Primitive divisors in arithmetic dynamics

Abstract

Let ϕ(z) ∈ Q(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 ∈ Q have infinite orbit under iteration of ϕ and write ϕ n (α) = A n /B n as a fraction in lowest terms. We prove that for all but finitely many n 0, the numerator A n has a primitive divisor, i.e., there is a prime p such that p | A n and p H A i for all i < n. More generally, we prove an analogous result when ϕ is defined over a number field and 0 is a preperiodic point for ϕ.

A local-global criterion for dynamics on P1

“…In this note we prove a dynamical analog of Scharaschkin's conjecture for dynamical systems on P 1 . The proof uses a variety of tools, including a Zsigmondy theorem for primitive divisors in dynamical systems that was recently proven by Ingram and Silverman [4].We recall from [3] the setup for the dynamical version of the BrauerManin obstruction.Notation. Let K be a number field, let X/K be a projective variety, and let ϕ : X → X be a K-morphism of infinite order.…”

“…In this note we prove a dynamical analog of Scharaschkin's conjecture for dynamical systems on P 1 . The proof uses a variety of tools, including a Zsigmondy theorem for primitive divisors in dynamical systems that was recently proven by Ingram and Silverman [4].…”

“…Theorem 6 (Ingram-Silverman [4]). Let K be a number field , let S be a finite set of primes, let β ∈ P 1 (K) be a point with infinite orbit, and let γ ∈ P 1 (K) be a preperiodic point.…”

“…We are going to use the dynamical Zsigmondy theorem from [4], whose statement we recall after making one definition.…”

ABC implies a Zsigmondy principle for ramification

“…The fact that there are infinitely many such primes follows from [Sil93]. Various authors (see [Zsi92,Elk91,Ric07,Kri13,IS09,FG11,GNT13] for example) have sought to show that not only are there infinitely many primes that divide φ n (α) − β for some n, but the stronger statement that there exists an N such that for all n > N, there is a prime that divides φ n (α)−β that does not divide φ m (α)−β for any m < n. If this is true, one might say that there are infinitely many primes dividing φ n (α) − β for some n because after a certain point each "new iterate" φ n (α) − β gives a "new prime" dividing φ n (α) − β. This is sometimes referred as the "Zsigmondy principle", after Zsigmondy [Zsi92] who studied these questions in the context of primitive divisors of a n − b n .…”