RATIONAL PARAMETRIZATIONS OF ALGEBRAIC-CURVES USING A CANONICAL DIVISOR

For an algebraic curve C with genus 0 the vector space L(D) where D is a divisor of degree 2 gives rise to a bijective morphism g from C to a conic C-2 in the projective plane. We present an algorithm that uses an integral basis for computing L(D) for a suitably chosen D. The adv antage of an integral basis is that it contains all the necessary info rmation about the singularities, so once the integral basis is known t he L(D) algorithm does not need work with the singularities anymore. I f the degree of C is odd, or more generally, if any odd degree rationa l divisor on C is known then we show how to construct a rational point on C-2. In such cases a rational parametrization, which means defined without algebraic extensions, of C-2 can be obtained. In the remainin g cases a parametrization of C-2 defined over a quadratic algebraic ex tension can be computed. A parametrization of C is obtained by composi ng the parametrization of C-2 with the inverse of the morphism g. (C) 1997 Academic Press Limited.