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On a problem posed by Steve Smale. (English) Zbl 1248.65047

S. Smale [Math. Intell. 20, No. 2, 7–15 (1998; Zbl 0947.01011)] posted 18 mathematical problems for the 21st century. The 17th problem asks for the existence of a deterministic algorithm computing an approximate solution of a system of \(n\) complex polynomials in \(n\) unknowns in polynomial time, on the average, in the size \(N\) of the input system. C. Beltrán and L. M. Pardo [London Mathematical Society Lecture Note Series 331, 1–35 (2006; Zbl 1107.65316)] gave a partial solution to the problem by proposing a randomized algorithm. The aim of the paper is to extend their results. Using a linear homotopy algorithm and a smoothed and condition-based analysis, the authors exhibit a deterministic algorithm and show that its complexity is \(N^{\mathcal{O}(\log\log N)}\). The result is a nearly solution to Smale’s 17th problem.

MSC:

65H04 Numerical computation of roots of polynomial equations
65H10 Numerical computation of solutions to systems of equations
65Y20 Complexity and performance of numerical algorithms
65E05 General theory of numerical methods in complex analysis (potential theory, etc.)
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
30C10 Polynomials and rational functions of one complex variable
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References:

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