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

Rufei Ren

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arXiv · 1903.01052

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

Rufei Ren

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

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BibTeX RIS

Rufei Ren. 2019-03-04. Primitive prime divisors in the critical orbits of one-parameter families of rational polynomials. https://arxiv.org/abs/1903.01052

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