Computation of All Rational Solutions of First-Order Algebraic ODEs

In this paper, we consider the class of first-order algebraic ordinary differential equations (AODEs), and study their rational solutions in three different approaches. A combinatorial approach gives a degree bound for rational solutions of a class of AODEs which do not have movable poles. Algebraic considerations yield an algorithm for computing rational solutions of quasilinear AODEs. And finally ideas from algebraic geometry combine these results to an algroithm for finding all rational solutions of a class of firstorder AODEs which covers all examples from the collection of Kamke. In particular, parametrizations of algebraic curves play an important role for a transformation of a parametrizable first-order AODE to a quasi-linear differential equation.