On the number of places of convergence for Newton’s method over number fields
Journal de Théorie des Nombres de Bordeaux, Volume 23 (2011) no. 2, pp. 387-401.

Let f be a polynomial of degree at least 2 with coefficients in a number field K, let x 0 be a sufficiently general element of K, and let α be a root of f. We give precise conditions under which Newton iteration, started at the point x 0 , converges v-adically to the root α for infinitely many places v of K. As a corollary we show that if f is irreducible over K of degree at least 3, then Newton iteration converges v-adically to any given root of f for infinitely many places v. We also conjecture that the set of places for which Newton iteration diverges has full density and give some heuristic and numerical evidence.

Soit f un polynôme de degré au moins 2 avec coefficients dans un corps de nombres K, soit x 0 un élément suffisamment général de K, et soit α une racine de f. Nous précisons des conditions pour lesquelles l’itération de Newton, commençant au point x 0 , converge v-adiquement vers la racine α pour un nombre infini de places v de K. Comme corollaire, nous montrons que si f est irréductible sur K de degré au moins 3, l’itération de Newton converge v-adiquement vers chaque racine de f pour un nombre infini de places v de K. Nous faisons aussi la conjecture que le nombre de places telles que l’itération de Newton ne converge pas a densité un et nous donnons des évidences heuristiques et numériques.

Received:
Published online:
DOI: 10.5802/jtnb.768
Classification: 37P05,  11B99
Keywords: Arithmetic Dynamics, Newton’s Method, Primitive Prime Factors
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     title = {On the number of places of convergence for {Newton{\textquoteright}s} method over number fields},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
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Xander Faber; José Felipe Voloch. On the number of places of convergence for Newton’s method over number fields. Journal de Théorie des Nombres de Bordeaux, Volume 23 (2011) no. 2, pp. 387-401. doi : 10.5802/jtnb.768. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.768/

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