具有等长圈的Rédei置换
Rédei permutations with cycles of the same length
摘要：
设$\mathbb{F}_q$是奇特征的有限域。我们研究Rédei函数，诱导置换$\mathbb{P}^1(\mathbb{F}_q)$的循环分解只包含周期长度$1$和$j$，为一个整数$j\geq 2$。当$j$是$4$或素数时，我们给出了这种类型的Rédei置换在$\mathbb{P}^1(\mathbb{F}_q)$上存在的充要条件，刻画了由$1$ -和$j$ -圈组成的Rédei置换，并确定了它们的总数。我们还提出了明确的公式为Rédei对合的基础上的不动点的数量，和程序来构建Rédei置换与规定数量的不动点和$j$ -周期的$j \in \{3,4,5\}$。
英文摘要：
Let $\mathbb{F}_q$ be a finite field of odd characteristic. We study Rédei functions that induce permutations over $\mathbb{P}^1(\mathbb{F}_q)$ whose cycle decomposition contains only cycles of length $1$ and $j$, for an integer $j\geq 2$. When $j$ is $4$ or a prime number, we give necessary and sufficient conditions for a Rédei permutation of this type to exist over $\mathbb{P}^1(\mathbb{F}_q)$, characterize Rédei permutations consisting of $1$- and $j$-cycles, and determine their total number. We also present explicit formulas for Rédei involutions based on the number of fixed points, and procedures to construct Rédei permutations with a prescribed number of fixed points and $j$-cycles for $j \in \{3,4,5\}$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
