Around the numeric–symbolic computation of differential Galois groups

Abstract

Let L ∈ K ( z ) [ ∂ ] be a linear differential operator, where K is an effective algebraically closed subfield of C . It can be shown that the differential Galois group of L is generated (as a closed algebraic group) by a finite number of monodromy matrices, Stokes matrices and matrices in local exponential groups. Moreover, there exist fast algorithms for the approximation of the entries of these matrices. In this paper, we present a numeric–symbolic algorithm for the computation of the closed algebraic subgroup generated by a finite number of invertible matrices. Using the above results, this yields an algorithm for the computation of differential Galois groups, when computing with a sufficient precision. Even though there is no straightforward way to find a “sufficient precision” for guaranteeing the correctness of the end result, it is often possible to check a posteriori whether the end result is correct. In particular, we present a non-heuristic algorithm for the factorization of linear differential operators.