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

TL;DR

This paper investigates the rarity of rational periodic points in polynomials of the form x^d + c over the rationals, establishing density results, necessary conditions, and new irreducibility findings related to such points.

Contribution

It proves that the density of polynomials with rational periodic points of any period is zero and links periodic points to arithmetic sequences, providing new irreducibility results.

Findings

Density of polynomials with rational periodic points is zero

Necessary conditions on c and d for rational periodic points

Lower bounds on primitive prime divisors in critical orbits

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 $f$.