Simple Forms and Rational Solutions of Pseudo-Linear Systems

Abstract

In this paper, we first provide a unified algorithm for computing simple forms for systems of pseudo-linear equations. We prove that the existing methods for linear differential and difference systems can be extended to handle more general pseudo-linear systems. We explain how the reduction to a simple form can be used to compute efficiently local data for a system of pseudo-linear equations. We then propose an alternative, again based on simple forms, to previous algorithms for computing rational solutions of pseudo-linear systems. Moreover we develop a new algorithm for computing rational solutions of systems in two variables composed of linear differential and difference equations. Finally, we show that this algorithm can be generalized to the case of a system of partial pseudo-linear equations. All the algorithms described in this paper have been implemented in Maple and some examples of computations are provided.