# Primitive prime divisors in the critical orbits of one-parameter families of rational polynomials

## TL;DR

This paper investigates the occurrence of primitive prime divisors in the critical orbits of one-parameter families of rational polynomials, establishing a uniform bound on the size of the Zsigmondy set for critical points.

## Contribution

It proves a uniform bound on the number of primitive prime divisors in the critical orbits of rational polynomial families when evaluated at rational critical points.

## Key findings

- Existence of a uniform bound for the Zsigmondy set size
- Bound depends only on the polynomial family, not on the parameter c
- Critical points play a key role in primitive prime divisor distribution

## Abstract

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