For the polynomial $f(z) \in \mathbb{ Q}[z]$, we consider the Zsigmondy set $\mathcal{ Z}(f,0)$ associated to the numerators of the sequence $\{ f^n(0)\}_{ n \geq 0}$. In this paper, we provide an upper bound on the largest element of $\mathcal{ Z}(f, 0)$. As an application, we show that the largest element of the set $\mathcal{ Z}(f,0)$ is bounded above by $6$ when $f(z) = z^d + z^e +c \in \mathbb{ Q}[z]$, with $d>e \geq 2$ and $|c|>2$. Furthermore, when $f(z) =z^d+c \in \mathbb{ Q}[z]$ with $|f(0)| > 2^{ \frac{ d}{ d-1}}$ and $d >2$, we also deduce a result of Krieger [Int. Math. Res. Not. IMRN, 23 (2013), pp. 5498-5525] as a consequence of our main result.
Categories: math.NT math.DS