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

**April 16, 1998**

**Abstract**

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