关于多项式映射的典范高度差距以及高度为0的多项式的预周期点的画像
On the Canonical Height Gap for Polynomial Maps and Portraits of Preperiodic Points of Polynomials with Height $0$
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中文总结 AI 辅助
研究\(\mathbb{P}^{1}\)上多项式映射的典范高度与朴素高度差，建立与次数\(d\)无关的精确界，以此确定\(\mathbb{Q}\)上高度为0的多项式映射的有理预周期点集并分类其画像。
中文摘要 AI 辅助
我们研究了射影直线\(\mathbb{P}^{1}\)上多项式映射的典范高度与朴素高度之间的差异。虽然这种高度差距的明确上界通常取决于有理映射的次数\(d\)，但我们为多项式映射建立了一个基本与\(d\)无关的精确界。作为应用，我们确定了在\(\mathbb{Q}\)上定义的高度为0的多项式映射的有理预周期点集，并对它们可实现的画像进行了分类。
英文摘要
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.
