Families of polynomials of every degree with no rational preperiodic points

Official URL: https://dx.doi.org/10.5802/CRMATH.173

Let K be a number field. Given a polynomial f (x) ∈ K [x] of degree d ≥ 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 : 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 = Q. In this article, given any integer d ≥ 2, we display infinitely many parametric families of polynomials of the form ft (x) = xd +c(t), c(t) ∈ K (t), with no rational preperiodic points for any t ∈ K.