### 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(a^{n}-1,b^{n}-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(f^{on}(x)-c(x),g^{on}(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 |

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### Keywords

- Composition
- Equidstribution
- Gcd
- Heights

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Algebra and Number Theory*,

*11*(6), 1437-1459. https://doi.org/10.2140/ant.2017.11.1437

**Greatest common divisors of iterates of polynomials.** / Hsia, Liang-Chung; Tucker, Thomas J.

Research output: Contribution to journal › Article

*Algebra and Number Theory*, vol. 11, no. 6, pp. 1437-1459. https://doi.org/10.2140/ant.2017.11.1437

}

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 -