摘要翻译：
本文给出了关于参数化域的三个相关结果。设C是特征为零的域上的有理曲线。设K是在Q上有限生成的域，使得它是C的定义域，但不是参数化域。已知K的二次扩张是C的参数化。首先，我们证明了K的二次扩张是C的参数化域，因此，我们证明了在参数Weil的descente方法中出现的见证变量总是一种与代数扩张有关的特殊曲线，称为超圆。对于给定的扩展，见证变量可能不是一个超圆，而对于另一个扩展，见证变量可能是一个超圆。我们利用这两个事实给出了一个算法来解决下面的最优再参数化问题。给定曲线C的双形参数化f(t),计算At+b的仿射重参数化，使得f(at+b)在域上的系数尽可能小。该算法的主要优点是不需要计算曲线上的任何有理点。
---
英文标题：
《Fields of Parametrization and Optimal Affine Reparametrization of
  Rational Curves》
---
作者：
Luis Felipe Tabera
---
最新提交年份：
2008
---
分类信息：

一级分类：Mathematics        数学
二级分类：Algebraic Geometry        代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
--
一级分类：Mathematics        数学
二级分类：Commutative Algebra        交换代数
分类描述：Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics
交换环，模，理想，同调代数，计算方面，不变理论，与代数几何和组合学的联系
--

---
英文摘要：
  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.
---
PDF链接：
https://arxiv.org/pdf/0810.5595