# An algorithm for computing invariants of differential Galois groups.

The purpose of this paper is to compute invariants of the differential Galois group. These invariants are rational solutions of a certain system of differential equations (the symmetric power system). However, this system is so large that it would be hard to directly solve it. In this paper we show how to get around this problem. We give two algorithms. First a heuristic that computes the invariants (as polynomials) and the "value" of these invariants (the first entry of the dual first integral that corresponds to the invariant). Then we give a complete (non-heuristic) algorithm that computes the invariants and gives the complete (not just the first entry) corresponding dual first integral as well. In the application, c.f. Liouvillian solutions of linear differential equations of order three and higher we need the invariant as a polynomial but we need the corresponding dual first integral as well. So in our application, we need to use the complete algorithm.

The output of the heuristic might be too large (meaning that this vector space could contain more than just all invariants), but the output of the complete algorithm is always correct because the dual first integrals are used to select those candidate invariants that are indeed true invariants.

However, the heuristic is still useful, because we can often use it to discard those degrees m for which there are no invariants of degree m, so that the complete algorithm only needs to be called in those cases where there probably do exist invariants.