Height inequality for rational maps and bounds for preperiodic points

다국어 초록 (Multilingual Abstract)

In this paper, we introduce the $D$-ratio of a rational map $f:\mathbb{P}^n \dashrightarrow \mathbb{P}^n$, defined over $\overline{\mathbb{Q}}$, whose indeterminacy locus is contained in a hyperplane $H$ on $\mathbb{P}^n$. The $D$-ratio $r(f;\overline{V})$ characterizes endomorphisms and provides a useful height inequality on $\mathbb{P}^n(\overline{\mathbb{Q}}) \setminus H$. We also provide a dynamical application: preperiodic points of dynamical systems of small $D$-ratio are of bounded height.

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