subset. It is clear how to proceed when the expression is a polynomial: Fully distribute with respect to all expansion variables, but collect as coefficient polynomials all terms that share the same exponents for the expansion variables. Derive uses a partially factored form, so the collected coefficient polynomials can be fortuitously partially factored. For rational expressions the expand function does partial fraction expansion because it is the most useful kind of rational expansion. However, most other computer algebra systems and examples in the literature focus on partial fraction expansion with respect to only one variable, where any other variables are considered mere parameters. For consistency with multivariate polynomial expansion, we wanted a useful and well-defined meaning for multivariate partial fraction expansion. This paper provides such a definition and a corresponding algorithm.