Numerical Calculation of the Multiplicity

of a Solution to Algebraic Equations

Authors:
Hidetsune Kobayashi, Hideo Suzuki and Yoshihiko Sakai

Journal:
Math. Comp. **67** (1998), 257-270

MSC (1991):
Primary 14Q99, 65H10; Secondary 14N05, 65H20

MathSciNet review:
1434942

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Abstract | References | Similar Articles | Additional Information

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.

**1.**J. G. Semple and L. Roth,*Introduction to Algebraic Geometry*, Oxford, at the Clarendon Press, 1949. MR**0034048****2.**B. L. van der Waerden,*Einführung in die algebraische Geometrie*, Springer-Verlag, Berlin-New York, 1973 (German). Zweite Auflage; Die Grundlehren der mathematischen Wissenschaften, Band 51. MR**0344245****3.**H. Kobayashi, H. Suzuki and Y. Sakai:*Separation of close roots by linear fractional transformation*, Proc. of ASCM, 1-10, Scientists INC, 1995.**4.**William Fulton,*Algebraic curves. An introduction to algebraic geometry*, W. A. Benjamin, Inc., New York-Amsterdam, 1969. Notes written with the collaboration of Richard Weiss; Mathematics Lecture Notes Series. MR**0313252****5.**Alden H. Wright,*Finding all solutions to a system of polynomial equations*, Math. Comp.**44**(1985), no. 169, 125–133. MR**771035**, 10.1090/S0025-5718-1985-0771035-4

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Additional Information

**Hidetsune Kobayashi**

Affiliation:
Department of Mathematics, Nihon University, 1-8 Kanda-surugadai, Tokyo 101, Japan

Email:
hikoba@math.cst.nihon-u.ac.jp

**Hideo Suzuki**

Affiliation:
Tokyo Polytechnic College, 2-32-1 Ogawa-nishi Kodaira, Tokyo, Japan

**Yoshihiko Sakai**

Affiliation:
Visual Science Laboratory, Inc., 2-21 Kanda-awajicho, Chiyoda, Tokyo, Japan

Email:
sakai@vsl.co.jp

DOI:
http://dx.doi.org/10.1090/S0025-5718-98-00906-5

Keywords:
Multiplicity,
Zeuthen's rule,
homotopy method,
system of algebraic equations

Received by editor(s):
July 13, 1995

Received by editor(s) in revised form:
August 28, 1996

Article copyright:
© Copyright 1998
American Mathematical Society