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

Abstract

For a rational polynomial $f$ and rational numbers $c,u$, we put $f_c(x):=f(x)+c$, and consider the Zsigmondy set $\mathcal{Z}(f_c,u)$ associated to the sequence $\{f_c^n(u)-u{\}}_{n\ge 0}$, where $f_c^n$ is the $n$-st iteration of $f_c$. In this paper, we prove that if $u$ is a rational critical point of $f$, then there exists an ${\mathbf{M}}_f>0$ such that ${\mathbf{M}}_f\ge {\max }_{c\in \mathbb{Q}}\{\#\mathcal{Z}(f_c,u)\}$.