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

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