On the existence of generally convergent algorithms

To motivate our result, consider Newton’s method N for solving the equationf(z) = 0, wheref is a complex polynomial, f(z) = X$0 aizi. We write N: 5$ X S + S, where Z& is the space of polynomials of degree %d and S is the Riemann sphere C U ~0. Then N(f, z) = Nf(z) = z f(z)/f’(z) is rational over C inf and z; that is, N can be formed from the complex rational operations (+ , , x , +) from the coefficients off and z. If z is sufficiently close to a zero J off, then the iterates zk = N)(z) converge to 5 as k tends to cc. However, as is well known there is an open set U in ?& X C (if d > 2) such that this convergence will not happen for (f, z) in 17. See, e.g., Smale (1985). In this paper it was conjectured that no such algorithm could be generally convergent. Curt McMullen settled the question by proving the following result.