Fields of Parametrization and Optimal Affine Reparametrization of Rational Curves

In this paper we present three related results on the subject of fields of parametrization. Let C be a rational curve over a field of characteristic zero. Let K be a field finitely generated over Q, such that it is a field of definition of C but not a field of parametrization. It is known that there are quadratic extensions of K that parametrize C. First, we prove that there are infinitely many quadratic extensions of K that are fields of parametrization of C. As a consequence, we prove that the witness variety, that appear in the context of the parametric Weil's descente method, is always a special curve related to algebraic extensions, called hypercircle. It is possible that the witness variety is not a hypercircle for the given extension, but for an alternative one. We use these two facts to present an algorithm to solve the following optimal reparametrization problem. Given a birational parametrization f(t) of a curve C, compute the affine reparametrization at+b such f(at+b) has coefficients over a field as small as possible. The main advantage of this algorithm is that it does not need to compute any rational point on the curve.