Puiseux Series Solutions with Real or Rational Coefficients of First Order Autonomous AODEs

Given an autonomous first order algebraic ordinary differential equation $F(y,y')=0$, we provide algorithms for computing formal Puiseux series solutions of $F(y,y')=0$ with real or rational coefficients. For this purpose we give necessary and sufficient conditions on the existence of such solutions by combining classical methods from algebraic geometry and the study of an associated differential equation. Since all formal Puiseux series solutions of such differential equations are convergent in a certain neighborhood, the solutions also define real solution functions.