On super-irreducible forms of linear differential systems with rational function coefficients

Consider a system of n linear first-order differential equations (d/dx)y=A(x)y in which A(x) is an n×n matrix of rational functions over a subfield F of the field C of complex numbers and let Γ={α1,…,αd}⊂C be a set of conjugate singularities of this system, i.e., poles of A(x) which are roots in C of some irreducible polynomial p(x) in F[x]. We propose an algorithm for transforming the given system into an equivalent system over F(x) which is super-irreducible in each element α∈Γ. This algorithm does not require working in the algebraic extension F(Γ) that appears when one applies Hilali–Waznerʹs algorithm (Numer. Math. 50 (1987) 429) successively with the individual singularities α1,…,αd. The transformation matrix as well as the resulting system have their coefficients in F(x) and all the computations are performed in F[x]/(p) instead of the splitting field of p.