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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