环Z/p^kZ上一类矩阵方程的解数

The solution number abouta class matrix equation over Z/p^kZ

设 R=Z/pk Z是模整数 pk的有限局部环 ,Di=O Di-Di O ,B=pμB是 R上任意取定的 2 si阶交错阵 ,Δ={Pi∈ GL2 si( R) |Pi Di Pi′-Di=B},其中 Di=diag{pri,… ,pri},0 <ri<k,ri<μ≤k,si≥ 1 .本文计算了 n( Δ) ,其中 n( Δ)表示集合

Let R=Z/pk Z is a finite local ring of module integer pk,Let D i=O Di - Di O ,Δ ={Pi∈ GL2 si ( R) | Pi D i Pi′- D i=B},and matrix B=pμBis a arbitrary alternate matrix with order2 siover R,where p is a prime and k>1 ,Di=diag{pri,… ,pri},0 <ri<k,ri<μ≤ k,si≥ 1 .The number of elements inΔ denoted by n(Δ) is computed.