Testing reducibility of linear differential operators: A group theoretic perspective

Let k[D] be the ring of differential operators with coefficients in a differential field k. We say that an element L of k[D] is reducible if L = L(1)oL(2) for L(1), L(2) is an element of K[D], L(1), L(2) is not an element of k. We show that for a certain class of differential operators (completely reducible operators) there exists a Berlekamp-style algorithm for factorization. Furthermore, we show that operators outside this class can never be irreducible and give an algorithm to test if an operator belongs to the above class. This yields a new reducibility test for linear differential operators. We also give applications of our algorithm to the question of determining Galois groups of linear differential equations.