The density of prime divisors in the arithmetic dynamics of quadratic polynomials
- Rafe Jones
- Mathematics
- 14 December 2006
Let f ∈ ℤ[x], and consider the recurrence given by an = f(an − 1), with a0 ∈ ℤ. Denote by P(f, a0) the set of prime divisors of this recurrence, that is, the set of primes dividing at least one…
Galois representations from pre-image trees: an arboreal survey
- Rafe Jones
- Mathematics
- 24 February 2014
Given a global field K and a rational function phi defined over K, one may take pre-images of 0 under successive iterates of phi, and thus obtain an infinite rooted tree T by assigning edges…
Settled polynomials over finite fields
- Rafe Jones, N. Boston
- Mathematics
- 1 June 2012
We study the factorization into irreducibles of iterates of a qua- dratic polynomial f over a nite eld. We call f settled when the factorization of its nth iterate for large n is dominated by…
Achievement Sets of Sequences
- Rafe Jones
- MathematicsThe American mathematical monthly
- 1 June 2011
This work examines the set of all sums of the form Σi∊Ixi, as I varies over subsets of the positive integers, and examines what sets can occur as achievement sets, and gives results on the topology of achievement sets.
Eventually stable rational functions
- Rafe Jones, Alon Y. Levy
- Mathematics
- 2 March 2016
For a field K, rational function phi in K(z) of degree at least two, and alpha in P^1(K), we study the polynomials in K[z] whose roots are given by the solutions to phi^n(z) = alpha, where phi^n…
Galois theory of iterated endomorphisms
- Rafe Jones, Jeremy A. Rouse
- Mathematics
- 15 June 2007
Given an abelian algebraic group A over a global field F, α ∈ A(F), and a prime ℓ, the set of all preimages of α under some iterate of [ℓ] generates an extension of F that contains all ℓ‐power…
Galois theory of quadratic rational functions
- Rafe Jones, M. Manes
- Mathematics
- 23 January 2011
For a number field K with absolute Galois group G_K, we consider the action of G_K on the infinite tree of preimages of a point in K under a degree-two rational function phi, with particular…
An iterative construction of irreducible polynomials reducible modulo every prime
- Rafe Jones
- Mathematics
- 13 December 2010
Arboreal Galois representations
- N. Boston, Rafe Jones
- Mathematics
- 16 January 2007
Let $$G_{\mathbb{Q}}$$ be the absolute Galois group of $$\mathbb{Q}$$, and let T be the complete rooted d-ary tree, where d ≥ 2. In this article, we study “arboreal” representations of…
The Density of Primes in Orbits of zd+c
- Spencer Hamblen, Rafe Jones, Kalyani Madhu
- Mathematics
- 26 March 2013
Given a polynomial f(z) = z^d + c over a global field K and a_0 in K, we study the density of prime ideals of K dividing at least one element of the orbit of a_0 under f. The density of such sets for…
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