On the Canonical Height Gap for Polynomial Maps and Portraits of Preperiodic Points of Polynomials with Height $0$

Abstract

We study the difference between the canonical height and the naive height for polynomial maps on $\mathbb{P}^{1}$. While explicit upper bounds on this height gap generally depend on the degree $d$ for rational maps, we establish a refined bound for polynomial maps that is essentially independent of $d$. As an application, we determine the set of rational preperiodic points for polynomial maps defined over $\mathbb{Q}$ of height $0$ and classify their realizable portraits.