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摘要翻译：
本文讨论了由Andradas、Recio和Sendra提出的一族空间有理曲线，以超圆为名，作为改进代数簇的有理参数化（尽可能简化有理函数的系数）的算法基石工具。实圆可以定义为实轴在复场中Moebius变换下的像。同样地，粗略地说，超圆可以定义为一条线（“${\mathbb{K}}$-轴”）在$n$度有限代数扩展$\mathbb{K}(\alpha)\thickaprox\mathbb{K}^n$下转换$\frac{at+b}{ct+d}:\mathbb{K}(\alpha)\到\mathbb{K}(\alpha)$。本文的目的是将圆的一些特殊性质推广到超圆的情形。我们通过$\MathBB{K}$-射影变换证明了超圆精确地是适当次数的有理法线。我们还得到了这些曲线无穷远处点的完整描述（推广了圆无穷远处的循环结构）。我们把超圆刻画为那些次等于周围仿射空间维数的曲线，它具有无穷多个${mathbb{K}}$有理点，并在无穷远处穿过这些点。此外，我们给出了超圆的参数化和蕴涵的显式公式。除了对这种非常特殊的曲线族的固有兴趣之外，对其性质的理解对参数化问题的简化也有直接的应用，如上节所示。
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英文标题：
《Generalizing circles over algebraic extensions》
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作者：
Tomas Recio, J. Rafael Sendra, Luis Felipe Tabera, Carlos Villarino
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最新提交年份：
2007
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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
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英文摘要：
This paper deals with a family of spatial rational curves that were introduced by Andradas, Recio and Sendra, under the name of hypercircles, as an algorithmic cornerstone tool in the context of improving the rational parametrization (simplifying the coefficients of the rational functions, when possible) of algebraic varieties. A real circle can be defined as the image of the real axis under a Moebius transformation in the complex field. Likewise, and roughly speaking, a hypercircle can be defined as the image of a line ("the ${\mathbb{K}}$-axis") in a $n$-degree finite algebraic extension $\mathbb{K}(\alpha)\thickapprox\mathbb{K}^n$ under the transformation $\frac{at+b}{ct+d}:\mathbb{K}(\alpha)\to\mathbb{K}(\alpha)$. The aim of this article is to extend, to the case of hypercircles, some of the specific properties of circles. We show that hypercircles are precisely, via $\mathbb{K}$-projective transformations, the rational normal curve of a suitable degree. We also obtain a complete description of the points at infinity of these curves (generalizing the cyclic structure at infinity of circles). We characterize hypercircles as those curves of degree equal to the dimension of the ambient affine space and with infinitely many ${\mathbb{K}}$-rational points, passing through these points at infinity. Moreover, we give explicit formulae for the parametrization and implicitation of hypercircles. Besides the intrinsic interest of this very special family of curves, the understanding of its properties has a direct application to the simplification of parametrizations problem, as shown in the last section.
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PDF链接：
https://arxiv.org/pdf/0704.1384
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