Liouvillian and algebraic solutions of second and third order linear differential equations (Q1332657)

Let \(F\) be an ordinary differential field of characteristic 0 and \(L\in F\langle y\rangle\) be a linear homogeneous polynomial. How can one find the Liouvillian solutions of \(L(y)=0\)? In the paper this problem is reduced to the problems of (1) factorization and (2) finding solutions \(u\) such that \(u'/u \in F\) of \(L\) and some polynomials associated with it (symmetric powers of \(L\)). Now there are the algorithms for the solution of the last problems for \(F= \mathbb{Q}(x)\) [see \textit{D. Yu. Grigor'ev}, J. Symb. Comput. 10, 7-37 (1990; Zbl 0728.68067) and \textit{M. F. Singer}, Am. J. Math. 103, 661-682 (1981; Zbl 0477.12026)]. For polynomials \(L\) of the second and third order the authors provide full investigation of the most difficult case when the solution \(u\) of \(L(y)=0\) is algebraic. They show that one can compute the minimal polynomial \(P(y)\in F[y]\) of \(u\). We note that the authors essentially used the tools of representation theory, invariant theory and computer algebra.