Calculating the Galois group of L1(L2(y))=0, L1,L2 completely reducible operators

In Calculating Galois groups of completely reducible linear operators, Compoint and Singer describe a decision procedure that computes the Galois group of a completely reducible linear differential operator with rational or algebraic function coefficients (i.e., a linear differential operator that is the least common left multiple of irreducible operators or, equivalently, one whose Galois group is a reductive group). At present, it is unknown how to calculate the Galois group of a general operator. In this paper, we push beyond the completely reducible case by showing how to compute the Galois group of an operator of the form L-1 circle L-2 where L-1 and L-2 are completely reducible and have rational function coefficients. We begin by showing how to compute the Galois group of an equation of the form L(y) = b with L completely reducible. This corresponds to the case of L-1 circle L-2 where L-1 = D - b'/b. We then show how one can reduce the general case to the above case and give several examples. (C) 1999 Elsevier Science B.V. All rights reserved.