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

Abstract

Abstract For a polynomial $f(x)\\in\\mathbb{Q}[x]$ and rational numbers c, u, we put $f_c(x)\\coloneqq f(x)+c$ , and consider the Zsigmondy set $\\calZ(f_c,u)$ associated to the sequence $\\{f_c^n(u)-u\\}_{n\\geq 1}$ , see Definition 1.1, where $f_c^n$ is the n-st iteration of fc. In this paper, we prove that if u is a rational critical point of f, then there exists an Mf > 0 such that $\\mathbf M_f\\geq \\max_{c\\in \\mathbb{Q}}\\{\\#\\calZ(f_c,u)\\}$ .