Computing the Lie algebra of the differential Galois group: the reducible case

TL;DR

This paper presents an algorithm to compute the Lie algebra of the differential Galois group for reducible linear differential systems by transforming them into a reduced form, bypassing the need to compute the Galois group directly.

Contribution

It introduces a method to transform block-triangular systems into a reduced form, enabling direct computation of the Lie algebra of the differential Galois group.

Findings

Algorithm successfully computes Lie algebra for reducible systems

Transforms systems into Kolchin-Kovacic reduced form

Provides a general approach for linear differential systems

Abstract

In this paper, we explain how to compute the Lie algebra of the differential Galois group of a reducible linear differential system. We achieve this by showing how to transform a block-triangular linear differential system into a Kolchin-Kovacic reduced form. We combine this with other reduction results to propose a general algorithm for computing a reduced form of a general linear differential system. In particular, this provides directly the Lie algebra of the differential Galois group without an a priori computation of this Galois group.