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Primitive prime divisors in the critical orbits of one-parameter families of rational polynomials
Mathematical Proceedings of the Cambridge Philosophical Society ( IF 0.6 ) Pub Date : 2021-01-25 , DOI: 10.1017/s0305004121000025
RUFEI REN

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 $\mathbf M_f\geq \max_{c\in \mathbb{Q}}\{\#\calZ(f_c,u)\}$.

中文翻译:

有理多项式单参数族临界轨道上的原始素数除数

对于多项式$f(x)\in\mathbb{Q}[x]$和有理数C,, 我们把$f_c(x)\coloneqq f(x)+c$,并考虑 Zsigmondy 集$\calZ(f_c,u)$与序列相关联$\{f_c^n(u)-u\}_{n\geq 1}$,见定义 1.1,其中$f_c^n$是个n-st 迭代FC. 在本文中,我们证明如果是一个理性的临界点F, 那么存在一个F> 0 使得$\mathbf M_f\geq \max_{c\in \mathbb{Q}}\{\#\calZ(f_c,u)\}$.
更新日期:2021-01-25
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