Existence and convergence of Puiseux series solutions for autonomous first order differential equations

Abstract

Given an autonomous first order algebraic ordinary differential equation F(y,y′)=0, we prove that every formal Puiseux series solution of F(y,y′)=0, expanded around any finite point or at infinity, is convergent. The proof is constructive and we provide an algorithm to describe all such Puiseux series solutions. Moreover, we show that for any point in the complex plane there exists a solution of the differential equation which defines an analytic curve passing through this point.