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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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참고문헌 (14)
참고문헌 신청
G. S. Call / 1993 / Canonical heights on varieties with morphisms / Compositio Math. 89(2):163~205
S. D. Cutkosky / 2004 / Resolution of Singularities, Graduate Studies in Mathematics, 63
R. Hartshorne / 1977 / Algebraic Geometry / Springer-Verlag
M. Hindry / 2000 / Diophantine Geometry, Graduate Texts in Mathematics, 201 / Springer-Verlag
H. Hironaka / 1964 / Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II / Ann. of Math. (2) 79:109~203
S. D. Cutkosky / 2004 / Resolution of Singularities, Graduate Studies in Mathematics, 63
R. Hartshorne / 1977 / Algebraic Geometry / Springer-Verlag
M. Hindry / 2000 / Diophantine Geometry, Graduate Texts in Mathematics, 201 / Springer-Verlag
H. Hironaka / 1964 / Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II / Ann. of Math. (2) 79:109~203
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