Skip to main content
Log in

A simple homotopy for solving deficient polynomial systems

  • Published:
Japan Journal of Applied Mathematics Aims and scope Submit manuscript

Abstract

Most systems of polynomials which arise in applications have fewer than the expected number of solutions. A simple homotopy is presented for finding all solutions of such a “deficient” system. Different from current homotopies used for such systems, only one parameter is needed to regularize the problem. Within some limits an arbitrary starting problem can be chosen, as long as its solution set is known.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. P. Brunovský and P. Meravý, Solving systems of polynomial equations by bounded and real homotopy. Numer. Math.,43 (1984), 397–418.

    Article  MATH  MathSciNet  Google Scholar 

  2. M. Chu, T. Y. Li and T. Sauer, A homotopy method for general λ-matrix problems. SIAM J. Matrix Anal. Appl.,9 (1988), 528–536.

    Article  MATH  MathSciNet  Google Scholar 

  3. T. Y. Li, On locating all zeros of an analytic function within a bounded domain by a revised Relves/Lyness method. SIAM J. Numer. Anal.,20 (1983), 865–871.

    Article  MATH  MathSciNet  Google Scholar 

  4. T. Y. Li, On Chow, Mallet-Paret and Yorke homotopy for solving systems of polynomials. Bull. Inst. Math. Acad. Sinica,11 (1983), 433–437.

    MATH  MathSciNet  Google Scholar 

  5. T. Y. Li and T. Sauer, Regularity results for solving systems of polynomials by homotopy methods. Numer. Math.,50 (1987), 283–289.

    Article  MathSciNet  Google Scholar 

  6. T. Y. Li, T. Sauer and J. Yorke, Numerical solution of a class of deficient polynomials systems. SIAM J. Numer. Anal.,24 (1987), 435–451.

    Article  MATH  MathSciNet  Google Scholar 

  7. T. Y. Li, T. Sauer and J. Yorke, The random product homotopy and deficient polynomial systems. Numer. Math.,51 (1987), 481–500.

    Article  MATH  MathSciNet  Google Scholar 

  8. T. Y. Li and T. Sauer, Homotopy methods for generalized eigenvalue problems. Linear Algebra Appl.,91 (1987), 65–74.

    Article  MATH  MathSciNet  Google Scholar 

  9. E. Lorenz, The local structure of a chaotic attractor in four dimensions. Physica,13D (1984), 90–104.

    MathSciNet  Google Scholar 

  10. P. Meravý, Symmetric homotopies for solving systems of polynomial equations. To appear, Mathematica Slovaca.

  11. A. Morgan, A homotopy for solving polynomial systems. Appl. Math. Comput.,18 (1986), 87–92.

    Article  MATH  MathSciNet  Google Scholar 

  12. A Morgan and A. Sommese, A homotopy for solving general polynomial systems that respectm-homogeneous structures. Applied Math. Comp.,24 (1987), 95–114.

    Google Scholar 

  13. D. Mumford, Algebraic Geometry I Complex Projective Varieties. Springer-Verlag, New York, 1976.

    MATH  Google Scholar 

  14. L.-W. Tsai and A. P. Morgan, Solving the kinematics of the most general six- and five-degree-of-freedom manipulators by continuation methods. ASME J. Mechanisms, Transmissions and Automation in Design,107 (1985), 48–57.

    Google Scholar 

  15. B. L. Van der Waerden, Algebra. Vol. 2. New York, Ungar, 1970.

    Google Scholar 

  16. A. Wright, Finding all solutions to a system of polynomial equations. Math. Comp.,44 (1985), 125–133.

    Article  MATH  MathSciNet  Google Scholar 

  17. W. Zulehner, A simple homotopy method for determining all isolated solutions to polynomial systems. Math. Comp.,50 (1988), 167–177.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Research was supported in part by NSF under Grand DMS-8701349.

About this article

Cite this article

Li, TY., Sauer, T. A simple homotopy for solving deficient polynomial systems. Japan J. Appl. Math. 6, 409 (1989). https://doi.org/10.1007/BF03167887

Download citation

  • Received:

  • DOI: https://doi.org/10.1007/BF03167887

Key words

Navigation