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.

翻译：Let F( z) $( z) 是一个任意的复杂多元数字 。 我们引入了本地根组问题, 以计算一组天然的 $\ varepsilon$- 根组, 在某个框区计算$( z) $( z) $( z) 在复杂的平面上 $( B_ 0 美元 ) 。 这可以被视为古典根隔离问题的延伸。 我们的贡献有两个方面: 我们为这一问题提供高效的分解算法, 我们根据根组的本地几何测量方法提供略微兼容性分析。 我们的计算模型假定, $( $) 的系数的任意优近似值是以一个神谕的方式提供的, 以阅读系数为代价。 我们的算法技术来自一个配套文件( 贝克尔等人, 201818年), 并基于 Pellet 测试、 Grareeffe 和 Newton Exerations, 并且独立于 Sch\ “ onhage” 分裂圆法。 我们的算法相对简单, 并且承诺在实践中会有效 。