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

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.

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.

DOI: 10.5802/jtnb.870
Classification: 37P05, 37P15
Keywords: Arithmetic Dynamics, Global Height Field, Newton’s Method, Density
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
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     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},
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Xander Faber; Adam Towsley. 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.

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