Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Solving real polynomial systems with real homotopies


Authors: T. Y. Li and Xiao Shen Wang
Journal: Math. Comp. 60 (1993), 669-680
MSC: Primary 65H20
DOI: https://doi.org/10.1090/S0025-5718-1993-1160275-5
MathSciNet review: 1160275
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: When a real homotopy is used for solving a polynomial system with real coefficients, bifurcation of some of the homotopy paths at singular points is inevitable. The main result of this paper shows that, generically, the solution set of a real homotopy contains no singular point other than a finite number of quadratic turning points. At a quadratic turning point, the bifurcation phenomenon is quite simple. It consists of two bifurcation branches with their tangent vectors being perpendicular to each other.


References [Enhancements On Off] (What's this?)

  • [1] E. L. Allgower, Bifurcations arising in the calculation of critical points via homotopy methods, Proceedings of the Conference at the University of Dortmund (H. D. M. T. Kupper and H. Weber, eds.), Birkhäuser Verlag, Basel, 1984, pp. 15-28. MR 821017 (88f:58014)
  • [2] E. L. Allgower, K. George, and R. Miranda, The method of resultants for computing real solutions of polynomial systems, SIAM J. Numer. Anal. 29 (1992), 831-844. MR 1163359 (93e:65079)
  • [3] P. Brunovsky and P. Meravy, Solving systems of polynomial equations by bounded and real homotopy, Numer. Math. 43 (1984), 379-418. MR 738385 (86c:58013)
  • [4] S. N. Chow, J. Mallet-Paret, and J. A. Yorke, A homotopy method for locating all zeros of a system of polynomials, Lecture Notes in Math., vol. 730, Springer-Verlag, Berlin and New York, 1979, pp. 77-88. MR 547982 (80k:58012)
  • [5] C. B. Garcia, An elimination method for finding all real isolated solutions to arbitrary square systems of polynomial equations, preprint.
  • [6] -, On the general solution to certain systems of polynomial equations, preprint.
  • [7] E. Henderson and H. B. Keller, Complex bifurcation from real paths, SIAM J. Appl. Math. 50 (1990), 460-482. MR 1043596 (91a:58135)
  • [8] D. Mumford, Algebraic geometry I: Complex projective varieties, Springer-Verlag, Berlin, Heidelberg, and New York, 1976. MR 0453732 (56:11992)
  • [9] P. Verlinden and A. Haegemans, About the real random product homotopy for solving systems of real polynomial equations, Bull. Soc. Math. Belg. Sér. (3) 41 (1989), 325-338. MR 1022757 (91e:65071)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65H20

Retrieve articles in all journals with MSC: 65H20


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1993-1160275-5
Article copyright: © Copyright 1993 American Mathematical Society

American Mathematical Society