Formes super-irr�ductibles des syst�mes diff�rentiels lin�aires

Abstract

In this paper, we generalise the criterion of J. Moser. A sequence of invariants related to a linear differential system is defined. An algorithm is given which reduces a differential system to a super-irreducible form. The computation of these invariants follows directly from this form. A more general classification of the singularity is thus obtained, cf. [11] where the link between this form and the Newton polygon of the differential system is studied. The algorithm given is implemented in computer algebra system (REDUCE). R6sum6. Dans cet article, nous donnons une g6n6ralisation du crit+re de J. Moser. Une suite d'invariants associ6s b. un syst6me diff6rentiel est d6finie ainsi qu'un algorithme permettant de r6duire le syst6me diff6rentiel sous une forme super-irr~ductible. Le calcul de ces invariants est alors imm6diat. Nous obtenons ainsi une classification plus g6n6rale de la singularit6, cf. Ell] off on 6tudie le lien entre cette forme et le polygone de Newton d'un syst6me diff6rentiel lin6aire. L'algorithme donn6 est programm6 dans un langage de calcul formel (REDUCE). Subject Classifications: AMS(MOS): 65L07; G: 1.7.

Gérard–Levelt membranes

“…The most classical tries to iteratively construct a suitable gauge transformation P , usually coefficient by coefficient in its series expansion. Featured methods rely on the linear algebra over C involved by (4), like Moser and continuators [4,15,21], whose methods are widely used nowadays in computer algebra, or other researchers such as [3,13], while [2] uses Lie group theoretic tools.…”

Higher-Order Linear Differential Systems with Truncated Coefficients

“…We will show that the existence and truncations problems are decidable for systems of this form. Our approach is based on the concept of simple systems [3] (which is related to the notion of super-irreducible forms of linear differential systems [13]). A system θy = A(x)y can always be rewritten as a system of the form…”

“…(i) The algorithm from [13] computes the so-called super-irreducible form of a given system (9). It was shown in [3] that if a system has the super-irreducible form then it can be written as a simple system.…”

“…It was shown in [3] that if a system has the super-irreducible form then it can be written as a simple system. The algorithm from [13] needs at the most (m − 1)q reduction steps (see, for example, the proof of Proposition 2.2 in [10]). At each step, a transformation matrix with span 1 is computed.…”

On Rational Solutions of Systems of Linear Differential Equations

“…We prove that any system can be reduced to an equivalent simple one. This reduction can be achieved using an adapted version of the super-reduction algorithm of Hilali and Wazner (1987) (see the Appendix). This yields an algorithm for computing indicial equations associated with a given system.…”