Liouvillian Solutions of Algebraic Ordinary Differential Equations of Order One of Genus Zero

This paper considers the class of autonomous algebraic ordinary differential equations(AODEs) of order one, and studies their Liouvillian general solutions. In particular, let F(y, w) = 0 be a rational algebraic curve over C. The authors give necessary and sufficient conditions for the autonomous first-order AODE F(y, y′) = 0 to have a Liouvillian solution over C. Moreover, the authors show that a Liouvillian solution α of this equation is either an algebraic function over C(x) or an algebraic f...