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On the Reduction of Singularly-Perturbed Linear Differential Systems

In this article, we recover singularly-perturbed linear differential systems from their turning points and reduce the rank of the singularity in the parameter 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 by Moser. Such algorithms have proved their utility in the symbolic resolution of the systems of linear functional equations, giving rise to the package ISOLDE, as well as in the perturbed algebraic eigenvalue problem. 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.