Liouvillian and Algebraic Solutions of Second and Third Order Linear Differential Equations

In this paper we show that the index of a 1-reducible subgroup of the diierential Galois group of an ordinary homogeneous linear diierential equation L(y) = 0 yields the best possible bound for the degree of the minimal polynomial of an algebraic solution of the Riccati equation associated to L(y) = 0. For an irreducible third order equation we show that this degree belongs to f3;6;9;21;36g. When the Galois group is a nite primitive group, we reformulate and generalize work of L. Fuchs to show how to compute the minimal polynomial of a solution instead of the minimal polynomial of the logarithmic derivative of a solution. These results lead to an eeective algorithm to compute Liouvillian solutions of second and third order linear diierential equations. 0. Introduction The computation of the algebraic solutions of a linear diierential equation L(y) = 0 over the eld of rational functions was a problem of great interest of the end of last century. and others worked on this problem and gave a solution for second order equations (cf. (Baldassarri and Dwork (1979)), the intoduction of Boulanger (1898), and Gray (1986)). Many of the earliest contributions to the representation theory of nite groups have been made in connection with diierential equation (e.g. Jordan's Theorem) and it was the starting point for the classiication of the nite primitive groups. In this paper we will focus on the ideas of Fuchs. In Fuchs (1878), Fuchs showed how the (then new) tools of invariant theory could be used to construct, in many cases, the minimal polynomial of an algebraic solution of a second order linear diierential equation. The more general question of nding the liouvillian solutions of a linear diierential equation, in which case the diierential Galois group can be innnite, leads to the theory of linear algebraic groups. But for a primitive unimodular Galois group, all liouvillian solutions are algebraic (cf. Ulmer (1992)) and in this case the approach of Fuchs can The second author would like to thank North Carolina State University for its hospitality and partial support during the preparation of this paper.