Newton’s method over global height fields
Journal de Théorie des Nombres de Bordeaux, Tome 26 (2014) no. 2, pp. 347-362.

Pour tout corps K 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 fKx sans facteur carré, parvienne à trouver une racine dans le complété v-adique pour une infinité de valeurs absolues v de K. De plus, nous montrons que si K est un corps global, la suite d’approximation de Newton ne converge pas v-adiquement pour une partie de densité positive de v.

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 fKx 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.

Reçu le : 2012-12-26
Accepté le : 2013-02-18
Publié le : 2015-03-09
DOI : https://doi.org/10.5802/jtnb.870
Classification : 37P05,  37P15
Mots clés: Arithmetic Dynamics, Global Height Field, Newton’s Method, Density
@article{JTNB_2014__26_2_347_0,
     author = {Xander Faber and Adam Towsley},
     title = {Newton'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 = {jtnb.centre-mersenne.org/item/JTNB_2014__26_2_347_0/}
}
Xander Faber; Adam Towsley. Newton’s method over global height fields. Journal de Théorie des Nombres de Bordeaux, Tome 26 (2014) no. 2, pp. 347-362. doi : 10.5802/jtnb.870. https://jtnb.centre-mersenne.org/item/JTNB_2014__26_2_347_0/

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