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 $\sum\limits_{l Keywords height;rational map;preperiodic points;jointly regular family; Language English Cited by References 1. S. D. Cutkosky, Resolution of Singularities, Graduate Studies in Mathematics, Vol 63, American Mathematics Society, 2004. 2. W. Fulton, Intersection Theory, Second edition, Springer-Verlag, Berlin, 1998. 3. R. Hartshorne, Algebraic Geometry, Springer, 1977. 4. H. Hironaka, Resolution of singularities of an algebraic variety over a field of charac-teristic zero. I, Ann. of Math. (2) 79 (1964), 109-203. 5. S. Kawaguchi, Canonical height functions for affine plane automorphisms, Math. Ann. 335 (2006), no. 2, 285-310. 6. S. Kawaguchi, Local and global canonical height functions for affine space regular automor-phisms, preprint, arXiv:0909.3573, 2009. 7. S. Lang, Fundamentals of Diophantine Geometry, Berlin, Heidelberg, New York, Springer 1983. 8. R. Lazarsfeld, Positivity in Algebraic Geometry. I, Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Bd. 48, Springer, New York, 2004. 9. C. Lee, The upper bound of height and regular affine automorphisms on$A^{n}\$, submitted, arXiv:0909.3107, 2009.

10.
C. Lee, The maximal ratio of coefficients of divisors and an upper bound for height for rational maps, submitted, arXiv:1002.3357, 2010.

11.
S. Marcello, Sur la dynamique arithmetique des automorphismes de l'espace affine, Bull. Soc. Math. France 131 (2003), no. 2, 229-257.

12.
D. G. Northcott, Periodic points on an algebraic variety, Ann. of Math. (2) 51 (1950), 167-177.

13.
I. Shafarevich, Basic Algebraic Geometry, Springer, 1994.

14.
J. H. Silverman, Height bounds and preperiodic points for families of jointly regular affine maps, Pure Appl. Math. Q. 2 (2006), no. 1, part 1, 135-145.

15.
J. H. Silverman, The Arithmetic of Dynamical Systems, Springer, 2007.

16.
J. H. Silverman and M. Hindry, Diophantine Geometry, Springer 2000.

17.
A. Weil, Arithmetic on algebraic varieties, Ann. of Math. (2) 53 (1951), 412-444.