Rational Liouvillian Solutions of Algebraic Ordinary Differential Equations of Order One

We give a necessary and sufficient condition for an autonomous first-order AODE to have a rational liouvillian solution. We also give an algorithm to compute a rational liouvillian general solution if it exists. The algorithm is based on the rational parametrization of the corresponding algebraic curve of the first-order autonomous AODE and the existence of a rational liouvillian element over ℂ . When the corresponding algebraic curve is rational, this method covers the known cases of rational solutions and radical solutions.