Squarefree Doubly Primitive Divisors in Dynamical Sequences

TL;DR

This paper investigates the existence of primes p for which certain differences in dynamical sequences have a p-adic valuation of exactly 1, linking number theory, dynamical systems, and conjectures like Vojta's.

Contribution

It establishes the existence of such primes for a broad class of dynamical sequences over number and function fields, extending previous results.

Findings

For all but finitely many pairs (m,n), there exists a prime p with specific valuation properties.

The results are unconditional in function fields due to Yamanoi's theorem.

The work generalizes earlier studies by Ingram-Silverman, Faber-Granville, and the authors.

Abstract

Let K be a number field or a function field of characteristic 0, let f be a K-rational function of degree greater than 1, and let a be an element of K. Let S be a finite set of places of K containing all the archimedean ones and the primes where f has bad reduction. After excluding all the natural counter-examples, we define a subset A(f,a) of pairs of integers (m,n) with m nonnegative and n positive, and show that for all but finitely many (m,n) in A(f,a) there is a prime p of K which is not in S such that the p-adic valuation of f^{m+n}(a)-f^m(a) is precisely equal to 1, and moreover a has portrait (m,n) under the action of f modulo p. This latter condition implies that the p-adic valuation of f^{u+v}(a)-f^u(a) is not positive if u is a nonnegative integer and v is a positive integer with u<m or v<n. Our proof assumes a conjecture of Vojta in the number field case and is unconditional in the function field case thanks to a deep theorem of Yamanoi. This paper extends earlier work of Ingram-Silverman, Faber-Granville, and of the authors.