# Liouvillian solutions of linear differential equations of order three and higher

This paper contains several results of a different kind, all of which are necessary ingredients for efficient computation of Liouvillian solutions of equations of order 3.

The first new result is theorem 2.1, which shows how the Liouvillian solutions can be obtained directly from the (semi)-invariants. This eliminates the main bottleneck in the computation. After that it is shown how to compute semi-invariants while avoiding another bottleneck (the computation of symmetric powers). We then show for order 3 how one can avoid computing semi-invariants as much as possible (getting the necessary information from invariants, which can be computed more easily) using constructions with (semi)-invariants (section 2.3), and by studying each group separately (section 4.3). This is yet another big efficiency improvement. Furthermore by this list of groups we know the degrees of the invariants we need to compute, and we can avoid another bottleneck (the problem that the appendix deals with) in most cases. In the appendix we treat the problem of how to find values of parameters for which polynomials can be factored.

There is a partial implementation (joint work with J.A. Weil) which we used to compute examples, but we never shaped up the code to the point where it can be distributed. However, there is now also a newer implementation that is available for download but it only treats the imprimitive case. In addition, Manuel Bronstein has written a complete implementation for order 3 that can be used online through his webpage.

Click here for verifying the correctness of the example in section 4.3.10. Here is the code for obtaining the operator for the FP28 example of order 4 and for verifying the Riccati polynomial in our paper. Note that the last two words "and higher" should be removed from the title of the paper because although the method in the paper works very well for order 3, it is not the best approach for higher order (see also the paper by Olivier Cormier in ISSAC'2001).

Here is another nice example, with Galois group H_{72}^{SL_3}.