van der Hoeven presents a numeric-symbolic algorithm for the computation of the closed algebraic subgroup generated by a finite number of invertible matrices, and which yields an algorithm for the computation of differential Galois groups, when computing with sufficient precision using the above results. The author also presents a nonheuristic algorithm for the factorization of linear differential operators.

In addition to the interesting algorithm, the author raises some interesting questions. Are there more efficient approaches for the reconstruction of elements in K if K=Qalg and when K is more general? Does there exist an efficient membership test that does not rely on probabilistic arguments? Can the approach in the paper be adapted to the computation of a basis for the usual topological closure of a finitely generated matrix group?

The paper is very interesting, and the algorithms might be used for other computations with algebraic matrix groups over C, and other fields of characteristic 0.