Primitive prime divisors in the forward orbit of a polynomial

TL;DR

This paper studies primitive prime divisors in the forward orbit of polynomial sequences over rationals, providing bounds on the Zsigmondy set and extending results for specific polynomial forms.

Contribution

It establishes an upper bound on the largest element of the Zsigmondy set for certain polynomials, advancing understanding of primitive prime divisors in polynomial orbits.

Findings

Largest element of Zsigmondy set bounded by 6 for specific polynomials

Bounded the Zsigmondy set for polynomials of the form z^d + z^e + c

Extended Krieger's result for polynomials with large constant term

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 $\{f^n(0)\}_{n \geq 0}$. 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.