Families of polynomials of every degree with no rational preperiodic points

Mohammad Sadek

doi:10.5802/crmath.173

Abstract

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.

Highlights

An arithmetic dynamical system over a number field K consists of a rational function f : Pn(K ) → Pn(K ) of degree at least 2 with coefficients in K where the mth iterate of f is defined recursively by f 1(x) = f (x) and f m(x) = f ( f m−1(x)) when m ≥ 2
Mohammad Sadek integer such that f N (P ) = P, the periodic point P is said to be of exact period N
There exists a bound B (D, n, d ) such that if K /Q is a number field of degree D, and f : Pn(K ) → Pn(K ) is a morphism of degree d ≥ 2 defined over K, the number of K -rational preperiodic points of f is bounded by B (D, n, d )

Summary

Introduction

A point P ∈ Pn(K ) is said to be a periodic point for f if there exists a positive integer m such that f m(P ) = P . There exists a bound B (D, n, d ) such that if K /Q is a number field of degree D, and f : Pn(K ) → Pn(K ) is a morphism of degree d ≥ 2 defined over K , the number of K -rational preperiodic points of f is bounded by B (D, n, d ). If N ≥ 4, there is no quadratic polynomial f (x) ∈ Q[x] with a rational point of exact period N .

Results

Conclusion