Liouvillian Solutions of Linear Differential Equations of Order Three and Higher

Singer and Ulmer (1997) gave an algorithm to compute Liouvillian (“closed-form") solutions of homogeneous linear differential equations. However, there were several efficiency problems that made computations often not practical. In this paper we address these problems. We extend the algorithm in van Hoeij and Weil (1997) to compute semi-invariants and a theorem in Singer and Ulmer (1997)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. These 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 three (order two equations are handled by the algorithm of Kovacic, 1986,see also Ulmer and Weil, 1996 or Fakler, 1997. In the Appendix, we present several methods to decide when a multivariate polynomial depending on parameters can admit linear factors, which is a necessary ingredient in the algorithm.