HEIGHT BOUND AND PREPERIODIC POINTS FOR JOINTLY REGULAR FAMILIES OF RATIONAL MAPS
2
조회
0
다운로드
1
인용
0
참고문헌
키워드
초록
Silverman [14] proved a height inequality for a jointly regular family of rational maps and the author [10] improved it for a jointly reg-ular pair. In this paper, we provide the same improvement for a jointly regular family: let h : <수식> be the logarithmic absolute height on the projective space, let r(f) be the D-ratio of a rational map f which is defined in [10] and let {f1,..., fk│ fl : A^n → A^n} be a finite set of poly-nomial maps which is defined over a number field K. If the intersection of the indeterminacy loci of f1,..., fk is empty, then there is a constant C such that <수식> where r = maxl=1,...,k (r(fl)).
Abstract
Silverman [14] proved a height inequality for a jointly regular family of rational maps and the author [10] improved it for a jointly reg-ular pair. In this paper, we provide the same improvement for a jointly regular family: let h : <수식> be the logarithmic absolute height on the projective space, let r(f) be the D-ratio of a rational map f which is defined in [10] and let {f1,..., fk│ fl : A^n → A^n} be a finite set of poly-nomial maps which is defined over a number field K. If the intersection of the indeterminacy loci of f1,..., fk is empty, then there is a constant C such that <수식> where r = maxl=1,...,k (r(fl)).
참고문헌 (17)
- S. D. Cutkosky, "Resolution of Singularities", Graduate Studies in Mathematics, 63 (2004), American Mathematics Society
- W. Fulton, "Intersection Theory, Second edition" (1998), Springer Verlag
- R. Hartshorne, "Algebraic Geometry" (1977), Springer
- H. Hironaka, "Resolution of singularities of an algebraic variety over a eld of charac- teristic zero. I", Ann. of Math, 79, pp.109~203 (1964)
- S. Kawaguchi, "Canonical height functions for affine plane automorphisms", Math. Ann, 335(2), pp.285~310 (2006)
- S. Kawaguchi, "Local and global canonical height functions for affine space regular automorphisms, preprint", arXiv:0909.3573, 2009, ~ (2009)
- S. Lang, "Fundamentals of Diophantine Geometry" (1983), Springer
- R. Lazarsfeld, "Positivity in Algebraic Geometry. I, Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Bd. 48" (2004), Springer
- C. Lee, "The upper bound of height and regular affine automorphisms on An", submitted, arXiv:0909.3107, ~ (2009)
- C. Lee, "The maximal ratio of coefficients of divisors and an upper bound for height for rational maps", submitted, arXiv:1002.3357, ~ (2010)
- S. Marcello, "Sur la dynamique arithmetique des automorphismes de l'espace affine", Bull. Soc. Math. France, 131(2), pp.229~257 (2003)
- D. G. Northcott, "Periodic points on an algebraic variety", Ann. of Math, 51, pp.167~177 (1950)
- I. Shafarevich, "Basic Algebraic Geometry" (1994), Springer
- J. H. Silverman, "Height bounds and preperiodic points for families of jointly regular affine maps", Pure Appl. Math. Q, 2(1), pp.135~145 (2006)
- J.H.Silverman, "The Arithmetic of Dynamical Systems" (2007), Springer
- J. H. Silverman, "Diophantine Geometry" (2000), Springer
- A. Weil, "Arithmetic on algebraic varieties", Ann. of Math, 53, pp.412~444 (1951)