有限域上交换矩阵的同时相似类
Simultaneous Similarity Classes of Commuting matrices over a finite field
摘要：
本文研究了有限域上多元多项式代数模同构类的计数问题。这与有限域上矩阵的交换元组直到同时相似的分类相同。设$c_{n,k}(q)$表示$n$ -维$\mathbb{F}_q[x_1,\dotsc,x_k]$ -模的同构类的个数.生成函数$\sum_k c_{n,k}(q)t^k$是一个有理函数。我们为$n\leq 4$计算此函数。我们发现它的系数是$q$中非负整数系数的多项式函数。
英文摘要：
This paper concerns the enumeration of isomorphism classes of modules of a polynomial algebra in several variables over a finite field. This is the same as the classification of commuting tuples of matrices over a finite field up to simultaneous similarity. Let $c_{n,k}(q)$ denote the number of isomorphism classes of $n$-dimensional $\mathbb{F}_q[x_1,\dotsc,x_k]$-modules. The generating function $\sum_k c_{n,k}(q)t^k$ is a rational function. We compute this function for $n\leq 4$. We find that its coefficients are polynomial functions in $q$ with non-negative integer coefficients.
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。
