论文标题
有限还原群中的力量的渐近学
Asymptotics of the powers in finite reductive groups
论文作者
论文摘要
令$ g $为$ \ mathbb f_q $定义的连接还原组。修复整数$ m \ geq 2 $，并考虑$ g $上的电源图$ x \ mapsto x^m $。我们用$ g（\ Mathbb f_q）^m $表示$ g（\ Mathbb f_q）$的图像，并估计它包含的常规半imple，semisimple和常规元素的比例是多少。我们证明，作为$ q \ to \ infty $，所有这些比例都是平等的，并提供了一个公式。我们还针对这些组$ \ text {gl}（n，q）$和$ \ text {u}（n，q）$更明确地计算了这一点。
Let $G$ be a connected reductive group defined over $\mathbb F_q$. Fix an integer $M\geq 2$, and consider the power map $x\mapsto x^M$ on $G$. We denote the image of $G(\mathbb F_q)$ under this map by $G(\mathbb F_q)^M$ and estimate what proportion of regular semisimple, semisimple and regular elements of $G(\mathbb F_q)$ it contains. We prove that as $q\to\infty$, all of these proportions are equal and provide a formula for the same. We also calculate this more explicitly for the groups $\text{GL}(n,q)$ and $\text{U}(n,q)$.
