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Primitive prime divisors in the critical orbits of one-parameter families of rational polynomials

Part of: Elementary number theory Sequences and sets Complex dynamical systems

Published online by Cambridge University Press:  25 January 2021

RUFEI REN
RUFEI REN*
Affiliation:
Department of Mathematics, Fudan University, 220 Handan Rd., Yangpu District, Shanghai200433, China. e-mail: [email protected]
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Abstract

For a polynomial $f(x)\in\mathbb{Q}[x]$ and rational numbers c, u, we put $f_c(x)\coloneqq f(x)+c$, and consider the Zsigmondy set $\calZ(f_c,u)$ associated to the sequence $\{f_c^n(u)-u\}_{n\geq 1}$, see Definition 1.1, where $f_c^n$ is the n-st iteration of fc. In this paper, we prove that if u is a rational critical point of f, then there exists an Mf > 0 such that $\mathbf M_f\geq \max_{c\in \mathbb{Q}}\{\#\calZ(f_c,u)\}$.

MSC classification

Primary: 11B37: Recurrences
Secondary: 37F10: Polynomials; rational maps; entire and meromorphic functions 11A41: Primes
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 171 , Issue 3 , November 2021 , pp. 569 - 584
Copyright
© Cambridge Philosophical Society 2021

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