Real Solving of Bivariate Polynomial Systems

  • Ioannis Z. Emiris
  • Elias P. Tsigaridas
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3718)


We propose exact, complete and efficient methods for 2 problems: First, the real solving of systems of two bivariate rational polynomials of arbitrary degree. This means isolating all common real solutions in rational rectangles and calculating the respective multiplicities. Second, the computation of the sign of bivariate polynomials evaluated at two algebraic numbers of arbitrary degree. Our main motivation comes from nonlinear computational geometry and computer-aided design, where bivariate polynomials lie at the inner loop of many algorithms. The methods employed are based on Sturm-Habicht sequences, univariate resultants and rational univariate representation. We have implemented them very carefully, using advanced object-oriented programming techniques, so as to achieve high practical performance. The algorithms are integrated in the public-domain C++ software library synaps, and their efficiency is illustrated by 9 experiments against existing implementations. Our code is faster in most cases; sometimes it is even faster than numerical approaches.


Algebraic Number Algebraic Curf Polynomial System Minimal Polynomial Arbitrary Degree 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Basu, S., Pollack, R., Roy, M.-F.: Algorithms in Real Algebraic Geometry. Algorithms and Computation in Mathematics, vol. 10. Springer, Heidelberg (2003)Google Scholar
  2. 2.
    Busé, L., Mourrain, B., Khalil, H.: Resultant-based methods for curves intersection problems (manuscript) (2005)Google Scholar
  3. 3.
    Cox, D., Little, J., O’Shea, D.: Using Algebraic Geometry. Graduate Texts in Mathematics, vol. 185. Springer, New York (1998)zbMATHGoogle Scholar
  4. 4.
    Davenport, J.H., Siret, Y., Tournier, E.: Computer Algebra. Academic Press, London (1988)zbMATHGoogle Scholar
  5. 5.
    Dos Reis, G., Mourrain, B., Rouillier, R., Trébuchet, P.: An environment for symbolic and numeric computation. In: Proc. Int. Conf. Math. Software, pp. 239–249. World Scientific, Singapore (2002)CrossRefGoogle Scholar
  6. 6.
    Dupont, L., Lazard, D., Lazard, S., Petitjean, S.: Near-optimal parameterization of the intersection of quadrics. In: Proc. ACM SoCG, pp. 246–255 (June 2003)Google Scholar
  7. 7.
    Emiris, I.Z., Tsigaridas, E.P.: Real algebraic numbers and polynomial systems of small degree (manuscript) (2005),
  8. 8.
    Emiris, I.Z., Kakargias, A.V., Teillaud, M., Tsigaridas, E.P., Pion, S.: Towards an open curved kernel. In: Proc. ACM SoCG, New York, pp. 438–446 (2004)Google Scholar
  9. 9.
    Emiris, I.Z., Tsigaridas, E.P.: Computing with real algebraic numbers of small degree. In: Albers, S., Radzik, T. (eds.) ESA 2004. LNCS, vol. 3221, pp. 652–663. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  10. 10.
    Gonzalez-Vega, L., Necula, I.: Efficient topology determination of implicitly defined algebraic plane curves. Comp. Aided Geom. Design 19(9), 719–743 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Guibas, L.J., Karavelas, M.I., Russel, D.: A computational framework for handling motion. In: Proc. 6th Work. Algor. Engin. & Experim (ALENEX) (2004)Google Scholar
  12. 12.
    Hemmer, M., Schömer, E., Wolpert, N.: Computing a 3-dimensional cell in an arrangement of quadrics: Exactly and actually! In: Proc. Annual ACM Symp. Comput. Geometry, pp. 264–273 (2001)Google Scholar
  13. 13.
    El Kahoui, M.: Computing with algebraic curves in generic position (2005) (submitted),
  14. 14.
    Keyser, J., Culver, T., Manocha, D., Krishnan, S.: MAPC: A library for efficient and exact manipulation of algebraic points and curves. In: Proc. Annual ACM Symp. Comput. Geometry, June 1999, pp. 360–369. ACM Press, New York (1999)Google Scholar
  15. 15.
    Keyser, J., Culver, T., Manocha, D., Krishnan, S.: ESOLID: A system for exact boundary evaluation. Comp. Aided Design 36(2), 175–193 (2004)CrossRefGoogle Scholar
  16. 16.
    Keyser, J., Ouchi, K., Rojas, M.: The Exact Rational Univariate Representation for Detecting Degeneracies. DIMACS: Series in Discrete Mathematics and Theoretical Computer Science. AMS Press (2004) (to appear)Google Scholar
  17. 17.
    Lickteig, T., Roy, M.-F.: Semi-algebraic complexity of quotients and sign determination of remainders. J. Complexity 12(4), 545–571 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Mourrain, B., Pavone, J.P., Trébuchet, P., Tsigaridas, E.: SYNAPS, a library for symbolic-numeric computation. In: 8th Int. Symp. on Effective Methods in Algebraic Geometry, MEGA, Italy (May 2005)Google Scholar
  19. 19.
    Mourrain, B., Trébuchet, Ph.: Algebraic methods for numerical solving. In: Proc. of the 3rd International Workshop on Symbolic and Numeric Algorithms for Scientific Computing 2001, Timisoara, Romania, pp. 42–57 (2002)Google Scholar
  20. 20.
    Mourrain, B., Vrahatis, M., Yakoubsohn, J.C.: On the complexity of isolating real roots and computing with certainty the topological degree. J. Complexity 18(2) (2002)Google Scholar
  21. 21.
    Rouillier, F.: Solving zero-dimensional systems through the rational univariate representation. Journal of Applicable Algebra in Engineering, Communication and Computing 9(5), 433–461 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Sakkalis, T.: Signs of algebraic numbers. Computers and Mathematics, 131–134 (1989)Google Scholar
  23. 23.
    Weispfenning, V.: Solving parametric polynomial equations and inequalities by symbolic algorithms. In: Proc. Computer Algebra in Science and Engineering. World Scientific, Singapore (1995)Google Scholar
  24. 24.
    Wolpert, N., Seidel, R.: On the Exact Computation of the Topology of Real Algebraic Curves. In: Proc. ACM SoCG, Pisa, pp. 107–115 (2004)Google Scholar
  25. 25.
    Yap, C.K.: Fundamental Problems of Algorithmic Algebra. Oxford University Press, New York (2000)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Ioannis Z. Emiris
    • 1
  • Elias P. Tsigaridas
    • 1
  1. 1.Department of Informatics and TelecommunicationsNational University of AthensHELLAS

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