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.

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