# Newton’s method over global height fields

Xander Faber^{[1]}; Adam Towsley^{[2]}

- [1] Department of Mathematics University of Hawaii Honolulu, HI
- [2] Department of Mathematics CUNY Graduate Center New York, NY

Journal de Théorie des Nombres de Bordeaux (2014)

- Volume: 26, Issue: 2, page 347-362
- ISSN: 1246-7405

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topFaber, Xander, and Towsley, Adam. "Newton’s method over global height fields." Journal de Théorie des Nombres de Bordeaux 26.2 (2014): 347-362. <http://eudml.org/doc/275682>.

@article{Faber2014,

abstract = {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 $f \in K \left[ x \right]$ will succeed in finding some 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 $v$.},

affiliation = {Department of Mathematics University of Hawaii Honolulu, HI; Department of Mathematics CUNY Graduate Center New York, NY},

author = {Faber, Xander, Towsley, Adam},

journal = {Journal de Théorie des Nombres de Bordeaux},

keywords = {Arithmetic Dynamics; Global Height Field; Newton’s Method; Density; arithmetic dynamics; global height field; Newton's method; density},

language = {eng},

month = {10},

number = {2},

pages = {347-362},

publisher = {Société Arithmétique de Bordeaux},

title = {Newton’s method over global height fields},

url = {http://eudml.org/doc/275682},

volume = {26},

year = {2014},

}

TY - JOUR

AU - Faber, Xander

AU - Towsley, Adam

TI - Newton’s method over global height fields

JO - Journal de Théorie des Nombres de Bordeaux

DA - 2014/10//

PB - Société Arithmétique de Bordeaux

VL - 26

IS - 2

SP - 347

EP - 362

AB - 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 $f \in K \left[ x \right]$ will succeed in finding some 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 $v$.

LA - eng

KW - Arithmetic Dynamics; Global Height Field; Newton’s Method; Density; arithmetic dynamics; global height field; Newton's method; density

UR - http://eudml.org/doc/275682

ER -

## References

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