Polynomials with many rational preperiodic points

TL;DR

This paper constructs polynomials over rationals with many rational preperiodic points and explores their shared complex preperiodic points with shifted versions, revealing new extremal behaviors.

Contribution

It demonstrates the existence of polynomials with a large number of rational preperiodic points and analyzes their common preperiodic points with shifted polynomials, a novel extremal property.

Findings

Existence of polynomials with at least d + floor(log2(d)) rational preperiodic points for all d ≥ 2.

Construction of polynomials with at least d^2 + d*floor(log2(d)) - 2d + 1 common complex preperiodic points with their shifts.

Identification of infinitely many degrees d where these extremal properties hold.

Abstract

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 $d\geq 2$, 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 $d + \lfloor \log_2(d)\rfloor$ rational preperiodic points. Furthermore, we show that for infinitely many integers $d$, the polynomials $f_d(x)$ and $f_d(x) + 1$ have at least $d^2 + d\lfloor \log_2(d)\rfloor - 2d + 1$ common complex preperiodic points.