有限域上二阶线性递推的轨道
Orbits of Second Order Linear Recurrences over Finite Fields
AI总结：
该研究聚焦有限域上二阶线性递推的轨道问题，通过分析阿贝尔群乘积元素的阶，探究非平凡轨道长度与其出现频率的关系，还推导了特征多项式根为乘法群生成元的条件，轨道对应模素数的斐波那契型递推。
摘要：
设$Q$为$GL_2(\mathbb{F}_q)$中的矩阵$\displaystyle \begin{pmatrix} a & b \\ 1 & 0 \end{pmatrix}$，其中$\mathbb{F}_q$为有限域，设$G$为由$Q$生成的有限循环群。我们研究$G$在集合$\mathbb{F}_q \times \mathbb{F}_q$上的作用，重点探讨$G$的非平凡轨道长度与其出现频率之间的若干关系。部分研究通过分析阿贝尔群中阶为素数幂的乘积元素的阶来开展。当$q$为素数且满足$b=1$时，这些轨道对应不同初始条件下模$q$的斐波那契型线性递推。我们还推导了$Q$的特征多项式的根为$\mathbb{F}_q^\times$生成元的若干条件，并附实例阐释该理论。
英文摘要：
Let $Q$ be the matrix $\displaystyle \begin{pmatrix} a & b \\ 1 & 0 \end{pmatrix}$ in $GL_2(\mathbb{F}_q)$ where $\mathbb{F}_q$ is a finite field, and let $G$ be the finite cyclic group generated by $Q$. We consider the action of $G$ on the set $\mathbb{F}_q \times \mathbb{F}_q$. In particular, we study certain relationships between the lengths of the non-trivial orbits of $G$, and their frequency of occurrence. This is done in part by investigating the order of elements of a product in an abelian group when the product has prime power order. For $q$ a prime and $b=1$, the orbits correspond to Fibonacci type linear recurrences modulo $q$ for different initial conditions. We also derive certain conditions under which the roots of the characteristic polynomial of $Q$ are generators of $\mathbb{F}_q^\times$. Examples are included to illustrate the theory.
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