Differential equations of order two with one singular point

The goal of this paper is to describe the set of polynomials r is an element of C[x] such that the linear differential equation y " = ry has Liouvillian solutions, where C 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 2n can be represented by a countable set of algebraic varieties of dimension n + 1. Some properties of those algebraic varieties are proved. (C) 1999 Academic Press.