# Computing parametrizations of rational algebraic curves

With a new fast method available for computing an integral basis, a number of computations with algebraic curves can now be done more efficiently in a different way.

The main result of this paper is to efficiently compute a generator of a function field of genus 0. Such a generator is a function with only 1 pole, with multiplicity 1. The idea is to divide the curve C into two parts A and B. Then one takes a function P that has only 1 pole on part A. This function probably has poles on part B as well. So P is OK on A, but not OK on part B. Now one can use an integral basis for part A, and an integral basis for part B, to eliminate the poles on part B without altering the situation on part A (fix what's broken, but don't touch what was OK).

To find the parametrization, one needs to express x and y in terms of the generator of the function field. This can be done by resultant computations. In section 3.2 a method is given to speed up this step significantly by reducing the number of variables (by substituting a value for that variable).

The method in this paper has one drawback, one first needs to pick a point on the curve (this could introduce an algebraic extension of the field of constants, of degree at most the degree of the curve). This problem will be fixed in [jsc1997a].

The method in these two papers is available in Maple V release 5. The method in this paper is implemented in the file iss94 in the algcurves package. Maple will use this method if it can find a rational point on the curve, otherwise it will use the method from [jsc1997a].

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