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

TL;DR

This paper investigates the conditions under which Newton's method converges v-adically to roots of polynomials over number fields, revealing infinite convergence at many places and proposing a conjecture on divergence density.

Contribution

It provides precise criteria for v-adic convergence of Newton's method over number fields and establishes infinite convergence at many places for irreducible polynomials of degree at least 3.

Findings

Newton iteration converges v-adically to roots at infinitely many places.

For irreducible polynomials of degree ≥ 3, convergence occurs at infinitely many places.

Heuristic and numerical evidence suggests divergence set has full density.

Abstract

Let f be a polynomial of degree at least 2 with coefficients in a number field K, let x_0 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 x_0, 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.