Points on algebraic curves and the parametrization problem

Abstract

A plane algebraic curve is given as the zeros of a bivariate polynomial. However, this implicit representation is badly suited for many applications, for instance in computer aided geometric design. What we want in many of these applications is a rational parametrization of an algebraic curve. There are several approaches to deciding whether an algebraic curve is rationally parametrizable and if so computing such a parametrization. In all these approaches we ultimately need some simple points on the curve. The field in which we can find such points crucially influences the coefficients in the resulting parametrization. We show how to find simple points over some practically interesting fields.Consequently, we are able to decide whether an algebraic curve defined over the rational numbers can be parametrized over the rationals or the reals. Some of these ideas also apply to parametrization of surfaces. If in the term geometric reasoning we do not only include the process of proving or disproving geometric statements, but also the analysis and manipulation of geometric objects, then algorithms for parametrization play an important role in this wider view of geometric reasoning.