Primitive prime divisors in the forward orbit of a polynomial

Abstract

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 $\{f^n(0){\}}_{n\ge 0}$. 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 $d>e\ge 2$ 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.

arXiv commentComments or suggestions are welcome. 14 pages