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

• [1] Department of Mathematics University of Georgia Athens, GA
• [2] Department of Mathematics University of Texas Austin, TX
• Volume: 23, Issue: 2, page 387-401
• ISSN: 1246-7405

top

## Abstract

top
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.

## How to cite

top

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 -

## References

top
1. Xander Faber and Andrew GranvillePrime factors of dynamical sequences. To appear in J. Reine Angew. Math.ArXiv:0903.1344v1. Zbl1290.11019
2. Patrick Ingram and Joseph H. Silverman, Primitive divisors in arithmetic dynamics. Math. Proc. Cambridge Philos. Soc. 146(2) (2009), 289–302. Zbl1242.11012MR2475968
3. Alain M. Robert, A course in $p$-adic analysis. Volume 198 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2000. Zbl0947.11035MR1760253
4. Joseph H. Silverman and José Felipe Voloch, A local-global criterion for dynamics on ${ℙ}^{1}$. Acta Arith. 137(3) (2009), 285–294. Zbl1243.37066MR2496466

top

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.