On the convergence of newton's method

Abstract Let Pd be the set of polynomials over the complex number of degree d with all its roots in the unit ball. For f ϵ 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 R2). For each d we consider Ad, the worst-case density (i.e., infimum) of Af for f ϵ Pd. S. Smale (1985) conjectured that Ad > 0 for all d ⪰ 3 (it was well known that A1 = A2 = 1) In this paper we prove that 1 d cd 2 log d ⩽ A d for some constant c. In particular, Ad > 0 for all d. Remark. Our definition of Ad differs slightly from that of Smale (1985) , but the conclusions hold for Ad as defined by Smale as well.