An O(n1+ϵ log b) algorithm for the complex roots problem

Given a univariate polynomial f(z) of degree n with complex coefficients, whose real and imaginary parts can be expressed as a ratio of two integers less than 2^m in magnitude, the root problem is to find all the roots of f(z) up to specified precision 2^-μ . Assuming the arithmetic model for computation, we provide, for any ε>0, an algorithm which has complexity O(n^1+ε log b), where b=m+μ. This improves on the previous best known algorithm for the problem which has complexity O(n² log b). We claim it that it follows from the fact that we can bound the precision required in all the arithmetic computations, that the complexity of our algorithm in the Boolean model of computation is O(n ^2+ε(n+b) log² b log log b)