Published online by Cambridge University Press: 25 January 2021
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
$\mathbf M_f\geq \max_{c\in \mathbb{Q}}\{\#\calZ(f_c,u)\}$.