Numerical calculation of the multiplicity of a solution to algebraic equations

Kobayashi, Hidetsune; Suzuki, Hideo; Sakai, Yoshihiko

doi:10.1090/S0025-5718-98-00906-5

Numerical calculation of the multiplicity of a solution to algebraic equations
HTML articles powered by AMS MathViewer

by Hidetsune Kobayashi, Hideo Suzuki and Yoshihiko Sakai PDF

Math. Comp. 67 (1998), 257-270 Request permission

Abstract:

A method to calculate numerically the multiplicity of a solution to a system of algebraic equations is presented. The method is an application of Zeuthen’s rule which gives the multiplicity of a solution as the multiplicity of a united point of an algebraic correspondence defined naturally by the system. The numerical calculation is applicable to a large scale system of algebraic equations which may have a solution that we cannot calculate the multiplicity by a symbolic computation.

References

Morgan Ward and R. P. Dilworth, The lattice theory of ova, Ann. of Math. (2) 40 (1939), 600–608. MR 11, DOI 10.2307/1968944
B. L. van der Waerden, Einführung in die algebraische Geometrie, Die Grundlehren der mathematischen Wissenschaften, Band 51, Springer-Verlag, Berlin-New York, 1973 (German). Zweite Auflage. MR 0344245
H. Kobayashi, H. Suzuki and Y. Sakai: Separation of close roots by linear fractional transformation, Proc. of ASCM, 1-10, Scientists INC, 1995.
William Fulton, Algebraic curves. An introduction to algebraic geometry, Mathematics Lecture Note Series, W. A. Benjamin, Inc., New York-Amsterdam, 1969. Notes written with the collaboration of Richard Weiss. MR 0313252
Alden H. Wright, Finding all solutions to a system of polynomial equations, Math. Comp. 44 (1985), no. 169, 125–133. MR 771035, DOI 10.1090/S0025-5718-1985-0771035-4