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Local Generic Position for Root Isolation of Zero-Dimensional Triangular Polynomial Systems

  • Jia Li
  • Jin-San Cheng
  • Elias P. Tsigaridas
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7442)

Abstract

We present an algorithm to isolate the real roots, and compute their multiplicities, of a zero-dimensional triangular polynomial system, based on the local generic position method. We also presentexperiments that demonstrate the efficiency of the method.

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References

  1. 1.
    Alonso, M.-E., Becker, E., Roy, M.-F., Wörmann, T.: Multiplicities and idempotents for zero dimensional systems. In: Algorithms in algebraic Geometry and Applications. Progress in Mathematics, vol. 143, pp. 1–20. Birkhäuser (1996)Google Scholar
  2. 2.
    Berberich, E., Kerber, M., Sagraloff, M.: Exact Geometric-Topological Analysis of Algebraic Surfaces. In: Teillaud, M. (ed.) Proc. of the 24th ACM Symp. on Computational Geometry (SoCG), pp. 164–173. ACM Press (2008)Google Scholar
  3. 3.
    Boulier, F., Chen, C., Lemaire, F., Moreno Maza, M.: Real Root Isolation of Regular Chains. In: ASCM 2009, pp. 1–15 (2009)Google Scholar
  4. 4.
    Cheng, J.S., Gao, X.S., Li, J.: Root isolation for bivariate polynomial systems with local generic position method. In: ISSAC 2009, pp. 103–110 (2009)Google Scholar
  5. 5.
    Cheng, J.S., Gao, X.S., Guo, L.: Root isolation of zero-dimensional polynomial systems with linear univariate representation. J. of Symbolic Computation (2011)Google Scholar
  6. 6.
    Cheng, J.S., Gao, X.S., Li, M.: Determining the Topology of Real Algebraic Surfaces. In: Martin, R., Bez, H.E., Sabin, M.A. (eds.) IMA 2005. LNCS, vol. 3604, pp. 121–146. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  7. 7.
    Cheng, J.S., Gao, X.S., Yap, C.K.: Complete Numerical Isolation of Real Roots in 0-dimensional Triangular Systems. JSC 44(7), 768–785 (2009)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Collins, G.E., Johnson, J.R., Krandick, W.: Interval arithmetic in cylindrical algebraic decomposition. Journal of Symbolic Computation 34, 145–157 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Eigenwillig, A., Kettner, L., Krandick, W., Mehlhorn, K., Schmitt, S., Wolpert, N.: A Descartes Algorithm for Polynomials with Bit-Stream Coefficients. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2005. LNCS, vol. 3718, pp. 138–149. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  10. 10.
    Emiris, I.Z., Mourrain, B., Tsigaridas, E.P.: The DMM bound: Multivariate (aggregate) separation bounds. In: ISSAC 2010, pp. 243–250. ACM, Germany (2010)Google Scholar
  11. 11.
    Fulton, W.: Introduction to intersection theory in algebraic geometry. CBMS Regional Conference Series in Mathematics, vol. 54. Conference Board of the Mathematical Sciences, Washington, DC (1984)Google Scholar
  12. 12.
    Gao, X.S., Chou, S.C.: On the theory of resolvents and its applications. Mathematics and Systems Science (1997)Google Scholar
  13. 13.
    Hong, H.: An Efficient Method for Analyzing the Topology of Plane Real Algebraic Curves. Mathematics and Computers in Simulation 42, 571–582 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Hong, H., Stahl, V.: Safe start region by fixed points and tightening. Computing 53(3-4), 323–335 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Lu, Z., He, B., Luo, Y., Pan, L.: An Algorithm of Real Root Isolation for Polynomial Systems. In: SNC 2005 (2005)Google Scholar
  16. 16.
    Rioboo, R.: Computation of the real closure of an ordered field. In: ISSAC 1992. Academic Press, San Francisco (1992)Google Scholar
  17. 17.
    Rouillier, F.: Solving zero-dimensional systems through the rational univariate representation. AAECC 9, 433–461 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Sagraloff, M.: When Newton meets Descartes: A Simple and Fast Algorithm to Isolate the Real Roots of a Polynomial. CoRR abs/1109.6279 (2011)Google Scholar
  19. 19.
    Xia, B., Zhang, T.: Real Solution Isolation Using Interval Arithmetic. Computers and Mathematics with Applications 52, 853–860 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Yap, C.: Fundamental Problems of Algorithmic Algebra. Oxford University Press, New York (2000)zbMATHGoogle Scholar
  21. 21.
    Yap, C., Sagraloff, M.: A simple but exact and efficient algorithm for complex root isolation. In: ISSAC 2011, pp. 353–360 (2011)Google Scholar
  22. 22.
    Zhang, Z.H., Fang, T., Xia, B.C.: Real solution isolation with multiplicity of 0-dimensional triangular systems. Science China: Information Sciences 54(1), 60–69 (2011)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Jia Li
    • 1
  • Jin-San Cheng
    • 2
  • Elias P. Tsigaridas
    • 3
  1. 1.Beijing Electronic Science and Technology InstituteChina
  2. 2.KLMM, AMSSChinese Academy of SciencesChina
  3. 3.POLSYS projectINRIA, LIP6/CNRSFrance

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