Primitive prime divisors in the forward orbit of a polynomial
Shanta Laishram, Sudhansu S. Rout, Prabhakar Yadav
TL;DR
This paper studies primitive prime divisors in the forward orbit of 0 under a polynomial map $f\in\mathbb{Q}[z]$, focusing on the Zsigmondy set $\mathcal{Z}(f,0)$. It develops an effective framework combining rigid divisibility properties of the numerator sequence and canonical height theory to obtain explicit bounds on the largest element of $\mathcal{Z}(f,0)$, including a uniform bound $6$ for polynomials of the form $f(z)=z^d+z^e+c$ with $d>e\ge2$ and $|c|>2$. The results also recover Krieger’s bound for $f(z)=z^d+c$ in the large-$|f(0)|$ regime and extend the analysis to several $|c|<2$ subcases, sometimes yielding finiteness or emptiness of $\mathcal{Z}(f,0)$. The work combines lower bounds for canonical heights via local height constants, resultant-based estimates, and a delicate height analysis to translate dynamical information into explicit numerical bounds, advancing effective results in arithmetic dynamics.
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.