Vol. 11, No. 6, 2017

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Greatest common divisors of iterates of polynomials

Liang-Chung Hsia and Thomas J. Tucker

Vol. 11 (2017), No. 6, 1437–1459
Abstract

Following work of Bugeaud, Corvaja, and Zannier for integers, Ailon and Rudnick prove that for any multiplicatively independent polynomials, a,b [x], there is a polynomial h such that for all n, we have

gcd(an 1,bn 1)|h

We prove a compositional analog of this theorem, namely that if f,g [x] are compositionally independent polynomials and c(x) [x], then there are at most finitely many λ with the property that there is an n such that (x λ) divides gcd(fn(x) c(x),gn(x) c(x)).

Keywords
gcd, composition, heights, equidstribution
Mathematical Subject Classification 2010
Primary: 37P05
Secondary: 14G25
Milestones
Received: 7 December 2016
Revised: 11 April 2017
Accepted: 13 May 2017
Published: 16 August 2017
Authors
Liang-Chung Hsia
Department of Mathematics
National Taiwan Normal University
Taipei
Taiwan
Thomas J. Tucker
Department of Mathematics
University of Rochester
Rochester, NY 14627
United States