论文标题
有限领域的合理功能的定期点
Periodic points of rational functions over finite fields
论文作者
论文摘要
对于$ q $ a Prime Power和$ ϕ $，有理功能具有$ \ Mathbb {f} _Q $中的系数，让$ p（q，ϕ）$为$ \ mathbb {p}^1（\ mathbb {f} _q）$的比例。如果$ d $是一个积极的整数，让$ q_d $是$ d！$的主要力量，让$ \ mathcal {p}（d，q）$是$ p（q，ϕ）$ as $ P（q，ϕ）$的预期价值，则是$ \ \ m nathbbbbbbbbbbbbbbbbbbbb {$ \ n prioncation函数的$ \ \ nmathbb {我们证明，如果$ d $是一个正整数不少于$ 2 $，则$ \ mathcal {p}（d，q）$趋向于0 as $ q $增加$ q_d $。该定理概括了我们以前的工作，该作品仅用于二次多项式，仅用于固定特征。为了推断这一结果，我们证明了关于在某些有限生成的代数中，在残留有限的DEDEKIND域中具有系数的理性函数动态系统的专长定理。这种专业定理概括了我们以前的工作，该工作仅用于维度一号的代数。
For $q$ a prime power and $ϕ$ a rational function with coefficients in $\mathbb{F}_q$, let $p(q,ϕ)$ be the proportion of $\mathbb{P}^1(\mathbb{F}_q)$ that is periodic with respect to $ϕ$. And if $d$ is a positive integer, let $Q_d$ be the set of prime powers coprime to $d!$ and let $\mathcal{P}(d,q)$ be the expected value of $p(q,ϕ)$ as $ϕ$ ranges over rational functions with coefficients in $\mathbb{F}_q$ of degree $d$. We prove that if $d$ is a positive integer no less than $2$, then $\mathcal{P}(d,q)$ tends to 0 as $q$ increases in $Q_d$. This theorem generalizes our previous work, which held only for quadratic polynomials, and only in fixed characteristic. To deduce this result, we prove a uniformity theorem on specializations of dynamical systems of rational functions with coefficients in certain finitely-generated algebras over residually finite Dedekind domains. This specialization theorem generalizes our previous work, which held only for algebras of dimension one.
