A characterization of reduced forms of linear differential systems

Ainhoa Aparicio-Monforte (1, 2) , Elie Compoint (3) , Jacques-Arthur Weil (4)

1 LIFL - Laboratoire d'Informatique Fondamentale de Lille
2 CALFOR - Calcul Formel
3 LPP - Laboratoire Paul Painlevé - UMR 8524
4 XLIM-DMI - DMI

A differential system [A] : Y' = AY, with A is an element of Mat(n, (k) over bar) is said to be in reduced form if A is an element of g((k) over bar) where g is the Lie algebra of the differential Galois group G of [A]. In this article, we give a constructive criterion for a system to be in reduced form. When G is reductive and unimodular, the system [A] is in reduced form if and only if all of its invariants (rational solutions of appropriate symmetric powers) have constant coefficients (instead of rational functions). When G is non-reductive, we give a similar characterization via the semi-invariants of G. In the reductive case, we propose a decision procedure for putting the system into reduced form which, in turn, gives a constructive proof of the classical Kolchin-Kovacic reduction theorem.

Domaines: Calcul formel [cs.SC]

Ainhoa Aparicio-Monforte, Elie Compoint, Jacques-Arthur Weil. A characterization of reduced forms of linear differential systems. Journal of Pure and Applied Algebra, 2013, 217 (8), pp.1504-1516. ⟨10.1016/j.jpaa.2012.11.007⟩. ⟨hal-00825852⟩