# On the Number of Places of Convergence for Newton's Method over Number Fields

@article{Faber2010OnTN, title={On the Number of Places of Convergence for Newton's Method over Number Fields}, author={X. W. C. Faber and Jos{\'e} Felipe Voloch}, journal={arXiv: Number Theory}, year={2010} }

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 alpha 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 alpha 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…

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CANADA E-mail address: xander@math.mcgill.ca URL: http://www.math.mcgill.ca/xander

- CANADA E-mail address: xander@math.mcgill.ca URL: http://www.math.mcgill.ca/xander