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

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## Abstract

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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 $v$.

## How to cite

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Faber, 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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7. X. Faber and J. F. Voloch, On the number of places of convergence for Newton’s method over number fields, J. Théor. Nombres Bordeaux, 23,2, (2011), 387–401. Zbl1223.37118MR2817936
8. S. Lang, Diophantine geometry, Interscience Tracts in Pure and Applied Mathematics, 11, Interscience Publishers (a division of John Wiley & Sons), New York-London, (1962). Zbl0115.38701MR142550
9. K. Nishizawa and M. FujimuraFamilies of rational maps and convergence basins of Newton’s method, Proc. Japan Acad. Ser. A Math. Sci., 68, 6, (1992), 143–147. Zbl0762.58019MR1179387
10. J. H. Silverman and J. F. Voloch, A local-global criterion for dynamics on ${ℙ}^{1}$, Acta Arith., 137, 3, (2009), 285–294. Zbl1243.37066MR2496466
11. A. Towsley, A Hasse principle for periodic points, Int. J. Number Theory, 9, 8 (2013), 2053–2068. Zbl1281.11100MR3145160

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