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

Xander Faber^{[1]}; José Felipe Voloch^{[2]}

- [1] Department of Mathematics University of Georgia Athens, GA
- [2] Department of Mathematics University of Texas Austin, TX

Journal de Théorie des Nombres de Bordeaux (2011)

- Volume: 23, Issue: 2, page 387-401
- ISSN: 1246-7405

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

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