Graphs and Combinatorics

, Volume 27, Issue 3, pp 419–430 | Cite as

A Note on the Complexity of Real Algebraic Hypersurfaces

Proceedings Paper

Abstract

Given an algebraic hypersurface \({\fancyscript{O}}\) in \({\mathbb{R}^d}\), how many simplices are necessary for a simplicial complex isotopic to \({\fancyscript{O}}\)? We address this problem and the variant where all vertices of the complex must lie on \({\fancyscript{O}}\). We give asymptotically tight worst-case bounds for algebraic plane curves. Our results gradually improve known bounds in higher dimensions; however, the question for tight bounds remains unsolved for d ≥ 3.

Keywords

Algebraic curves Algebraic surfaces Triangulation Isotopy 

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References

  1. 1.
    Alberti L., Mourrain B., Técourt J.-P.: Isotopic triangulation of a real algebraic surface. J. Symb. Comp. 44, 1291–1310 (2009)CrossRefMATHGoogle Scholar
  2. 2.
    Barnette D., Edelson A.: All 2-manifolds have finitely many minimal triangulations. Israel J. Math. 67, 123–128 (1989)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Basu S., Pollack R., Roy M.-F.: Algorithms in Real Algebraic Geometry, Algorithms and Computation in Mathematics, vol. 10. 2nd edn. Springer, Berlin (2006)Google Scholar
  4. 4.
    Berberich, E., Kerber, M., Sagraloff, M.: An Efficient Algorithm for the Stratification and Triangulation of Algebraic Surfaces. Computational Geometry: Theory and Applications, vol. 43, pp. 257–278 (special issue on SoCG’08, 2010)Google Scholar
  5. 5.
    de Berg M., van Kreveld M., Overmars M., Schwarzkopf O.: Computational Geometry: Algorithms and Applications. 2nd edn. Springer, Berlin (2000)MATHGoogle Scholar
  6. 6.
    Boissonnat, J.-D., Teillaud, M. (eds): Effective Computational Geometry for Curves and Surfaces. Springer, Berlin (2006)Google Scholar
  7. 7.
    Bredon G.: Topology and Geometry. Springer, New York (1997)MATHGoogle Scholar
  8. 8.
    Caviness, B.F., Johnson, J.R. (eds): Quantifier Elimination and Cylindrical Algebraic Decomposition, Texts and Monographs in Symbolic Computation. Springer, Berlin (1998)Google Scholar
  9. 9.
    Cheng, J., Lazard, S., Penaranda, L., Pouget, M., Rouillier, F., Tsigaridas, E.: On the topology of planar algebraic curves, pp. 361–370. In: SCG ’09: Proceedings of the 25th Annual Symposium on Computational Geometry. ACM, New York, NY, USA (2009)Google Scholar
  10. 10.
    Cox D., Little J., O’Shea D.: Using Algebraic Geometry. Springer, New York (1998)MATHGoogle Scholar
  11. 11.
    Diochnos D.I., Emiris I.Z., Tsigaridas E.P.: On the asymptotic and practical complexity of solving bivariate systems over the reals. J. Symb. Comp. 44, 818–835 (2009)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Eigenwillig, A., Kerber, M., Wolpert, N.: Fast and exact geometric analysis of real algebraic plane curves, pp. 151–158. In: Proceedings of the 2007 International Symposium on Symbolic and Algebraic Computation (ISSAC 2007)Google Scholar
  13. 13.
    Fulton W.: Intersection Theory. Springer, New York (1998)MATHGoogle Scholar
  14. 14.
    Kerber, M.: Geometric Algorithms for Algebraic Curves and Surfaces. Ph.D. thesis. Universität des Saarlandes, Germany (2009)Google Scholar
  15. 15.
    Lavrenchenko S.A.: Irreducible triangulations of the torus. J. Math. Sci. 51, 2537–2543 (1990)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Nakamoto A., Ota K.: Note on irreducible triangulations of surfaces. J. Graph Theory 20, 227–233 (1995)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer 2011

Authors and Affiliations

  1. 1.IST (Institute of Science and Technology) AustriaKlosterneuburgAustria
  2. 2.Max-Planck-Institute for InformaticsSaarbrückenGermany

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