Families of polynomials of every degree with no rational preperiodic points

TL;DR

This paper constructs infinite families of polynomials of any degree over number fields that have no rational preperiodic points, providing new examples related to a conjecture on uniform bounds of preperiodic points.

Contribution

It introduces infinitely many parametric families of polynomials of arbitrary degree with no rational preperiodic points, expanding known examples beyond degrees divisible by 2 or 3.

Findings

Constructed infinite families for all degrees d ≥ 2

Demonstrated no rational preperiodic points for these families

Extended known examples beyond special degrees and fields

Abstract

Let $K$ be a number field. Given a polynomial $f(x)\in K[x]$ of degree $d\ge 2$, it is conjectured that the number of preperiodic points of $f$ is bounded by a uniform bound that depends only on $d$ and $[K:\mathbb Q]$. However, the only examples of parametric families of polynomials with no preperiodic points are known when $d$ is divisible by either $2$ or $3$ and $K=\mathbb Q$. In this article, given any integer $d\ge 2$, we display infinitely many parametric families of polynomials of the form $f_t(x)=x^d+c(t)$, $c(t)\in K(t)$, with no rational preperiodic points for any $t\in K$.