Computing the Lie Algebra of the Differential Galois Group of a Linear Differential System

Abstract

We consider a linear differential system [A] : y'=A, y}, where A has with coefficients in C(x). The differential Galois group G of [A] is a linear algebraic group which measures the algebraic relations among solutions. Although there exist general algorithms to compute $G$, none of them is either practical or implemented. This paper proposes an algorithm to compute the Lie algebra g of G when [A] is absolutely irreducible. The algorithm is implemented in Maple.