A differential system $[A] : \; Y'=AY$, with $A\in \mathrm{ Mat}(n, \bar{ k})$ is said to be in reduced form if $A\in \mathfrak{ g}(\bar{ k})$ where $\mathfrak{ g}$ 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.
Categories: math.CA cs.SC math.RT