Representation of Algebraic Curves * )

If algebraic varieties like curves or surfaces are to be manipulated by computers, it is essential to be able to represent these geometric objects in an appropriate way. For some applications an implicit representation by algebraic equations is desirable, whereas for others an explicit or parametric representation is more suitable. Therefore, transformation algorithms from one representation to the other are of utmost importance. We investigate the transformation of an implicit representation of a plane algebraic curve into a parametric representation. In the course of the transformation algorithm the coefficient field has to be extended algebraically. If the known parametrization algorithms are used uncritically, the algebraic extensions get so high that any computation becomes practically impossible. Our goal is to keep the degree of the necessary algebraic extension as small as possible.