On Newton´s method for polynomials

Let Pd be the set of polynomials over the complex numbers of degree d with all its roots in the unit ball. For f ∈ Pd, let Γf be the set of points for which Newton´s method converges to a root, and let Af ≡ |Γf ∩ B2(O)|/|B2(O)|, i.e. the density of Γf in the ball of radius 2. For each d we consider Ad, the worst-case density Af for f ∈ Pd. In |S|, S. Smale conjectured that Ad ≫ 0 for all d ≥ 3 (it was wellknown that A1 = A2 = 1). In this paper we prove that (1/d)cd2 log d ≤ Ad for some constant c. In particular, Ad ≫ 0 for all d.