On the Number of Places of Convergence for Newton’s Method over Number Fields

Let f be a polynomial of degree at least 2 with coefficients in a number field K, let x0 be a sufficiently general element of K, and let α be a root of f . We give precise conditions under which Newton iteration, started at the point x0, converges v-adically to the root α for infinitely many places v of K. As a corollary we show that if f is irreducible over K of degree at least 3, then Newton iteration converges v-adically to any given root of f for infinitely many places v. We also conjecture that the set of places for which Newton iteration diverges has full density and give some heuristic and numerical evidence.