Monomial Maps on P^2 and their Arithmetic Dynamics
Monomial Maps on P^2 and their Arithmetic Dynamics
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P^2 上的单项式映射及其算术动力学
DOI：
10.1016/j.jnt.2011.06.012
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发表时间：
2011
影响因子：
0.7
通讯作者：
Aryeh Gregor and Yu Yasufuku
中科院分区：
文献类型：
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作者：
Holly Krieger;Aaron Levin;Zachary Scherr;Thomas Tucker;Yu Yasufuku;and Michael Zieve;Yu Yasufuku;安福 悠;Yu Yasufuku;Thomas Scanlon and Yu Yasufuku;Yu Yasufuku;Yu Yasufuku;Yu Yasufuku;Aryeh Gregor and Yu Yasufuku
我们说p上的有理映射是单项式映射，如果它可以在某些坐标系中表示为[F0:⋯:Fn]，其中每个fi1都是单项式。研究了P2上单项式映射的算术动力学问题。特别是，正如Silverman（1993）在P1上探索的理性映射，我们确定轨道何时只包含有限多个积分点。我们的第一个结果是，如果p2上的一个单项式映射的某个迭代是一个多项式，那么第一个这样的迭代是1,2,3,4,6,8或12。然后我们完全确定所有的单项式映射，其轨道总是包含有限多个积分点。我们的条件是基于单项式的指数。在所有轨道上有有限多个积分点的情况下，我们还证明了分子和分母的大小是可比较的。证明的主要成分是线性代数，如佩龙-弗罗贝尼乌斯定理。
We say that a rational map on Pnis a monomial map if it can be expressed in some coordinate system as [F0:⋯:Fn] where each Fiis a monomial. We consider arithmetic dynamics of monomial maps on P2. In particular, as Silverman (1993) explored for rational maps on P1, we determine when orbits contain only finitely many integral points. Our first result is that if some iterate of a monomial map on P2is 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.