Families of polynomials of every degree with no rational preperiodic points

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 $t\in K$.