Monomial maps on P2 and their arithmetic dynamics

JOURNAL OF NUMBER THEORY

卷:131

Article

Gregor, Aryeh¹  Yasufuku, Yu² 

[1] NYU, New York, NY 10012 USA

[2] City Univ New York Grad Ctr, New York, NY 10016 USA

关键词: Algebraic dynamics;    Orbits;    Integral points;    Perron's theorem;    Monomials;    Vojta's conjecture;    Cyclotomic theory;   

DOI  :  10.1016/j.jnt.2011.06.012

来源: Elsevier

【 摘 要 】

We say that a rational map on P-n is a monomial map if it can be expressed in some coordinate system as [F-0 : ... : F-n] where each F-1 is a monomial. We consider arithmetic dynamics of monomial maps on P-2. In particular, as Silverman (1993) explored for rational maps on P-1, we determine when orbits contain only finitely many integral points. Our first result is that if some iterate of a monomial map on P2 is a polynomial, then the first such iterate is 1, 2, 3, 4, 6, 8, or 12. We then completely determine all monomial maps whose orbits always contain just finitely many integral points. Our condition is based on the exponents in the monomials. In cases when there are finitely many integral points in all orbits, we also show that the sizes of the numerators and the denominators are comparable. The main ingredients of the proofs are linear algebra, such as Perron-Frobenius theorem. (C) 2011 Elsevier Inc. All rights reserved.

【 授权许可】

Free