Fields of Parametrization and Optimal Affine Reparametrization of Rational Curves

TL;DR

This paper investigates fields of parametrization for rational curves, proving the existence of infinitely many quadratic parametrizations, and introduces an algorithm for optimal affine reparametrization without needing rational points.

Contribution

It establishes the infinite nature of quadratic fields of parametrization and presents a novel algorithm for optimal reparametrization avoiding rational point computation.

Findings

Witness variety is always a hypercircle related to algebraic extensions.

There are infinitely many quadratic extensions that parametrize a given rational curve.

The proposed algorithm finds optimal affine reparametrizations without rational point computation.

Abstract

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.