On the convergence of newton's method

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(0)|/|B2(0)|, i.e. the density of Γf in the ball of radius 2 (where | | denotes Lebesgue measure on C viewed as R). For each d we consider Ad, the worst-case density (i.e. infimum) of Af for f ∈ Pd. In [S], S. Smale conjectured that Ad > 0 for all d ≥ 3 (it was well-known that A1 = A2 = 1). In this paper we prove that ( 1