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.
Similar content being viewed by others
References
P. Brunovský and P. Meravý, Solving systems of polynomial equations by bounded and real homotopy. Numer. Math.,43 (1984), 397–418.
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.
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.
T. Y. Li, On Chow, Mallet-Paret and Yorke homotopy for solving systems of polynomials. Bull. Inst. Math. Acad. Sinica,11 (1983), 433–437.
T. Y. Li and T. Sauer, Regularity results for solving systems of polynomials by homotopy methods. Numer. Math.,50 (1987), 283–289.
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.
T. Y. Li, T. Sauer and J. Yorke, The random product homotopy and deficient polynomial systems. Numer. Math.,51 (1987), 481–500.
T. Y. Li and T. Sauer, Homotopy methods for generalized eigenvalue problems. Linear Algebra Appl.,91 (1987), 65–74.
E. Lorenz, The local structure of a chaotic attractor in four dimensions. Physica,13D (1984), 90–104.
P. Meravý, Symmetric homotopies for solving systems of polynomial equations. To appear, Mathematica Slovaca.
A. Morgan, A homotopy for solving polynomial systems. Appl. Math. Comput.,18 (1986), 87–92.
A Morgan and A. Sommese, A homotopy for solving general polynomial systems that respectm-homogeneous structures. Applied Math. Comp.,24 (1987), 95–114.
D. Mumford, Algebraic Geometry I Complex Projective Varieties. Springer-Verlag, New York, 1976.
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.
B. L. Van der Waerden, Algebra. Vol. 2. New York, Ungar, 1970.
A. Wright, Finding all solutions to a system of polynomial equations. Math. Comp.,44 (1985), 125–133.
W. Zulehner, A simple homotopy method for determining all isolated solutions to polynomial systems. Math. Comp.,50 (1988), 167–177.
Author information
Authors and Affiliations
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
Received:
DOI: https://doi.org/10.1007/BF03167887