Computing Galois Groups of Completely Reducible Differential Equations

At present we do not know a general algorithm that will compute the Galois group of a linear differential equation with coefficients in a differential field k, even when k = Q̄(x), where Q̄ is the algebraic closure of the rational numbers. In contrast, algorithms for calculating the Galois group of a polynomial with coefficients in Q or Q̄(x) have been known for a long time (van der Waerden, 1953; Pohst and Zassenhaus, 1989; Cohen, 1993). The key idea behind these methods is to represent the splitting field of a polynomial in terms of generators and relations. The Galois group is then the set of permutations of the generators that preserve the relations. In the differential case, the analogue of the splitting field is called the Picard–Vessiot extension and the Galois group is defined as the group of differential automorphisms leaving elements of the base field fixed. The obstruction to mimicking the ideas from the Galois theory of polynomials is that, at present, we do not know how to effectively present a general Picard–Vessiot extension in terms of generators and relations. In this paper, we will show that for differential equations whose Galois group is reductive, one can effectively present the corresponding Picard–Vessiot extension and from this presentation compute the Galois group. In Compoint (1996a,b), the first author showed that if a Picard–Vessiot extension has a reductive unimodular Galois group then the relations defining this extension come from the invariants of the Galois group. To be more specific, let k be a differential field of characteristic zero with algebraically closed field C of constants and let Y ′ = AY be a differential equation where A is an n × n matrix with entries in k. Let G ⊂ SL(n) be the Galois group and let its action on the polynomial ring C[Y1,1, . . . , Yn,n] be defined by letting each element of G act on the n × n matrix [Yi,j ] by multiplication on the