Newton's Method Over Global Height Fields
Number Theory
2013-02-15 v2 Dynamical Systems
Abstract
Newton's method is used to approximate roots of complex valued functions f by creating a sequence of points that converges to a root of f in the usual topology. For any field K equipped with a set of pairwise inequivalent absolute values satisfying a product formula, we completely describe the conditions under which Newton's method applied to a squarefree polynomial f with K-coefficients will succeed in finding a root of f in the v-adic topology for infinitely many places v of K. Furthermore, we show that if K is a finite extension of the rationals or of the rational function field over a finite field, then the Newton approximation sequence fails to converge v-adically for a positive density of places v.
Cite
@article{arxiv.1212.6409,
title = {Newton's Method Over Global Height Fields},
author = {Xander Faber and Adam Towsley},
journal= {arXiv preprint arXiv:1212.6409},
year = {2013}
}