Families of polynomials of every degree with no rational preperiodic points
Mohammad Sadek · Comptes Rendus Mathématique · 2021
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 f t ( x ) = x d + c ( t ) , c ( t ) ∈ K ( t ) , with no rational preperiodic points for any t ∈ K .