Greatest common divisors of iterates of polynomials

Hsia, Liang-Chung; Tucker, Thomas

doi:10.2140/ant.2017.11.1437

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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

DOI: 10.2140/ant.2017.11.1437

Abstract

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

gcd (a^{n} - 1, b^{n} - 1) | h

We prove a compositional analog of this theorem, namely that if $f, g \in ℂ [x]$ are compositionally independent polynomials and $c (x) \in ℂ [x]$ , then there are at most finitely many $λ$ with the property that there is an $n$ such that $(x - λ)$ divides $gcd (f^{\circ n} (x) - c (x), g^{\circ n} (x) - c (x))$ .

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

Vol. 11, No. 6, 2017