ETD

• AODE
• Differential Algebra
• Rational Curves
• Rational Parametrization
• Rational Solution
• Rational Surface

In this thesis, I approach to algebraic ODEs from Differential Algebra's point of view. I look for rational solutions of AODE, I present an algebro-geometric method to decide the existence of rational solutions of a first-order algebraic ODE and if they exist an algorithm to compute them. This method depends heavily on rational parametrizations, in particular for autonomous equations on the parametrization of algebraic curves, and for non-autonomous equations on the parametrization of algebraic surfaces. In the last case, I prove the correspondence between rational solutions of a parametrizable algebraic ODE and rational solutions of a first-order linear autonomous differential system of two equations in two variables. I provide an algorithm to compute rational solutions of such system based on its invariant algebraic curves. I also study a group of affine transformations which preserves the rational solvability, in order to reduce, when possible, an algebraic ODE to an easier one. Moreover I present the results of the implementation of all these algorithms in two computer algebra system: CoCoA and Singular.