HEIGHT BOUND AND PREPERIODIC POINTS FOR JOINTLY REGULAR FAMILIES OF RATIONAL MAPS

Title & Authors
HEIGHT BOUND AND PREPERIODIC POINTS FOR JOINTLY REGULAR FAMILIES OF RATIONAL MAPS
Lee, Chong-Gyu;

Abstract
Silverman [14] proved a height inequality for a jointly regular family of rational maps and the author [10] improved it for a jointly regular pair. In this paper, we provide the same improvement for a jointly regular family: let h : $\small{{\mathbb{P}}_{\mathbb{Q}}^n{\rightarrow}{{\mathbb{R}}}$ be the logarithmic absolute height on the projective space, let r(f) be the D-ratio of a rational map f which is de ned in [10] and let {$\small{f_1,{\ldots},f_k|f_l:\mathbb{A}^n{\rightarrow}\mathbb{A}^n}$} bbe finite set of polynomial maps which is defined over a number field K. If the intersection of the indeterminacy loci of $\small{f_1,{\ldots},f_k}$ is empty, then there is a constant C such that $\small{ \sum\limits_{l=1}^k\frac{1}{def\;f_\iota}h(f_\iota(P))}$>$\small{(1+\frac{1}{r})f(P)-C}$ for all $\small{P{\in}\mathbb{A}^n}$ where r= $\small{max_{\iota=1},{\ldots},k(r(f_l))}$.
Keywords
height;rational map;preperiodic points;jointly regular family;
Language
English
Cited by
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