A characterization of reduced forms of linear differential systems

International audienceA 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