A rational version of Moser's algorithm

It is important to know whether a linear system of differential equations haa a regular or an irregular singularity at a given point zo of the complex domain C. Moser has given an algorithm to solve this problem. His algorithm needs to compute with the individual singularities of the system and requires computations in algebraic extensions of the constant field for the coefficients of the system. This may represents an important drawback from a practical point of view. This paper describes a rational version of Moser’s algorithm which has been implemented in the Maple computer algebra system for systems of differential equations with coefficients in Q(z). It never needs to compute with the individual singularities of the system and avoids any algebraic extensions. In addition, our algorithm reduces a linear system of differential equations with coefficients in Q(x) to an “irreducible” form which is particularly convenient if one wishes to compute invariants at singularities.