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Squarefree doubly primitive divisors in dynamical sequences

Published online by Cambridge University Press:  27 June 2017

DRAGOS GHIOCA ,
KHOA D. NGUYEN  and
THOMAS J. TUCKER
DRAGOS GHIOCA
Affiliation:
Department of Mathematics, (Room 121) 1984 Mathematics Road, University of British Columbia, Vancouver, BC V6T 1Z2 e-mails: dghioca@math.ubc.edu, dknguyen@math.ubc.ca
KHOA D. NGUYEN
Affiliation:
Department of Mathematics, (Room 121) 1984 Mathematics Road, University of British Columbia, Vancouver, BC V6T 1Z2 e-mails: dghioca@math.ubc.edu, dknguyen@math.ubc.ca
THOMAS J. TUCKER
Affiliation:
Department of Mathematics, University of Rochester, Rochester, NY 14627 e-mail: thomas.tucker@rochester.edu
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Abstract

Let K be a number field or a function field of characteristic 0, let φ ∈ K(z) with deg(φ) ⩾ 2, and let α ∈ ℙ1(K). Let S be a finite set of places of K containing all the archimedean ones and the primes where φ has bad reduction. After excluding all the natural counterexamples, we define a subset A(φ, α) of ℤ⩾0 × ℤ>0 and show that for all but finitely many (m, n) ∈ A(φ, α) there is a prime 𝔭 ∉ S such that ord𝔭(φm+n(α)−φm(α)) = 1 and α has portrait (m, n) under the action of φ modulo 𝔭. This latter condition implies ord𝔭(φu+v(α)−φu(α)) ⩽ 0 for (u, v) ∈ ℤ⩾0 × ℤ>0 satisfying u < m or v < n. Our proof assumes a conjecture of Vojta for ℙ1 × ℙ1 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 the authors.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 164 , Issue 3 , May 2018 , pp. 551 - 572
Copyright
Copyright © Cambridge Philosophical Society 2017 

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