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

Abstract

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∘ 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∘ 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.