A Standard Basis Free Algorithm for Computing the Tangent Cones of a Space Curve

  • Parisa Alvandi
  • Marc Moreno Maza
  • Éric Schost
  • Paul Vrbik
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9301)


We outline a method for computing the tangent cone of a space curve at any of its points. We rely on the theory of regular chains and Puiseux series expansions. Our approach is novel in that it explicitly constructs the tangent cone at arbitrary and possibly irrational points without using a standard basis.


Computational algebraic geometry tangent cone regular chain Puiseux series 


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  1. 1.
    Alvandi, P., Chen, C., Marcus, S., Maza, M.M., Schost, É., Vrbik, P.: Doing algebraic geometry with the regularchains library. In: Hong, H., Yap, C. (eds.) ICMS 2014. LNCS, vol. 8592, pp. 472–479. Springer, Heidelberg (2014)Google Scholar
  2. 2.
    Alvandi, P., Chen, C., Maza, M.M.: Computing the limit points of the quasi-component of a regular chain in dimension one. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2013. LNCS, vol. 8136, pp. 30–45. Springer, Heidelberg (2013)Google Scholar
  3. 3.
    Chen, C., Maza, M.M.: Algorithms for computing triangular decomposition of polynomial systems. J. Symb. Comput. 47(6), 610–642 (2012)Google Scholar
  4. 4.
    Cox, D., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms, 1st edn. Spinger (1992)Google Scholar
  5. 5.
    Cox, D., Little, J., O’Shea, D.: Using Algebraic Geometry. Graduate Text in Mathematics, vol. 185. Springer, New York (1998)Google Scholar
  6. 6.
    Fulton, W.: Introduction to intersection theory in algebraic geometry. CBMS Regional Conference Series in Mathematics, vol. 54. Published for the Conference Board of the Mathematical Sciences, Washington, DC (1984)Google Scholar
  7. 7.
    Fulton, W.: Algebraic curves. Advanced Book Classics. Addison-Wesley (1989)Google Scholar
  8. 8.
    Marcus, S., Maza, M.M., Vrbik, P.: On fulton’s algorithm for computing intersection multiplicities. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2012. LNCS, vol. 7442, pp. 198–211. Springer, Heidelberg (2012)Google Scholar
  9. 9.
    Mora, F.: An algorithm to compute the equations of tangent cones. In: Calmet, J. (ed.) Computer Algebra. LNCS, vol. 144, pp. 158–165. Springer, Berlin Heidelberg (1982)Google Scholar
  10. 10.
    Mora, T., Pfister, G., Traverso, C.: An introduction to the tangent cone algorithm issues in robotics and non-linear geometry. Advances in Computing Research 6, 199–270 (1992)Google Scholar
  11. 11.
    Mumford, D.: The Red Book of Varieties and Schemes, 2nd edn., Springer-Verlag (1999)Google Scholar
  12. 12.
    Shafarevich, I.R.: Basic algebraic geometry. 1, 2nd edn. Springer, Berlin (1994)Google Scholar
  13. 13.
    Vrbik, P.: Computing Intersection Multiplicity via Triangular Decomposition. PhD thesis, The University of Western Ontario (2014)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Parisa Alvandi
    • 1
  • Marc Moreno Maza
    • 1
  • Éric Schost
    • 1
  • Paul Vrbik
    • 2
  1. 1.Department of Computer ScienceUniversity of Western OntarioLondonCanada
  2. 2.School of Mathematical and Physical SciencesThe University of Newcastle AustraliaCallaghanAustralia

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