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Generalizing circles over algebraic extensions


Authors: T. Recio, J. R. Sendra, L. F. Tabera and C. Villarino
Journal: Math. Comp. 79 (2010), 1067-1089
MSC (2000): Primary 14Q05; Secondary 14M20
Published electronically: December 14, 2009
MathSciNet review: 2600556
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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.


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Additional Information

T. Recio
Affiliation: Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria, 39071, Santander, Spain
Email: tomas.recio@unican.es

J. R. Sendra
Affiliation: Departamento de Matemáticas, Universidad de Alcalá, 28871, Alcalá de Henares, Spain
Email: rafael.sendra@uah.es

L. F. Tabera
Affiliation: Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria, 39071, Santander, Spain
Email: taberalf@unican.es

C. Villarino
Affiliation: Departamento de Matemáticas, Universidad de Alcalá, 28871, Alcalá de Henares, Spain
Email: carlos.villarino@uah.es

DOI: https://doi.org/10.1090/S0025-5718-09-02284-4
Received by editor(s): November 2, 2006
Received by editor(s) in revised form: August 2, 2008
Published electronically: December 14, 2009
Additional Notes: The authors are partially supported by the project MTM2005-08690-C02-01/02
The second and fourth authors were partially supported by CAM-UAH2005/053 “Dirección General de Universidades de la Consejería de Educación de la CAM y la Universidad de Alcalá”.
The third author was also supported by a FPU research grant. Ministerio de Educación y Ciencia
Article copyright: © Copyright 2009 American Mathematical Society