On super-irreducible forms of linear differential systems with rational function coefficients.
Let \(F\) be a subfield of the field \(\mathbb C,\) and consider the system of equations \[ \frac{dy}{dx}=A(x)y, \] where \(A(x)\) is a matrix whose entries are in \(F(x)\) (the field of rational functions with coefficients in \(F\)). Suppose that \(x=\alpha\in \mathbb C\) is a pole of \(A(x).\) \textit{J. Moser} [Math. Z. 72, 379--398 (1960; Zbl 0117.04902)] gave an algorithm to decide whether \(x=\alpha\) is a regular singularity. This algorithm yields a coefficient matrix called the Moser irreducible form, which has the minimal pole order among matrices equivalent to \(A(x)\) under transformations \(y=T(x)z\) with entries in \(F(\alpha)[x].\) The notion of super-irreducible forms, which was introduced by \textit{A. Hilali} and \textit{A. Wazner} [Numer. Math. 50, 429--449 (1987; Zbl 0627.65091)], is a generalization of that of Moser irreducible forms. Let \(\Gamma=\{\alpha_1,\dots, \alpha_d\}\subset \mathbb C\) be a set of poles of \(A(x)\) which are the roots of some irreducible polynomial \(p(x)\) in \(F[x].\) In this paper, the author proposes an algorithm for computing an equivalent system with coefficients in \(F(x)\) which is super-irreducible in each \(\alpha_j\in\Gamma.\) In this algorithm, all computations are performed in \(F[x]/(p).\)
scientific article; zbMATH DE number 2032609