## Finding all solutions to a system of polynomial equations

HTML articles powered by AMS MathViewer

- by Alden H. Wright PDF
- Math. Comp.
**44**(1985), 125-133 Request permission

## 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)$^\ast$. 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.

## References

- Ralph Abraham and Joel Robbin,
*Transversal mappings and flows*, W. A. Benjamin, Inc., New York-Amsterdam, 1967. An appendix by Al Kelley. MR**0240836**
S. N. Chow, J. Mallet-Paret & J. A. Yorke, "A homotopy method for locating all zeros of a system of polynomials," - Shui Nee Chow, John Mallet-Paret, and James A. Yorke,
*Finding zeroes of maps: homotopy methods that are constructive with probability one*, Math. Comp.**32**(1978), no. 143, 887–899. MR**492046**, DOI 10.1090/S0025-5718-1978-0492046-9 - F. J. Drexler,
*A homotopy method for the calculation of all zeros of zero-dimensional polynomial ideals*, Developments in statistics, Vol. 1, Academic Press, New York, 1978, pp. 69–93. MR**505422** - Franz-Josef Drexler,
*Eine Methode zur Berechnung sämtlicher Lösungen von Polynomgleichungssystemen*, Numer. Math.**29**(1977/78), no. 1, 45–58 (German, with English summary). MR**483386**, DOI 10.1007/BF01389312 - C. B. García and Tien-Yian Li,
*On the number of solutions to polynomial systems of equations*, SIAM J. Numer. Anal.**17**(1980), no. 4, 540–546. MR**584729**, DOI 10.1137/0717046 - C. B. García and W. I. Zangwill,
*Finding all solutions to polynomial systems and other systems of equations*, Math. Programming**16**(1979), no. 2, 159–176. MR**527572**, DOI 10.1007/BF01582106 - Morris W. Hirsch,
*Differential topology*, Graduate Texts in Mathematics, No. 33, Springer-Verlag, New York-Heidelberg, 1976. MR**0448362** - R. W. Klopfenstein,
*Zeros of nonlinear functions*, J. Assoc. Comput. Mach.**8**(1961), 366–373. MR**130113**, DOI 10.1145/321075.321080 - Masakazu Kojima and Shinji Mizuno,
*Computation of all solutions to a system of polynomial equations*, Math. Programming**25**(1983), no. 2, 131–157. MR**686876**, DOI 10.1007/BF02591768
B. L. van der Waerden, "Die Alternative bei nichtlinearen Gleichungen,"

*Functional Differential Equations and Approximation of Fixed Points*(Proceedings, Bonn, 1978), Lecture Notes in Math., Vol. 730, Springer-Verlag, Berlin and New York, 1979.

*Nachrichten der Gesellschaft der Wissenschaften zu Göttingen*, Math. Phys. Klasse, 1928, pp. 77-87.

## Additional Information

- © Copyright 1985 American Mathematical Society
- 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