An Efficient and Exact Subdivision Algorithm for Isolating Complex Roots of a Polynomial and its Complexity Analysis

We introduce an exact subdivision algorithm CEVAL for isolating complex roots of a square-free polynomial. The subdivision predicates are based on evaluating the original polynomial or its derivatives, and hence is easy to implement. It can be seen as a generalization of a previous real root isolation algorithm called EVAL. Under suitable conditions, the algorithm is applicable for general analytic functions. We provide a complexity analysis of our algorithm on the benchmark problem of isolating all complex roots of a square-free polynomial with Gaussian integer coefficients. The analysis is based on a novel technique called δ-clusters. This analysis shows, somewhat surprisingly, that the simple EVAL algorithm matches (up to logarithmic factors) the bit complexity bounds of current practical exact algorithms such as those based on Descartes, Continued Fraction or Sturm methods. Furthermore, the more general CEVAL also achieves the same complexity.