Generalizing circles over algebraic extensions
HTML articles powered by AMS MathViewer
- by T. Recio, J. R. Sendra, L. F. Tabera and C. Villarino PDF
- Math. Comp. 79 (2010), 1067-1089 Request permission
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.
References
- Cesar Alonso, Jaime Gutierrez, and Tomas Recio, A rational function decomposition algorithm by near-separated polynomials, J. Symbolic Comput. 19 (1995), no. 6, 527–544. MR 1370620, DOI 10.1006/jsco.1995.1030
- Carlos Andradas and Tomás Recio, Plotting missing points and branches of real parametric curves, Appl. Algebra Engrg. Comm. Comput. 18 (2007), no. 1-2, 107–126. MR 2280313, DOI 10.1007/s00200-006-0032-7
- Carlos Andradas, Tomás Recio, and J. Rafael Sendra, A relatively optimal rational space curve reparametrization algorithm through canonical divisors, Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation (Kihei, HI), ACM, New York, 1997, pp. 349–355. MR 1810004, DOI 10.1145/258726.258844
- Carlos Andradas, Tomás Recio, and J. Rafael Sendra, Base field restriction techniques for parametric curves, Proceedings of the 1999 International Symposium on Symbolic and Algebraic Computation (Vancouver, BC), ACM, New York, 1999, pp. 17–22. MR 1802062, DOI 10.1145/309831.309845
- David Cox, John Little, and Donal O’Shea, Ideals, varieties, and algorithms, 2nd ed., Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1997. An introduction to computational algebraic geometry and commutative algebra. MR 1417938
- Joe Harris, Algebraic geometry, Graduate Texts in Mathematics, vol. 133, Springer-Verlag, New York, 1992. A first course. MR 1182558, DOI 10.1007/978-1-4757-2189-8
- Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
- D. Hilbert and A. Hurwitz, Über die diophantischen Gleichungen vom Geschlecht Null, Acta Math. 14 (1890), no. 1, 217–224 (German). MR 1554798, DOI 10.1007/BF02413323
- D. Manocha and J. Canny, Rational curves with polynomial parametrization, Computer Aided Design 23 (1991), no. 9, 653–653.
- Tomas Recio and J. Rafael Sendra, Real reparametrizations of real curves, J. Symbolic Comput. 23 (1997), no. 2-3, 241–254. Parametric algebraic curves and applications (Albuquerque, NM, 1995). MR 1448697, DOI 10.1006/jsco.1996.0086
- Tomás Recio, J. Rafael Sendra, Luis Felipe Tabera, and Carlos Villarino, Fast computation of the implicit ideal of a hypercircle, Actas de AGGM 2006, 2006, pp. 258–265.
- Tomas Recio, J. Rafael Sendra, and Carlos Villarino, From hypercircles to units, ISSAC 2004, ACM, New York, 2004, pp. 258–265. MR 2126952, DOI 10.1145/1005285.1005323
- J. Rafael Sendra and Carlos Villarino, Optimal reparametrization of polynomial algebraic curves, Internat. J. Comput. Geom. Appl. 11 (2001), no. 4, 439–453. MR 1852578, DOI 10.1142/S0218195901000572
- J. Rafael Sendra and Carlos Villarino, Algebraically optimal parametrizations of quasi-polynomial algebraic curves, J. Algebra Appl. 1 (2002), no. 1, 51–74. MR 1907738, DOI 10.1142/S0219498802000045
- J. Rafael Sendra and Franz Winkler, Symbolic parametrization of curves, J. Symbolic Comput. 12 (1991), no. 6, 607–631. MR 1141543, DOI 10.1016/S0747-7171(08)80144-7
- J. Rafael Sendra and Franz Winkler, Parametrization of algebraic curves over optimal field extensions, J. Symbolic Comput. 23 (1997), no. 2-3, 191–207. Parametric algebraic curves and applications (Albuquerque, NM, 1995). MR 1448694, DOI 10.1006/jsco.1996.0083
- Luis Felipe Tabera, Fields of parametrization and optimal affine reparametrization of rational curves, Preprint, arXiv:0810.5595.
- Robert J. Walker, Algebraic Curves, Princeton Mathematical Series, vol. 13, Princeton University Press, Princeton, N. J., 1950. MR 0033083
- André Weil, Adèles et groupes algébriques, Séminaire Bourbaki, Vol. 5, Soc. Math. France, Paris, 1995, pp. Exp. No. 186, 249–257 (French). MR 1603471
- http://www.algebra.uni-linz.ac.at/Nearrings/
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
- MR Author ID: 260673
- 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
- MR Author ID: 683262
- Email: carlos.villarino@uah.es
- 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 - © Copyright 2009 American Mathematical Society
- Journal: Math. Comp. 79 (2010), 1067-1089
- MSC (2000): Primary 14Q05; Secondary 14M20
- DOI: https://doi.org/10.1090/S0025-5718-09-02284-4
- MathSciNet review: 2600556