Abstract
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)).
| Original language | English |
|---|---|
| Pages (from-to) | 1437-1459 |
| Number of pages | 23 |
| Journal | Algebra and Number Theory |
| Volume | 11 |
| Issue number | 6 |
| DOIs | |
| Publication status | Published - 2017 Jan 1 |
Fingerprint
Keywords
- Composition
- Equidstribution
- Gcd
- Heights
ASJC Scopus subject areas
- Algebra and Number Theory
Cite this
Greatest common divisors of iterates of polynomials. / Hsia, Liang-Chung; Tucker, Thomas J.
In: Algebra and Number Theory, Vol. 11, No. 6, 01.01.2017, p. 1437-1459.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Greatest common divisors of iterates of polynomials
AU - Hsia, Liang-Chung
AU - Tucker, Thomas J.
PY - 2017/1/1
Y1 - 2017/1/1
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
VL - 11
SP - 1437
EP - 1459
JO - Algebra and Number Theory
JF - Algebra and Number Theory
SN - 1937-0652
IS - 6
ER -