2010/03/05 by Xander Faber, José Felipe Voloch, Faber, Xander +1
Computer Science · Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Iterative Methods for Nonlinear Equations #Polynomial and algebraic computation #math.DS #math.NT #msc:11B99 #msc:37P05
paper · pdf · doi:10.48550/arxiv.1003.1236
9 pages; minor changes from the previous version; to appear in Journal de Théorie des Nombres de Bordeaux
arxiv created 2010/10/10 · arxiv updated 2010/10/12
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 alpha be a root of f. We give precise conditions under which Newton iteration, started at the point x0, converges v-adically to the root alpha 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.