Finding all solutions to a system of polynomial equations

Author:
Alden H. Wright

Journal:
Math. Comp. **44** (1985), 125-133

MSC:
Primary 12D05; Secondary 65H10

DOI:
https://doi.org/10.1090/S0025-5718-1985-0771035-4

MathSciNet review:
771035

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Abstract | References | Similar Articles | Additional Information

Abstract: Given a polynomial equation of degree *d* over the complex domain, the Fundamental Theorem of Algebra tells us that there are *d* solutions, assuming that the solutions are counted by multiplicity. These solutions can be approximated by deforming a standard *n*th degree equation into the given equation, and following the solutions through the deformation. This is called the homotopy method. The Fundamental Theorem of Algebra can be proved by the same technique.

In this paper we extend these results and methods to a system of *n* polynomial equations in *n* complex variables. We show that the number of solutions to such a system is the product of the degrees of the equations (assuming that infinite solutions are included and solutions are counted by multiplicity). The proof is based on a homotopy, or deformation, from a standard system of equations with the same degrees and known solutions. This homotopy provides a computational method of approximating all solutions. Computational results demonstrating the feasibility of this method are also presented.

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1985-0771035-4

Keywords:
Systems of nonlinear equations,
homotopy methods,
systems of polynomial equations

Article copyright:
© Copyright 1985
American Mathematical Society