On the number of places of convergence for Newton’s method over number fields

Faber, Xander, and Voloch, José Felipe. "On the number of places of convergence for Newton’s method over number fields." Journal de Théorie des Nombres de Bordeaux 23.2 (2011): 387-401. <http://eudml.org/doc/219698>.

@article{Faber2011,
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.},
affiliation = {Department of Mathematics University of Georgia Athens, GA; Department of Mathematics University of Texas Austin, TX},
author = {Faber, Xander, Voloch, José Felipe},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Arithmetic Dynamics; Newton’s Method; Primitive Prime Factors; arithmetic dynamics; Newton's method; primitive prime factors},
language = {eng},
month = {6},
number = {2},
pages = {387-401},
publisher = {Société Arithmétique de Bordeaux},
title = {On the number of places of convergence for Newton’s method over number fields},
url = {http://eudml.org/doc/219698},
volume = {23},
year = {2011},
}

TY - JOUR
AU - Faber, Xander
AU - Voloch, José Felipe
TI - On the number of places of convergence for Newton’s method over number fields
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2011/6//
PB - Société Arithmétique de Bordeaux
VL - 23
IS - 2
SP - 387
EP - 401
AB - 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.
LA - eng
KW - Arithmetic Dynamics; Newton’s Method; Primitive Prime Factors; arithmetic dynamics; Newton's method; primitive prime factors
UR - http://eudml.org/doc/219698
ER -