For any field 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 will succeed in finding some root of in the -adic topology for infinitely many places of . Furthermore, we show that if 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 -adically for a positive density of places .
Pour tout corps muni d’un ensemble de valeurs absolues satisfaisant la formule du produit, nous décrivons complètement les conditions pour que la méthode de Newton, appliquée à un polynôme sans facteur carré, parvienne à trouver une racine dans le complété -adique pour une infinité de valeurs absolues de . De plus, nous montrons que si est un corps global, la suite d’approximation de Newton ne converge pas -adiquement pour une partie de densité positive de .
Mots-clés : Arithmetic Dynamics, Global Height Field, Newton’s Method, Density
@article{JTNB_2014__26_2_347_0, author = {Faber, Xander and Towsley, Adam}, title = {Newton{\textquoteright}s method over global height fields}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {347--362}, publisher = {Soci\'et\'e Arithm\'etique de Bordeaux}, volume = {26}, number = {2}, year = {2014}, doi = {10.5802/jtnb.870}, mrnumber = {3320483}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/jtnb.870/} }
TY - JOUR AU - Faber, Xander AU - Towsley, Adam TI - Newton’s method over global height fields JO - Journal de théorie des nombres de Bordeaux PY - 2014 SP - 347 EP - 362 VL - 26 IS - 2 PB - Société Arithmétique de Bordeaux UR - http://archive.numdam.org/articles/10.5802/jtnb.870/ DO - 10.5802/jtnb.870 LA - en ID - JTNB_2014__26_2_347_0 ER -
%0 Journal Article %A Faber, Xander %A Towsley, Adam %T Newton’s method over global height fields %J Journal de théorie des nombres de Bordeaux %D 2014 %P 347-362 %V 26 %N 2 %I Société Arithmétique de Bordeaux %U http://archive.numdam.org/articles/10.5802/jtnb.870/ %R 10.5802/jtnb.870 %G en %F JTNB_2014__26_2_347_0
Faber, Xander; Towsley, Adam. Newton’s method over global height fields. Journal de théorie des nombres de Bordeaux, Volume 26 (2014) no. 2, pp. 347-362. doi : 10.5802/jtnb.870. http://archive.numdam.org/articles/10.5802/jtnb.870/
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