关于同时共轭下有限域上幂零矩阵元组的个数
On Numbers of Tuples of Nilpotent Matrices over Finite Fields under Simultaneous Conjugation
摘要：
同时共轭域上幂零矩阵元组的分类问题被认为是“无望的”。然而，对于任何给定的有限域上的矩阵阶，相关轨道的数目总是有限的。本文用与华[5]相同的方法给出了绝对不可分解轨道个数的一个闭公式，这些轨道在域扩张上是不可分裂的.因此，这些数字总是多项式的基数域的整数系数。它被证明，这些系数总是非负的。
英文摘要：
The problem of classifying tuples of nilpotent matrices over a field under simultaneous conjugation is considered "hopeless". However, for any given matrix order over a finite field, the number of concerned orbits is always finite. This paper gives a closed formula for the number of absolutely indecomposable orbits using the same methodology as Hua [5]; those orbits are non-splittable over field extensions. As a consequence, those numbers are always polynomials in the cardinality of the base field with integral coefficients. It is conjectured that those coefficients are always non-negative.
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。
