Newton's Method Over Global Height Fields

TL;DR

This paper investigates the conditions under which Newton's method converges to roots of polynomials over global height fields in various topologies, revealing limitations in convergence for certain fields and places.

Contribution

It provides a complete characterization of Newton's method convergence over global height fields and identifies cases where convergence fails at many places.

Findings

Newton's method converges in the v-adic topology under specific conditions.

Failure of convergence occurs at a positive density of places for certain fields.

The results apply to fields like finite extensions of rationals and rational function fields.

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.