About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 690341, 7 pages
http://dx.doi.org/10.1155/2013/690341
Research Article

The Viro Method for Construction of Piecewise Algebraic Hypersurfaces

1Department of Information and Computing Science, Zhejiang Gongshang University, Hangzhou 310018, China
2Institute of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China

Received 30 July 2013; Accepted 22 August 2013

Academic Editor: Lawrence Narici

Copyright © 2013 Yisheng Lai et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. Y. S. Lai, R. H. Wang, and J. M. Wu, “Real zeros of the zero-dimensional parametric piecewise algebraic variety,” Science in China A, vol. 52, no. 4, pp. 817–832, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. Y. S. Lai, R. H. Wang, and J. M. Wu, “Solving parametric piecewise polynomial systems,” Journal of Computational and Applied Mathematics, vol. 236, no. 5, pp. 924–936, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. R.-H. Wang, X. Q. Shi, Z. X. Luo, and Z. X. Su, Multivariate spline and its Applications, Science Press, Beijing, China, Kluwer Academic Publishers, New York, NY, USA, 2001. View at MathSciNet
  4. O. Ya. Viro, “Gluing of algebraic hypersurfaces, smoothing of singularities and constructionof curves,” in Proceedings of the Leningrad International Topological Conference, pp. 149–197, Nauka, Leningrad, Russia, 1983 (Russian).
  5. O. Ya. Viro, “Gluing of plane real algebraic curves and constructions of curves of degrees 6 and 7,” in Topology (Leningrad, 1982), vol. 1060 of Lecture Notes in Mathematics, pp. 187–200, Springer, Berlin, Germany, 1984. View at Zentralblatt MATH · View at MathSciNet
  6. O. Ya. Viro, “Patchworking real algebraic varieties,” http://www.math.sunysb.edu/~oleg/pw.pdf.
  7. E. Shustin, “Critical points of real polynomials, subdivisions of Newton polyhedra and topology of real algebraic hypersurfaces,” Transactions of the American Mathematical Society, vol. 173, no. 2, pp. 203–223, 1996.
  8. E. Shustin, “Gluing of singular and critical points,” Topology, vol. 37, no. 1, pp. 195–217, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. E. Shustin, “Real plane algebraic curves with prescribed singularities,” Topology, vol. 32, no. 4, pp. 845–856, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. F. Bihan, “Viro method for the construction of real complete intersections,” Advances in Mathematics, vol. 169, no. 2, pp. 177–186, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. J. J. Risler, “Construction d’hypersurfaces réelles [d'après Viro],” in Séminaire Bourbaki, vol. 1992/93 of Astérique no. 216, Exp. 763:3, pp. 69–86, 1993.
  12. I. Itenberg and E. Shustin, “Singular points and limit cycles of planar polynomial vector fields,” Duke Mathematical Journal, vol. 102, no. 1, pp. 1–37, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. I. Itenberg and E. Shustin, “Viro theorem and topology of real and complex combinatorial hypersurfaces,” Israel Journal of Mathematics, vol. 133, no. 1, pp. 189–238, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. B. Sturmfels, “Viro's theorem for complete intersections,” Annali della Scuola Normale Superiore di Pisa, vol. 21, no. 3, pp. 377–386, 1994. View at Zentralblatt MATH · View at MathSciNet
  15. G. Farin, Curves and Surfaces for Computer-Aided Geometric Design, A Practical Guide (Computer Science and Scientific Computing), Academic Press, San Diego, Calif, USA, 4th edition, 1997. View at MathSciNet
  16. C. M. Hoffmann, Geometric and Solid Modeling, Morgan Kaufmann, San Mateo, Calif, USA, 1989.
  17. J. Hoschek and D. Lasser, Fundamentals of Computer Aided Geometric Design, A.K. Peters, Boston, Mass, USA, 1993. View at MathSciNet
  18. Y. S. Lai, W. P. Du, and R. H. Wang, “The Viro method for construction of Bernstein-Bézier algebraic hypersurface piece,” Science China Mathematics, vol. 55, no. 6, pp. 1269–1279, 2012. View at Publisher · View at Google Scholar · View at MathSciNet