Let $F(z)$ be an arbitrary complex polynomial. We introduce the local root clustering problem, to compute a set of natural $\varepsilon$-clusters of roots of $F(z)$ in some box region $B_0$ in the complex plane. This may be viewed as an extension of the classical root isolation problem. Our contribution is two-fold: we provide an efficient certified subdivision algorithm for this problem, and we provide a bit-complexity analysis based on the local geometry of the root clusters. Our computational model assumes that arbitrarily good approximations of the coefficients of $F$ are provided by means of an oracle at the cost of reading the coefficients. Our algorithmic techniques come from a companion paper (Becker et al., 2018) and are based on the Pellet test, Graeffe and Newton iterations, and are independent of Sch\"onhage's splitting circle method. Our algorithm is relatively simple and promises to be efficient in practice.

中文翻译：

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复杂多项式根聚类的复杂度分析

令$ F（z）$为任意复多项式。我们引入局部根类聚类问题，以计算复杂平面中某些盒子区域$ B_0 $中的根根$ F（z）$的自然$ \ varepsilon $-簇。这可以看作是经典根隔离问题的扩展。我们的贡献有两个方面：我们提供了一个有效的认证细分算法来解决此问题，并且我们根据根簇的局部几何形状提供了位复杂度分析。我们的计算模型假定，通过预言机以$ f $的形式任意提供了$ F $的系数的近似值，但以读取系数为代价。我们的算法技术来自随附的论文（Becker等人，2018年），基于Pellet检验，Graeffe和Newton迭代，并且独立于Sch \“ onhage” s裂圆法。我们的算法相对简单，并且有望在实践中高效。