有限局部环上对称矩阵的计数定理(I)

设$A_{n}(R)$是有限局部环$Z/p^{k}Z$上$n$阶对称矩阵的集合, 这里$n\geq 2$. $p$是大于$2$素数, $p\equiv1({\rm mod}4)$ 且$k>1$. 通过确定有限局部环$Z/p^{k}Z$上对称矩阵的标准型, 计算出$A_{n}(R)$在线性群${\rm GL}_{n}(R)$作用下的轨道数, 从而计算出由特定对称矩阵确定的正交群的阶以及与特定对称矩阵在同一轨道的对称矩阵的阶.

Anzahl Theorems in Symmetric Matrices over Finite Local Rings (I)

Let $A_{n}(R)$ be the set of symmetric matrices over $Z/p^{k}Z$ with order $n$, where $n\geq 2$, $p$ is a prime, $p>2$ and $p\equiv1({\rm mod}4)$, $k>1$. By determining the normal form of $n$ by $n$ symmetric matrices over $Z/p^{k}Z$, we compute the number of the orbits of $ A_{n}(R)$ and then compute the order of the orthogonal group determined by the special symmetric matrix. Finally we get the number of the symmetric matrices which are in the same orbit with the special symmetric matrix.