Skip to main content

Intrinsic Near Quadratic Complexity Bounds for Real Multivariate Root Counting

  • Conference paper
  • First Online:
Algorithms — ESA’ 98 (ESA 1998)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1461))

Included in the following conference series:

  • 962 Accesses

This research was completed at City University of Hong Kong and partially funded by a National Science Foundation Mathematical Sciences Postdoctoral Fellowship.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bernshtein, D. N., “The Number of Roots of a System of Equations”, Functional Analysis and its Applications (translated from Russian), Vol. 9, No. 2, (1975), pp. 183–185.

    Article  MATH  Google Scholar 

  2. Basu, S., Pollack, R., and Roy, M.-F., “On the Combinatorial and Algebraic Complexity of Quantifier Elimination,” J. ACM 43 (1996), no. 6, pp. 1002–1045.

    Article  MATH  MathSciNet  Google Scholar 

  3. Bini, Dario and Pan, Victor Y. Polynomial and Matrix Computations, Volume 1: Fundamental Algorithms, Progress in Theoretical Computer Science, Birkhäuser, 1994.

    Google Scholar 

  4. Canny, John F., “Generalized Characteristic Polynomials,” Symbolic and Algebraic Computation (Rome, 1988), pp. 293–299, Lecture Notes in Comput. Sci., 358, Springer, Berlin, 1989.

    Google Scholar 

  5. _____, “Improved Algorithms for Sign Determination and Existential Quantifier Elimination,” Comput. J. 36 (1993), no. 5, pp. 409–418.

    Article  MATH  MathSciNet  Google Scholar 

  6. Emiris, Ioannis Z., “Sparse Elimination and Applications in Kinematics,” Ph.D. dissertation, Computer Science Division, U. C. Berkeley (December, 1994).

    Google Scholar 

  7. Emiris, Ioannis Z., “On the Complexity of Sparse Elimination,” J. Complexity 12 (1996), no. 2, pp. 134–136.

    Article  MATH  MathSciNet  Google Scholar 

  8. Emiris, Ioannis Z. and Canny, John F., “Efficient Incremental Algorithms for the Sparse Resultant and the Mixed Volume,” Journal of Symbolic Computation, vol. 20 (1995), pp. 117–149.

    Article  MATH  MathSciNet  Google Scholar 

  9. Ewald, Günter, Combinatorial Convexity and Algebraic Geometry, Graduate Texts in Mathematics, 168, Springer-Verlag, New York, 1996.

    MATH  Google Scholar 

  10. Gel’fand, I. M., Kapranov, M. M., and Zelevinsky, A. V., Discriminants, Resultants and Multidimensional Determinants, Birkhäuser, Boston, 1994.

    Google Scholar 

  11. Gritzmann, Peter and Klee, Victor, “On the Complexity of Some Basic Problems in Computational Convexity II: Volume and Mixed Volumes,” Polytopes: Abstract, Convex, and Computational (Scarborough, ON, 1993), pp. 373–466, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 440, Kluwer Acad. Publ., Dordrecht, 1994.

    Google Scholar 

  12. Gritzmann, Peter and Sturmfels, Bernd, “Minkowski Addition of Polytopes: Computational Complexity and Applications to Gröbner Bases,” SIAM J. Discrete Math. 6 (1993), no. 2, pp. 246–269.

    Article  MATH  MathSciNet  Google Scholar 

  13. Manocha, Dinesh and Canny, John “Real Time Inverse Kinematics for General 6R Manipulators,” Technical Report, University of California, Berkeley, 1992.

    Google Scholar 

  14. Mourrain, Bernard and Pan, Victor, “Asymptotic Acceleration of Solving Multivariate Polynomial Systems of Equations,” STOC’ 98, ACM Press, 1998.

    Google Scholar 

  15. Mumford, David, “Algebraic Geometry I: Complex Algebraic Varieties,” reprint of 1976 edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995, x+186 pp.

    MATH  Google Scholar 

  16. Preparata, Franco P. and Shamos, Michael Ian, Computational Geometry: An Introduction, Texts and Monographs in Computer Science, Springer-Verlag, New York-Berlin, 1985.

    Google Scholar 

  17. Renegar, Jim, “On the Worst Case Arithmetic Complexity of Approximating Zeros of Systems of Polynomials,” Technical Report, School of Operations Research and Industrial Engineering, Cornell University.

    Google Scholar 

  18. Rojas, J. Maurice, “Toric Laminations, Sparse Generalized Characteristic Polynomials, and a Refinement of Hilbert’s Tenth Problem,” Foundations of Computational Mathematics, selected papers of a conference, held at IMPA in Rio de Janeiro, January 1997, Springer-Verlag (1997).

    Google Scholar 

  19. _____, “Linear Algebra for Large Nonlinear Algebra over the Reals,” extended abstract, Sixth SIAM Conference on Applied Linear Algebra, Snowbird, Utah, October, 1997.

    Google Scholar 

  20. _____, “Affine Elimination Theory,” extended abstract, Proceedings of a Conference in Honor of David A. Buchsbaum, October, 1997, Northeastern University.

    Google Scholar 

  21. _____, “Toric Intersection Theory for Affine Root Counting,” Journal of Pure and Applied Algebra, to appear.

    Google Scholar 

  22. _____, “Twisted Chow Forms and Toric Perturbations for Degenerate Polynomial Systems,” submitted.

    Google Scholar 

  23. Roy, Marie-Françoise, “Basic Algorithms in Real Algebraic Geometry and their Complexity: from Sturm’s Theorem to the Existential Theory of Reals,” Lectures in Real Geometry (Madrid, 1994), pp. 1–67, de Gruyter Exp. Math., 23, de Gruyter, Berlin, 1996.

    Google Scholar 

  24. Sturmfels, Bernd, “Sparse Elimination Theory,” In D. Eisenbud and L. Robbiano, editors, Proc. Computat. Algebraic Geom. and Commut. Algebra 1991, pages 377–396, Cortona, Italy, 1993, Cambridge Univ. Press.

    Google Scholar 

  25. _____, “On the Newton Polytope of the Resultant,” Journal of Algebraic Combinatorics, 3: 207–236, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  26. _____, “Introduction to Resultants,” Notes from a lecture presented at an AMS short course on Applications of Computational Algebraic Geometry, San Diego, January 6–7, 1997.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1998 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Rojas, J.M. (1998). Intrinsic Near Quadratic Complexity Bounds for Real Multivariate Root Counting. In: Bilardi, G., Italiano, G.F., Pietracaprina, A., Pucci, G. (eds) Algorithms — ESA’ 98. ESA 1998. Lecture Notes in Computer Science, vol 1461. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-68530-8_11

Download citation

  • DOI: https://doi.org/10.1007/3-540-68530-8_11

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-64848-2

  • Online ISBN: 978-3-540-68530-2

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics