Using basis computation to determine pseudo-multiplicative independence

Abstract

This paper describes a procedure for determining when a set of rational functions are pseudo-multiplicatively independent, i.e. when no non-trivial power product of the rational functions can be a constant. The method used is to derive a multiplicative basis of factors of the numerators and denominators of the rational functions; the basis is then used to derive a system of homogeneous linear equations which will have a non-trivial solution if and only if the rational functions are pseudo-multiplicatively dependent. Algorithms which find a basis for a set of Gaussian rational functions and the corresponding linear equations are presented in detail and bounds on their theoretical computing times are derived.