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Let $K$ be a number field. Given a polynomial $f(x)\in K[x]$ of degree $d\ge2$, it is conjectured that the number of preperiodic points of $f$ is boundedby a uniform bound that depends only on $d$ and $[K:\mathbb Q]$. However, theonly examples of parametric families of polynomials with no preperiodic pointsare known when $d$ is divisible by either $2$ or $3$ and $K=\mathbb Q$. In thisarticle, given any integer $d\ge 2$, we display infinitely many parametricfamilies of polynomials of the form $f_t(x)=x^d+c(t)$, $c(t)\in K(t)$, with norational preperiodic points for any $t\in K$.

Families of polynomials of every degree with no rational preperiodic points