Greatest common divisors of iterates of polynomials
| Autor: | Thomas J. Tucker, Liang Chung Hsia |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2016 |
| Předmět: |
Discrete mathematics
Polynomial Algebra and Number Theory Mathematics - Number Theory Mathematics::Number Theory 010102 general mathematics 14G25 heights equidstribution Dynamical Systems (math.DS) Composition (combinatorics) Lambda 01 natural sciences Combinatorics Iterated function composition 0103 physical sciences FOS: Mathematics 010307 mathematical physics gcd Number Theory (math.NT) 0101 mathematics Mathematics - Dynamical Systems Mathematics 37P05 |
| Zdroj: | Algebra Number Theory 11, no. 6 (2017), 1437-1459 |
| Popis: | Following work of Bugeaud, Corvaja, and Zannier for integers, Ailon and Rudnick prove that for any multiplicatively independent polynomials, $a, b \in {\mathbb C}[x]$, there is a polynomial $h$ such that for all $n$, we have \[ \gcd(a^n - 1, b^n - 1) \mid h\] We prove a compositional analog of this theorem, namely that if $f, g \in {\mathbb C}[x]$ are nonconstant compositionally independent polynomials and $c(x) \in {\mathbb C}[x]$, then there are at most finitely many $\lambda$ with the property that there is an $n$ such that $(x - \lambda)$ divides $\gcd(f^{\circ n}(x) - c(x), g^{\circ n}(x) - c(x))$. Comment: 22 pages |
| Databáze: | OpenAIRE |
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