Note on algebraic solutions of differential equations with known finite Galois group
Given a linear differential equation with known finite differential Galois group, we discuss methods to construct the minimal polynomial of a solution. We first outline a well known general method involving a basis transformation of the basis of formal solutions at a singular point. In the second part we construct directly the minimal polynomial of an eigenvector of the monodromy matrix at a singular point. The method is very efficient for irreducible second and third order linear differential equations where a one dimensional eigenspace of some monodromy matrix always exists.
algebraic solutions linear differential equations differential galois theory liouvillian solutions