Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Computing the singularities of rational surfaces

Authors: S. Pérez-Díaz, J. R. Sendra and C. Villarino
Journal: Math. Comp. 84 (2015), 1991-2021
MSC (2010): Primary 14Q10; Secondary 14J17, 68W30
Published electronically: October 9, 2014
MathSciNet review: 3335901
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Given a rational projective parametrization $ \mathcal {P}(\mathfrak{s},\mathfrak{t},\mathfrak{v})$ of a rational projective surface $ \mathcal {S}$ we present an algorithm such that, with the exception of a finite set (maybe empty) $ \mathfrak{B}$ of projective base points of $ \mathcal {P}$, decomposes the projective parameter plane as $ {\mathbb{P}}^2(\mathbb{K})\setminus \mathfrak{B}=\bigcup _{k=1}^{\ell } \mathfrak{S}_k$ such that, if $ (\mathfrak{s}_0:\mathfrak{t}_0:\mathfrak{v}_0)\in \mathfrak{S}_k$, then $ \mathcal {P}(\mathfrak{s}_0,\mathfrak{t}_0,\mathfrak{v}_0)$ is a point of $ \mathcal {S}$ of multiplicity $ k$.

References [Enhancements On Off] (What's this?)

  • [1] Falai Chen, Wenping Wang, and Yang Liu, Computing singular points of plane rational curves, J. Symbolic Comput. 43 (2008), no. 2, 92-117. MR 2357078 (2009a:14035),
  • [2] Joe Harris, Algebraic Geometry: A First Course, Graduate Texts in Mathematics, vol. 133, Springer-Verlag, New York, 1995. Corrected reprint of the 1992 original. MR 1416564 (97e:14001)
  • [3] Xiaohong Jia, Falai Chen, and Jiansong Deng, Computing self-intersection curves of rational ruled surfaces, Comput. Aided Geom. Design 26 (2009), no. 3, 287-299. MR 2493078 (2010a:65032),
  • [4] Hyungju Park, Effective computation of singularities of parametric affine curves, J. Pure Appl. Algebra 173 (2002), no. 1, 49-58. MR 1912959 (2003f:14068),
  • [5] Sonia Pérez-Díaz, Computation of the singularities of parametric plane curves, J. Symbolic Comput. 42 (2007), no. 8, 835-857. MR 2345839 (2008g:14116),
  • [6] Sonia Pérez-Díaz and J. Rafael Sendra, Computation of the degree of rational surface parametrizations, J. Pure Appl. Algebra 193 (2004), no. 1-3, 99-121. MR 2076380 (2005d:14090),
  • [7] Sonia Pérez-Díaz and J. Rafael Sendra, Partial degree formulae for rational algebraic surfaces, ISSAC'05, ACM, New York, 2005, pp. 301-308. MR 2280561,
  • [8] Sonia Pérez-Díaz and J. Rafael Sendra, A univariate resultant-based implicitization algorithm for surfaces, J. Symbolic Comput. 43 (2008), no. 2, 118-139. MR 2357079 (2009b:14112),
  • [9] S. Pérez-Díaz, J. R. Sendra, and C. Villarino, A first approach towards normal parametrizations of algebraic surfaces, Internat. J. Algebra Comput. 20 (2010), no. 8, 977-990. MR 2747411 (2011m:68303),
  • [10] R. Rubio, J. M Serradilla, and M. P. Vélez, Detecting real singularities of a space curve from a real rational parametrization, J. Symbolic Comput. 44 (2009), no. 5, 490-498. MR 2499924 (2010h:14095),
  • [11] J. Rafael Sendra and Franz Winkler, Tracing index of rational curve parametrizations, Comput. Aided Geom. Design 18 (2001), no. 8, 771-795. MR 1857997 (2002h:65022),
  • [12] J. Rafael Sendra, Franz Winkler, and Sonia Pérez-Díaz, Rational Algebraic Curves, Algorithms and Computation in Mathematics, vol. 22, Springer, Berlin, 2008. A computer algebra approach. MR 2361646 (2009a:14073)
  • [13] Igor R. Shafarevich, Basic Algebraic Geometry. 1. Varieties in Projective Space, 2nd ed., Springer-Verlag, Berlin, 1994. Translated from the 1988 Russian edition and with notes by Miles Reid. MR 1328833 (95m:14001)
  • [14] F. Winkler, Polynomial Algorithms in Computer Algebra, Texts and Monographs in Symbolic Computation, Springer-Verlag, Vienna, 1996. MR 1408683 (97j:68063)
  • [15] O. Zariski, P. Samuel, Commutative Algebra, Volume I. Graduate Texts in Mathematics, vol. 28, Springer-Verlag, 1975.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 14Q10, 14J17, 68W30

Retrieve articles in all journals with MSC (2010): 14Q10, 14J17, 68W30

Additional Information

S. Pérez-Díaz
Affiliation: Dpto. de Física y Matemáticas, Universidad de Alcalá, E-28871 Madrid, Spain

J. R. Sendra
Affiliation: Dpto. de Física y Matemáticas, Universidad de Alcalá, E-28871 Madrid, Spain

C. Villarino
Affiliation: Dpto. de Física y Matemáticas, Universidad de Alcalá, E-28871 Madrid, Spain

Received by editor(s): December 25, 2011
Received by editor(s) in revised form: January 23, 2013, June 5, 2013, and October 27, 2013
Published electronically: October 9, 2014
Additional Notes: This work was partially supported by the Spanish Ministerio de Ciencia e Innovación under the project MTM2008-04699-C03-01 and by the Ministerio de Economía y Competitividad under the project MTM2011-25816-C02-01; the authors are members of the Research Group ASYNACS (Ref. CCEE2011/R34).
Article copyright: © Copyright 2014 American Mathematical Society

American Mathematical Society