Skip to Main Content

Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.78.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Solving real polynomial systems with real homotopies
HTML articles powered by AMS MathViewer

by T. Y. Li and Xiao Shen Wang PDF
Math. Comp. 60 (1993), 669-680 Request permission

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
  • Eugene L. Allgower, Bifurcations arising in the calculation of critical points via homotopy methods, Numerical methods for bifurcation problems (Dortmund, 1983) Internat. Schriftenreihe Numer. Math., vol. 70, Birkhäuser, Basel, 1984, pp. 15–28. MR 821017
  • Eugene L. Allgower, Kurt Georg, and Rick Miranda, The method of resultants for computing real solutions of polynomial systems, SIAM J. Numer. Anal. 29 (1992), no. 3, 831–844. MR 1163359, DOI 10.1137/0729051
  • Pavol Brunovský and Pavol Meravý, Solving systems of polynomial equations by bounded and real homotopy, Numer. Math. 43 (1984), no. 3, 397–418. MR 738385, DOI 10.1007/BF01390182
  • Shui Nee Chow, John Mallet-Paret, and James A. Yorke, A homotopy method for locating all zeros of a system of polynomials, Functional differential equations and approximation of fixed points (Proc. Summer School and Conf., Univ. Bonn, Bonn, 1978) Lecture Notes in Math., vol. 730, Springer, Berlin, 1979, pp. 77–88. MR 547982
    • C. B. Garcia, An elimination method for finding all real isolated solutions to arbitrary square systems of polynomial equations, preprint. —, On the general solution to certain systems of polynomial equations, preprint.
  • M. E. Henderson and H. B. Keller, Complex bifurcation from real paths, SIAM J. Appl. Math. 50 (1990), no. 2, 460–482. MR 1043596, DOI 10.1137/0150027
  • David Mumford, Algebraic geometry. I, Grundlehren der Mathematischen Wissenschaften, No. 221, Springer-Verlag, Berlin-New York, 1976. Complex projective varieties. MR 0453732
  • P. Verlinden and A. Haegemans, Real homotopy for solving systems of real polynomial equations, Bull. Soc. Math. Belg. Sér. B 41 (1989), no. 3, 325–338. MR 1022757
Similar Articles
  • Retrieve articles in Mathematics of Computation with MSC: 65H20
  • Retrieve articles in all journals with MSC: 65H20
Additional Information
  • © Copyright 1993 American Mathematical Society
  • Journal: Math. Comp. 60 (1993), 669-680
  • MSC: Primary 65H20
  • DOI: https://doi.org/10.1090/S0025-5718-1993-1160275-5
  • MathSciNet review: 1160275