  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 $\{f_c^n(u)-u\}_{n\geq 0}$, 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 $\mathbf M_f\geq \max_{c\in
\mathbb{Q}}\{\#\mathcal{Z}(f_c,u)\}$.
