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**Complexity of Bezout’s theorem. V: Polynomial time.**
*(English)*
Zbl 0846.65022

Summary: [For part III see J. Complexity 9, No. 1, 4-14 (1993; Zbl 0846.65018). For part IV see SIAM J. Numer. Anal. 33, No. 1, 128-148 (1996; Zbl 0843.65035).]

We show that there are algorithms which find an approximate zero of a system of polynomial equations and which function in polynomial time on the average. The number of arithmetic operations is \(cN^{4s}\), where \(N\) is the input size and \(c\) a universal constant.

We show that there are algorithms which find an approximate zero of a system of polynomial equations and which function in polynomial time on the average. The number of arithmetic operations is \(cN^{4s}\), where \(N\) is the input size and \(c\) a universal constant.

### MSC:

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

12Y05 | Computational aspects of field theory and polynomials (MSC2010) |

65Y20 | Complexity and performance of numerical algorithms |

30C15 | Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral) |

### Keywords:

complexity; Bezout’s theorem; algorithms; approximate zero; system of polynomial equations; polynomial time; number of arithmetic operations
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\textit{M. Shub} and \textit{S. Smale}, Theor. Comput. Sci. 133, No. 1, 141--164 (1994; Zbl 0846.65022)

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### References:

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