特征为2的有限域上对称矩阵方程解的计数公式及q超几何级数表达

The Number of Solutions to the Symmetric Matrix Equation over a Finite Field of Ch_o=2 and Representation of Its q-hypergeometric Series

设Fq是有q＝2t个元的有限域.本文利用Fq上奇异辛几何和奇异伪辛几何理论，给出当A，C是Fq上对称矩阵时，Fq上适合XAXT＝C的解存在的充要条件以及秩k的解X和解X的个数的明显公式，并且用q超几何级数简化表达解数公式．

Let Fq be a finite field with q=2t elements. In this paper, using singular symplectic and singular pseudo-symplactic geometry over Fq, we have given the sufficient and necessary condition of there exists solutions and the number of solutions X of rank k and solutions X to the equation XAXT=B over Fq. when A and B are symmetric matrices. Finally. we have obtained simple representations of enumerational formulas by q-hypergeometric series.