On rational periodic points of $x^d+c$

Abstract

We consider the polynomials $\displaystyle f(x)=x^d+c$, where $d\ge 2$ and $c\in\mathbb Q$. It is conjectured that if $d=2$, then $f$ has no rational periodic point of exact period $N\ge 4$. In this note, fixing some integer $d\ge 2$, we show that the density of such polynomials with a rational periodic point of any period among all polynomials $f(x)=x^d+c$, $c\in\Q$, is zero. Furthermore, we establish the connection between polynomials $f$ with periodic points and two arithmetic sequences. This yields necessary conditions that must be satisfied by $c$ and $d$ in order for the polynomial $f$ to possess a rational periodic point of exact period $N$, and a lower bound on the number of primitive prime divisors in the critical orbit of $f$ when such a rational periodic point exists. The note also introduces new results on the irreducibility of iterates of $f$.