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… 

Tables from this paper

Newton's Method Over Global Height Fields
Newton's method is used to approximate roots of complex valued functions f by creating a sequence of points that converges to a root of f in the usual topology. For any field K equipped with a set of
Primitive prime divisors in the critical orbit of z^d+c
We prove the finiteness of the Zsigmondy set associated to the critical orbit of f(z) = z^d+c for rational values of c by finding an effective bound on the size of the set. For non-recurrent critical
A large arboreal Galois representation for a cubic postcritically finite polynomial
We give a complete description of the arboreal Galois representation of a certain postcritically finite cubic polynomial over a large class of number fields and for a large class of basepoints. This
Periods of rational maps modulo primes
Let K be a number field, let $${\varphi \in K(t)}$$ be a rational map of degree at least 2, and let $${\alpha, \beta \in K}$$ . We show that if α is not in the forward orbit of β, then there is a
Two New Iterated Maps for Numerical Nth Root Evaluation
In this paper we propose two original iterated maps to numerically approximate the nth root of a real number. Comparisons between the new maps and the famous Newton-Raphson method are carried out,
The Hidden Geometry of the Babylonian Square Root Method
We propose and demonstrate an original geometric argument for the ancient Babylonian square root method, which is analyzed and compared to the Newton-Raphson method. Based on simple geometry and
Primitive Prime Divisors for Unicritical Polynomials

References

SHOWING 1-9 OF 9 REFERENCES
Prime factors of dynamical sequences
Abstract Let (t) (t) have degree d 2. For a given rational number x0, define xn1 (xn) for each n 0. If this sequence is not eventually periodic, and if does not lie in one of two explicitly
A Local-Global Criterion for Dynamics on P^1
Let K be a number field or a function field, let F:P^1 --> P^1 be a rational map of degree at least two defined over K, let P be a point in P^1(K) having infinite F-orbit, and let Z be a finite
Primitive divisors in arithmetic dynamics
Abstract Let ϕ(z) ∈ (z) be a rational function of degree d ≥ 2 with ϕ(0) = 0 and such that ϕ does not vanish to order d at 0. Let α ∈ have infinite orbit under iteration of ϕ and write ϕn(α) = An/Bn
A Course in p-adic Analysis
1 p-adic Numbers.- 2 Finite Extensions of the Field of p-adic Numbers.- 3 Construction of Universal p-adic Fields.- 4 Continuous Functions on Zp.- 5 Differentiation.- 6 Analytic Functions and
Graduate Texts in Mathematics
Graduate Texts in Mathematics bridge the gap between passive study and creative understanding, offering graduate-level introductions to advanced topics in mathematics. The volumes are carefully
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
Silverman and José Felipe Voloch . A localglobal criterion for dynamics on P 1
  • Acta Arith
  • 2009
Silverman . Primitive divisors in arithmetic dynamics
  • Math . Proc . Cambridge Philos . Soc .
  • 2009
Prime factors of dynamical sequences. arXiv:0903.1344v1, preprint
  • Prime factors of dynamical sequences. arXiv:0903.1344v1, preprint
  • 2010