A Characterization of Reduced Forms of Linear Differential Systems

TL;DR

This paper provides a constructive criterion and decision procedure for transforming linear differential systems into reduced form based on the properties of their differential Galois group, extending classical reduction theorems.

Contribution

It introduces a new criterion for reduced form characterization and a decision procedure, especially for reductive and non-reductive Galois groups.

Findings

Criterion for reduced form based on invariants and semi-invariants

Decision procedure for systems with reductive Galois groups

Extension of Kolchin-Kovacic reduction theorem

Abstract

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.