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

Haruki Imamura

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arXiv · 2607.10160

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

Haruki Imamura

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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.

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Haruki Imamura. 2026-07-11. On the Canonical Height Gap for Polynomial Maps and Portraits of Preperiodic Points of Polynomials with Height $0$. https://arxiv.org/abs/2607.10160

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