Khoa D. Nguyen

ABC implies primitive prime divisors in arithmetic dynamics

Let K be a number field, let ϕ(x) ∈ K(x) be a rational function of degree d> 1, and let α ∈ K be a wandering point such that ϕ n (α) �= 0 for all n> 0. We prove that if the abc-conjecture holds for K, then for all but finitely many positive integers n, there is a prime p of K such that vp(ϕ n (α)) > 0a ndvp(ϕ m (α)) 0 for all positive integers m 0 with the stronger condition vp(ϕ n (α)) = 1. We prove the same result unconditionally for function fields of characteristic 0 when ϕ is not isotrivial. Let K be a number field or function field, let ϕ(x) ∈ K(x) be a rational function of degree d> 1, and let α ∈ K. We denote the n-iterate of ϕ as ϕ n . It is often the case that for all but finitely many n, there is a prime that is a divisor of ϕ n (α), but is not a divisor of ϕ m (α )f or any m<n . This problem was considered by Bang [2], Zsigmondy [38], and Schinzel [27 ]i n the context of the multiplicative group. More recently, many authors have considered the problem in other cases. Most of these results apply either when 0 is preperiodic under ϕ (see [11, 16], for example) or when 0 is a ramification point of ϕ (see [8, 20, 25]). Much work has been done on this problem in the setting of elliptic curves (refer to [10, 15], for example). However, here we do not have an underlying algebraic group. In this paper, we show that similar results

Portraits of preperiodic points for rational maps

Let

Evidence for the Dynamical Brauer-Manin Criterion

K

A Case of the Dynamical André–Oort Conjecture

be a function field over an algebraically closed field

Dynamical anomalous subvarieties: Structure and bounded height theorems ☆

k

On modules of integral elements over finitely generated domains

of characteristic

Some Finiteness Results on Monogenic Orders in Positive Characteristic

0

Dynamics of Split Polynomial Maps: Uniform Bounds for Periods and Applications

, let

The Hermite-Joubert Problem and a Conjecture of Brassil-Reichstein

\varphi\in K(z)

Simultaneously preperiodic points for families of polynomials in normal form

be a rational function of degree at least equal to