# Rational Parametrizations of Algebraic Curves using a Canonical Divisor

This purpose of this paper is to compute L(D) where D is
-1 times a canonical divisor. This gives a rational (i.e.
without algebraic extensions) bijective morphism to a
conic, which can be used to parametrize the curve using
an algebraic extension of degree <= 2.
The problem of computing this L(D) is divided into several
smaller subproblems. First L(D_infty) is computed where
D_infty is the divisor of the line at infinity. This is
done by a rather tricky but efficient algorithm
which uses
a sort of "divide and conquer" technique. Afterwards,
the ramification points are treated not by actually
computing with those points (which would seem to be the
most natural approach), but to use derivatives of
elements of the integral basis. This turns out to be
much faster.
Then to compute the inverse morphism, tricks similar to
the ones in my ISSAC'95 paper are used in order to speed
up the resultant computations.
The implementation of this parametrization algorithm is
available in Maple V release 5. To view the code
see the file ratpar in
the algcurves package.
Download this paper.