Bounding the basins of attraction of Newton's method

Chee Yap's book on "Fundamental Problems of Algorithmic Algebra", chapter 6 deals with this problem precisely.

Yap cites Friedman's "On the convergence of Newton's method." and Smale's "The fundamental theorem of algebra and complexity theory" as references for the following result (Yap, Theorem 6.37):

Let $f(X) \in \mathbb{Z}[X]$ be square free with $m = \deg f$ and $M = 1 + \|f\|_{\infty}$. Then Newton iteration for $f(X)$ is guaranteed to converge to root $X^*$ provied the initial approximation is at most $$\delta_0 = [m^{3m+9}(1+M)^{6m}]^{-1} $$ from $X^*$

Where $\|f\|_{\infty}$ is the maximum coefficient of the polynomial $f(x)$.

Hope that helps.