In this paper, we give a necessary and sufficient condition for an algebraic ODE to have an algebraic general solution. For a first order autonomous ODE, we give an optimal bound for the degree of its algebraic general solutions and a polynomial-time algorithm to compute an algebraic general solution if it exists. Here an algebraic ODE means that an ODE given by a differential polynomial.

In 1903 E.Maillet proved that a formal solution of an algebraic ordinary differential equation has some Gevrey order 8 < oo. In 1989, B.Malgrange gave a bound of s for convergent ordinary differential equations. Here we extend these results for formal differential equations of Gevrey type. Our method is based in an algorithmic use of the Newton-Puiseux Polygon for differential equations.

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The Newton polygon construction for ODEs, and Malgrange-Ramis polygon for partial differential equations in one variable are generalized in order to give an algorithm to find solutions of a linear partial differential equation at a singularity. The solutions found involve exponentials, logarithms and Laurent power series with exponents contained in a strongly convex cone.