Generalizing circles over algebraic extensions

Abstract:

This paper deals with a family of spatial rational curves that were introduced in 1999 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 an $n$-degree finite algebraic extension $\mathbb {K}(\alpha )\thickapprox \mathbb {K}^n$ under the transformation $\frac {at+b}{ct+d}:\mathbb {K}(\alpha )\rightarrow \mathbb {K}(\alpha )$.

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.