多项式的几何与通过路径提升的求根方法
Geometry of Polynomials and Root-Finding via Path-Lifting
AI总结：
本文利用多项式的拓扑、组合和几何性质以及解析结果（主要是覆盖结构和畸变估计），分析了一种路径提升方法用于寻找近似零点，并证明了该算法始终收敛（除有限个初始点外），其迭代次数仅依赖于几何量而非多项式次数。
摘要：
利用多项式拓扑、组合和几何性质与解析结果（主要是覆盖结构和畸变估计）之间的相互作用，我们分析了一种寻找近似零点的路径提升方法，类似于Smale、Shub、Kim等人研究的方法。对于任意多项式，该简单算法总是收敛到一个根，除了位于给定半径圆上的有限个初始点。具体来说，我们分析的算法迭代公式为 $$z - \frac{f(z)-t_kf(z_0)}{f'(z)}$$，其中$t_k$构成一个递减的实数序列，$z_0$选在包含所有根的圆上。我们证明，定位多项式$f$的近似零点所需的迭代次数仅依赖于$\log|f(z_0)/ρ_ζ|$（其中$ρ_ζ$是将0映射到根$ζ$的$f^{-1}$分支的收敛半径）以及$f(z_0)$与某些临界值之间夹角的对数。先前相关算法的复杂度结果线性依赖于这些夹角的倒数。注意，算法的复杂度不直接依赖于$f$的次数，而仅依赖于临界值的几何性质。此外，对于具有不同根的多项式$f$，在包含所有根的圆上所有起始点所需的平均步数被一个常数乘以$\log(1/ρ_ζ)$的平均值所界定。对于单位圆盘内具有$d$个根的所有多项式$f$，$\log(1/ρ_ζ)$的平均值为${\mathcal{O}}({d})$。该算法很容易推广到寻找多项式的所有根（无需缩减）；这样做最多将复杂度增加$d$倍。
英文摘要：
Using the interplay between topological, combinatorial, and geometric properties of polynomials and analytic results (primarily the covering structure and distortion estimates), we analyze a path-lifting method for finding approximate zeros, similar to those studied by Smale, Shub, Kim, and others. Given any polynomial, this simple algorithm always converges to a root, except on a finite set of initial points lying on a circle of a given radius. Specifically, the algorithm we analyze consists of iterating $$z - \frac{f(z)-t_kf(z_0)}{f'(z)}$$ where the $t_k$ form a decreasing sequence of real numbers and $z_0$ is chosen on a circle containing all the roots. We show that the number of iterates required to locate an approximate zero of a polynomial $f$ depends only on $\log|f(z_0)/ρ_ζ|$ (where $ρ_ζ$ is the radius of convergence of the branch of $f^{-1}$ taking $0$ to a root $ζ$) and the logarithm of the angle between $f(z_0)$ and certain critical values. Previous complexity results for related algorithms depend linearly on the reciprocals of these angles. Note that the complexity of the algorithm does not depend directly on the degree of $f$, but only on the geometry of the critical values. Furthermore, for any polynomial $f$ with distinct roots, the average number of steps required over all starting points taken on a circle containing all the roots is bounded by a constant times the average of $\log(1/ρ_ζ)$. The average of $\log(1/ρ_ζ)$ over all polynomials $f$ with $d$ roots in the unit disk is ${\mathcal{O}}({d})$. This algorithm readily generalizes to finding all roots of a polynomial (without deflation); doing so increases the complexity by a factor of at most $d$.
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