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.

DOI: 10.5802/jtnb.768
Classification: 37P05, 11B99
Keywords: Arithmetic Dynamics, Newton’s Method, Primitive Prime Factors
Faber, Xander 1; Voloch, José Felipe 2

1 Department of Mathematics University of Georgia Athens, GA
2 Department of Mathematics University of Texas Austin, TX
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Faber, Xander; Voloch, José Felipe. 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. http://archive.numdam.org/articles/10.5802/jtnb.768/

[1] Xander Faber and Andrew Granville Prime factors of dynamical sequences. To appear in J. Reine Angew. Math. ArXiv:0903.1344v1. | Zbl

[2] Patrick Ingram and Joseph H. Silverman, Primitive divisors in arithmetic dynamics. Math. Proc. Cambridge Philos. Soc. 146(2) (2009), 289–302. | MR | Zbl

[3] Alain M. Robert, A course in p-adic analysis. Volume 198 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2000. | MR | Zbl

[4] Joseph H. Silverman and José Felipe Voloch, A local-global criterion for dynamics on 1 . Acta Arith. 137(3) (2009), 285–294. | EuDML | MR | Zbl

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