Liouvillian solutions of linear differential equations of order three and higher

M. van Hoeij, J.-F. Ragot, F. Ulmer, J.-A. Weil

In [SUl97], Singer and Ulmer gave an algorithm to compute Liouvillian ("closed-form") solutions of homogeneous linear differential equations. However, there were several efficiency problems that made computations not practical. In this paper we address these problems. We extend the algorithm in [hwmega] to compute semi-invariants and a theorem in [SUl97] in such a way that, by computing one semi-invariant that factors into linear forms, one gets all coefficients of the minimal polynomial of an algebraic solution of the Riccati equation, instead of only one coefficient. This way we no longer need to do large computations to obtain the remaining coefficients. These remaining coefficients come "for free" as a byproduct of our algorithm for computing semi-invariants. We specifically detail the algorithm in the cases of equations of order 3 (order 2 equations are handled by the Kovacic algorithm [Kov86,UWe96,Fak97]).

In the appendix, we present several methods to decide when a multivariate polynomial depending on parameters can admit linear factors, which is a keystone in the algorithm.