On the reduction of singularly-perturbed linear differential systems

Abstract

In this article, we recover singularly-perturbed linear differential systems from their turning points and reduce their parameter singularity's rank to its minimal integer value. Our treatment is Moser-based; that is to say it is based on the reduction criterion introduced for linear singular differential systems in [Moser, 1960]. Such algorithms have proved their utility in the symbolic resolution of the systems of linear functional equations [Barkatou, 1989;Barkatou et al., 2008Barkatou et al., , 2009, giving rise to the package ISOLDE [Barkatou et al., 2013], as well as in the perturbed algebraic eigenvalue problem [Jeannerod et al., 1999]. In particular, we generalize the Moser-based algorithm described in [Barkatou, 1995]. Our algorithm, implemented in the computer algebra system Maple, paves the way for efficient symbolic resolution of singularly-perturbed linear differential systems as well as further applications of Moser-based reduction over bivariate (differential) fields [Barkatou et al., 2014].