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Primitive divisors in arithmetic dynamics

Published online by Cambridge University Press:  01 March 2009

PATRICK INGRAM
Affiliation:
Department of Mathematics, University of Toronto, Toronto, Ontario M5S 2E4, Canada. e-mail: pingram@math.utoronto.ca
JOSEPH H. SILVERMAN
Affiliation:
Mathematics Department, Box 1917 Brown University, Providence, RI 02912, U.S.A. e-mail: jhs@math.brown.edu

Abstract

Let ϕ(z) ∈ (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 ∈ have infinite orbit under iteration of ϕ and write ϕn(α) = An/Bn as a fraction in lowest terms. We prove that for all but finitely many n ≥ 0, the numerator An has a primitive divisor, i.e., there is a prime p such that p | An and pAi 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.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2008

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