Differential Equations of Order Two with One Singular Point

The goal of this paper is to describe the set of polynomials r C[ x ] such that the linear differential equationy′′ = ry has Liouvillian solutions, whereC is an algebraically closed field of characteristic 0. It is known that the differential equation has Liouvillian solutions only if the degree of r is even. Using differential Galois theory we show that the set of such polynomials of degree 2 n can be represented by a countable set of algebraic varieties of dimension n + 1. Some properties of those algebraic varieties are proved.