Polynomials with many rational preperiodic points

John R. Doyle, Trevor Hyde

Published 2022-01-27Version 1

In this paper we study two questions related to exceptional behavior of preperiodic points of polynomials in $\mathbb{Q}[x]$. We show that for all sufficiently large integers $d$, there exists a polynomial $f_d(x) \in \mathbb{Q}[x]$ with $2\leq \mathrm{deg}(f_d) \leq d$ such that $f_d(x)$ has at least $d + \lfloor \log_{16}(d)\rfloor$ rational preperiodic points. Furthermore, we show that for infinitely many $d \geq 2$, the polynomials $f_d(x)$ and $f_d(x) + 1$ have at least $d^2 + d\lfloor \log_{16}(d)\rfloor - 2d + 1$ common complex preperiodic points. To prove the existence of such polynomials we introduce the technique of dynamical compression.