Optimal Parametrization of Algebraic Curves 1

Franz Winkler
RISC-Linz, J. Kepler Universität Linz, A-4040 Linz, Austria

April 16, 1998


There are various algorithms known for deciding the parametrizability (rationality) of a plane algebraic curve, and if the curve is rational, actually computing a parametrization. Any parametrizable curve actually has infinitely many different parametrizations of different complexity. Optimality criteria such as low degrees in the parametrization or low degree field extensions are met by some parametrization algorithms. We discuss several such optimality criteria, and also show how these criteria can be met efficiently by symbolic algorithms. In addition to curves over algebraically closed fields, we discuss real curves. Given a parametrizable plane curve over the complex numbers, we decide whether it is in fact real. Furthermore, we discuss methods for actually computing a real parametrization for a parametrizable real curve.


IMACS ACA'98 Electronic Proceedings