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**Numerically determining solutions of systems of polynomial equations.**
*(English)*
Zbl 0651.65042

The authors describe some homotopy methods for which the computational work to find all solutions of a system of n polynomial equations in n unknowns is proportional to the actual number of solutions instead of being proportional to the total degree. In this respect two theorems assuring smoothness and accessibility properties of the homotopy path from the trivial solution of a random system to a solution of the initial system are stated and a sketch of the proofs is given.

Reviewer: F.Luban

### MSC:

65H10 | Numerical computation of solutions to systems of equations |

### References:

[1] | Eugene Allgower and Kurt Georg, Simplicial and continuation methods for approximating fixed points and solutions to systems of equations, SIAM Rev. 22 (1980), no. 1, 28 – 85. · Zbl 0432.65027 · doi:10.1137/1022003 |

[2] | 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. · Zbl 0427.65034 |

[3] | Franz-Josef Drexler, Eine Methode zur Berechnung sämtlicher Lösungen von Polynomgleichungssystemen, Numer. Math. 29 (1977/78), no. 1, 45 – 58 (German, with English summary). · Zbl 0352.65023 · doi:10.1007/BF01389312 |

[4] | 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. · Zbl 0409.65026 · doi:10.1007/BF01582106 |

[5] | Tien-Yien Li, On Chow, Mallet-Paret and Yorke homotopy for solving system of polynomials, Bull. Inst. Math. Acad. Sinica 11 (1983), no. 3, 433 – 437. · Zbl 0538.65030 |

[6] | A. P. Morgan and L.-W. Tsai, Solving the kinematics of the most general six- and five-degree-of-freedom manipulators by continuation methods, ASME J. of Mechanisms, Transmissions and Automation in Design 107 (1985), 48-57. |

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