Algebraic Differential Equations - Rational Solutions and Beyond

An algebraic ordinary differential equation (AODE) is a polynomial relation between the unknown function and its derivatives. This polynomial defines an algebraic hypersurface, the solution surface. Here we consider AODEs of order 1. From rational parametrizations of the solution surface, we can decide the rational solvability of the given AODE, and in fact compute the general rational solution. This method depends crucially on curve and surface parametrization and the determination of rational invariant algebraic curves. Transforming the ambient space by some group of transformations, we get a classification of AODEs, such that equivalent equations share the property of rational solvability. In particular we discuss affine and birational transformation groups. We also discuss the extension of this method to non-rational parametrizations and solutions.