TY - JOUR

T1 - Greatest common divisors of iterates of polynomials

AU - Hsia, Liang Chung

AU - Tucker, Thomas J.

N1 - Funding Information:
The first author was partially supported by MOST Grant 105-2628-M-002-003. The second author was partially supported by NSF Grant DMS-0101636. MSC2010: primary 37P05; secondary 14G25. Keywords: gcd, composition, heights, equidstribution.
Funding Information:
We would like to thank Dragos Ghioca, Keping Huang, Alina Ostafe, Juan Rivera-Letelier, Umberto Zannier, Shou-Wu Zhang, and Mike Zieve for helpful conversations. We’d also like to thank the anonymous referee for many helpful comments and corrections. The first named author would like to thank his coauthor for his hospitality during the visit to the math department of University of Rochester in the summer of 2014 when this project was initiated.

PY - 2017

Y1 - 2017

N2 - Following work of Bugeaud, Corvaja, and Zannier for integers, Ailon and Rudnick prove that for any multiplicatively independent polynomials, a,bεℂℂ[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,gεℂℂ[x] are compositionally independent polynomials and c(x)εℂℂ[x], then there are at most finitely many λ with the property that there is an n such that (x-λ) divides gcd(fon(x)-c(x),gon(x)-c(x)).

AB - Following work of Bugeaud, Corvaja, and Zannier for integers, Ailon and Rudnick prove that for any multiplicatively independent polynomials, a,bεℂℂ[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,gεℂℂ[x] are compositionally independent polynomials and c(x)εℂℂ[x], then there are at most finitely many λ with the property that there is an n such that (x-λ) divides gcd(fon(x)-c(x),gon(x)-c(x)).

KW - Composition

KW - Equidstribution

KW - Gcd

KW - Heights

UR - http://www.scopus.com/inward/record.url?scp=85028043813&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85028043813&partnerID=8YFLogxK

U2 - 10.2140/ant.2017.11.1437

DO - 10.2140/ant.2017.11.1437

M3 - Article

AN - SCOPUS:85028043813

VL - 11

SP - 1437

EP - 1459

JO - Algebra and Number Theory

JF - Algebra and Number Theory

SN - 1937-0652

IS - 6

ER -