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arXiv 1601.07293math.NTmath.DS
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关于定义在$\mathbb{F}_p(t)$上的有理函数的预周期点

On preperiodic points of rational functions defined over $\mathbb{F}_p(t)$

Jung Kyu Canci, Laura Paladino

备注：
arXiv admin note: substantial text overlap with arXiv:1403.2293
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摘要：

令$P\in\mathbb{P}_1(\mathbb{Q})$是系数在$\mathbb{Z}$中的一元多项式的周期点。用初等技巧可以看出，$P$的最小周期至多是$2$。最近，我们证明了这一事实的推广到所有有理函数定义在${\mathbb{Q}}$具有良好的减少无处不在（即在任何有限的地方$\mathbb{Q}$）。在$\mathbb{Z}$中系数的一元多项式的集合可以被特征化，直到PGL $_2({\mathbb{Z}})$中元素的共轭，作为定义在$\mathbb{Q}$上的所有有理函数的集合，在$\mathbb{Q}$中具有完全分歧的不动点，并且处处具有良好的约化。令$p$为素数，令${\mathbb{F}}_p$为具有$p$元素的字段。本文研究了有理整体函数域${\mathbb{F}}_p(t)$上的有理函数在每个有限位置都具有良好的约化.我们证明了${\mathbb{F}}_p(t)\cup \{\infty\}$中周期点和预周期点的轨道基数的一些界限。

英文摘要：

Let $P\in\mathbb{P}_1(\mathbb{Q})$ be a periodic point for a monic polynomial with coefficients in $\mathbb{Z}$. With elementary techniques one sees that the minimal periodicity of $P$ is at most $2$. Recently we proved a generalization of this fact to the set of all rational functions defined over ${\mathbb{Q}}$ with good reduction everywhere (i.e. at any finite place of $\mathbb{Q}$). The set of monic polynomials with coefficients in $\mathbb{Z}$ can be characterized, up to conjugation by elements in PGL$_2({\mathbb{Z}})$, as the set of all rational functions defined over $\mathbb{Q}$ with a totally ramified fixed point in $\mathbb{Q}$ and with good reduction everywhere. Let $p$ be a prime number and let ${\mathbb{F}}_p$ be the field with $p$ elements. In the present paper we consider rational functions defined over the rational global function field ${\mathbb{F}}_p(t)$ with good reduction at every finite place. We prove some bounds for the cardinality of orbits in ${\mathbb{F}}_p(t)\cup \{\infty\}$ for periodic and preperiodic points.

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