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
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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 -