Primitive Divisors in Arithmetic Dynamics

Abstract

Let F(z) be a rational function in Q(z) of degree at least 2 with F(0) = 0 and such that F does not vanish to order d at 0. Let b be a rational number having infinite orbit under iteration of F, and write F^n(b) = 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 divides A_n and p does not divide A_i for all i < n. More generally, we prove an analogous result when F is defined over a number field and 0 is a periodic point for F.

Version 2 is substantial revision. The proof of the main theorem has been simplified and strengthened. (16 pages)

References in corpus (5)

• The density of prime divisors in the arithmetic dynamics of quadratic polynomials
• Primitive divisors of elliptic divisibility sequences
• Heights and preperiodic points of polynomials over function fields
• Equidistribution and integral points for Drinfeld modules
• A finiteness theorem for canonical heights attached to rational maps over function fields

Cited by in corpus (11)

• Portraits of preperiodic points for rational maps
• Primitive Divisors, Dynamical Zsigmondy Sets, and Vojta's Conjecture
• Moduli Spaces for Dynamical Systems with Portraits
• Index Divisibility in Dynamical Sequences and Cyclic Orbits Modulo $p$
• Multiplicative dependence among iterated values of rational functions modulo finitely generated groups
• The Arithmetic of Consecutive Polynomial Sequences over Finite Fields
• Mean Divisibility of Multinomial coefficients
• Index divisibility in the orbit of 0 for integral polynomials
• Zsigmondy's theorem and primitive divisors of the Lucas and Lehmer sequences in polynomial rings
• Primitive prime divisors in the critical orbits of one-parameter families of rational polynomials
• Heights and periodic points for one-parameter families of Hénon maps