• DOI:10.5802/JTNB.768
• Corpus ID: 12934737

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}
}

• X. Faber, J. Voloch
• Published 5 March 2010
• Mathematics
• arXiv: Number Theory

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 conjecture that the set of places for which Newton iteration diverges has full density and give some heuristic and numerical evidence.