Parametrization over prescribed coefficient fields.
Franz Winkler
Abstract:
In parametrizing an algebraic curves we often actually want
the parametrization to have coefficients in some desirable
field. For instance, if the curve equation $f(x,y)=0$ has
integral coefficients, we might want to determine a parametrization
with rational coefficients, if possible.
Or in the context of computer aided geometric design, we might
want to parametrize a real curve in such a way that the
parametrization has only real coefficients.
Some surfaces $S(x,y,z)=0$ can be parametrized by considering
one of the variables as a parameter $t$, determining a uniform
parametrization of the curves $S(x,y,t)=0$, and extracting
from this a parametrization of the surface.
In this context we want to compute parametrizations of curves
with coefficients in $Q(t)$, i.e. rational functions in $t$.
We describe algorithmic approaches to the question of parametrization
over such prescribed coefficient fields.
