Computing the Galois Group of a Linear Differential Equation of Order Four

In 1978 J. Kovacic described an efficient algorithm for computing liouvillian solutions of a linear homogeneous differential equation of order two over a field C(x), where x′ = 1 and C is an algebraically closed field of characteristic 0. During the years from 1990 to 1994 M. Singer and F. Ulmer published several papers in which they describe efficient algorithms for determining the Galois group of such a differential equation of order two or three and computing liouvillian solutions using this group. In this paper we present results concerning Galois groups of order four linear differential equations. In particular we construct a list of irreducible linear algebraic subgroups of SL(4, C) where C is an algebraically closed field of characteristic zero. This list is complete up to conjugation, and in the finite primitive case, up to projective equivalence. Then, in keeping with the spirit of the work of Kovacic, Singer and Ulmer we use representation theory to distinguish between the groups in this list. Received: December 21, 1998; revised version: May 20, 2000