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 $\mathbb{K}(α)\thickapprox\mathbb{K}^n$ under the transformation $\frac{at+b}{ct+d}:\mathbb{K}(α)\to\mathbb{K}(α)$.
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.
二元实多项式的零点集是$\mathbb R^2$中的曲线。对于其系数的一般选择，这是一条非奇异曲线，是适当嵌入$\mathbb R^2$中的圆和线的集合。对于给定次数的多项式，这些圆和直线的拓扑排列是什么？这个问题出现在19世纪的作品Harnack和希尔伯特，并列入希尔伯特到他的第16个问题。从那时起，取得了一些部分成果。然而，完整的答案是已知的，只有多项式的次数为5或更少。本文提出了一个新的部分结果的解决方案的第16希尔伯特问题。证明利用了Kronheimer和Mrowka在$\mathbb C P^2$中对Thom猜想的证明。
The symplectic nature of the space of dormant indigenous bundles on algebraic curves
我们研究了正特征域$K$上休眠曲线分类的模叠加的辛性质，即，适当的双曲曲线在$K$配备了一个休眠的本地束。本文的中心对象是以下两个德利涅-芒福德堆栈。一个是模栈${^\circledcirc \mathfrak{M}^{^\mathrm{Zzz...}}_{g,K}}$的余切丛${^\circledcirc T^{\vee ^\mathrm{Zzz...}}_{g,K}}$，它将$g$类的$K$上的普通休眠曲线分类。另一种是在带有固有束的$K$上分类普通休眠曲线的模量栈${^\circledcirc \mathfrak{S}^{^\mathrm{Zzz...}}_{g,K}}$。这些Deligne-Mumford栈分别具有典型的辛结构。本文的主要结果是：正则同构${^\circledcirc T^{\vee ^\mathrm{Zzz...}}_{g,K}} \rightarrow {^\circledcirc \mathfrak{S}^{^\mathrm{Zzz...}}_{g,K}}$保持辛结构。这一结果可以看作是S. Kawai（在题为“The symplectic nature of the space of projective connections on Riemann surfaces”的论文中），P. Arés-Gastesi，I. Biswas和B.卢斯托最后，作…
