Sylvester矩阵、子结式、辗转相除法的三者关系(第三部分)

作者：盐析白兔 | 2024-05-23 17:34:16

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2. 执行辗转相除法第二步

$F_{1} = Q_{2} \times F_{2} + F_{3}\ \ \ \ \ \ \ \ \ \ \deg\left( F_{1} \right) = 7\ \ \ \ \ \ \deg\left( F_{2} \right) = 6\ \ \ \ \ \ \deg\left( F_{3} \right) = 5$

$\left| S^{'} \right| =$

\begin{matrix} \begin{matrix} F_{1} \\ F_{1} \\ F_{2} \\ F_{2} \\ F_{2} \\ F_{2} \\ F_{2} \\ F_{2} \\ F_{2} \\ F_{1} \\ F_{1} \\ F_{1} \\ F_{1} \\ F_{1} \\ F_{1} \end{matrix}

$\begin{matrix} \begin{matrix} F_{1} \\ F_{1} \\ F_{2} \\ F_{2} \\ F_{2} \\ F_{2} \\ F_{2} \\ F_{2} \\ F_{2} \\ F_{1} \\ F_{1} \\ F_{1} \\ F_{1} \\ F_{1} \\ F_{1} \end{matrix}$ & \left|

\begin{matrix} b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} \\ 0 & 0 & b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} \end{matrix}

$\begin{matrix} b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} \\ 0 & 0 & b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} \end{matrix}$ \right| \end{matrix} =

\begin{matrix} \begin{matrix} F_{1} \\ F_{1} \\ F_{2} \\ F_{2} \\ F_{2} \\ F_{2} \\ F_{2} \\ F_{2} \\ F_{2} \\ F_{3} \\ F_{3} \\ F_{3} \\ F_{3} \\ F_{3} \\ F_{3} \end{matrix}

$\begin{matrix} \begin{matrix} F_{1} \\ F_{1} \\ F_{2} \\ F_{2} \\ F_{2} \\ F_{2} \\ F_{2} \\ F_{2} \\ F_{2} \\ F_{3} \\ F_{3} \\ F_{3} \\ F_{3} \\ F_{3} \\ F_{3} \end{matrix}$ & \left|

\begin{matrix} b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} \\ 0 & 0 & 0 & 0 & d_{5} & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & d_{5} & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & d_{5} & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & d_{5} & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & d_{5} & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & d_{5} & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} \end{matrix}

$\begin{matrix} b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} \\ 0 & 0 & 0 & 0 & d_{5} & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & d_{5} & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & d_{5} & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & d_{5} & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & d_{5} & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & d_{5} & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} \end{matrix}$ \right| \end{matrix}

S^{^{'}} = F_{1} F_{1} F_{2} F_{2} F_{2} F_{2} F_{2} F_{2} F_{2} F_{1} F_{1} F_{1} F_{1} F_{1} F_{1} b_{7} 00000000000000 b_{6} b_{7} 0000000000000 b_{5} b_{6} c_{6} 000000 b_{7} 00000 b_{4} b_{5} c_{5} c_{6} 00000 b_{6} b_{7} 0000 b_{3} b_{4} c_{4} c_{5} c_{6} 0000 b_{5} b_{6} b_{7} 000 b_{2} b_{3} c_{3} c_{4} c_{5} c_{6} 000 b_{4} b_{5} b_{6} b_{7} 00 b_{1} b_{2} c_{2} c_{3} c_{4} c_{5} c_{6} 00 b_{3} b_{4} b_{5} b_{6} b_{7} 0 b_{0} b_{1} c_{1} c_{2} c_{3} c_{4} c_{5} c_{6} 0 b_{2} b_{3} b_{4} b_{5} b_{6} b_{7} 0 b_{0} c_{0} c_{1} c_{2} c_{3} c_{4} c_{5} c_{6} b_{1} b_{2} b_{3} b_{4} b_{5} b_{6} 000 c_{0} c_{1} c_{2} c_{3} c_{4} c_{5} b_{0} b_{1} b_{2} b_{3} b_{4} b_{5} 0000 c_{0} c_{1} c_{2} c_{3} c_{4} 0 b_{0} b_{1} b_{2} b_{3} b_{4} 00000 c_{0} c_{1} c_{2} c_{3} 00 b_{0} b_{1} b_{2} b_{3} 000000 c_{0} c_{1} c_{2} 000 b_{0} b_{1} b_{2} 0000000 c_{0} c_{1} 0000 b_{0} b_{1} 00000000 c_{0} 00000 b_{0} = F_{1} F_{1} F_{2} F_{2} F_{2} F_{2} F_{2} F_{2} F_{2} F_{3} F_{3} F_{3} F_{3} F_{3} F_{3} b_{7} 00000000000000 b_{6} b_{7} 0000000000000 b_{5} b_{6} c_{6} 000000000000 b_{4} b_{5} c_{5} c_{6} 00000000000 b_{3} b_{4} c_{4} c_{5} c_{6} 0000 d_{5} 00000 b_{2} b_{3} c_{3} c_{4} c_{5} c_{6} 000 d_{4} d_{5} 0000 b_{1} b_{2} c_{2} c_{3} c_{4} c_{5} c_{6} 00 d_{3} d_{4} d_{5} 000 b_{0} b_{1} c_{1} c_{2} c_{3} c_{4} c_{5} c_{6} 0 d_{2} d_{3} d_{4} d_{5} 00 0 b_{0} c_{0} c_{1} c_{2} c_{3} c_{4} c_{5} c_{6} d_{1} d_{2} d_{3} d_{4} d_{5} 0 000 c_{0} c_{1} c_{2} c_{3} c_{4} c_{5} d_{0} d_{1} d_{2} d_{3} d_{4} d_{5} 0000 c_{0} c_{1} c_{2} c_{3} c_{4} 0 d_{0} d_{1} d_{2} d_{3} d_{4} 00000 c_{0} c_{1} c_{2} c_{3} 00 d_{0} d_{1} d_{2} d_{3} 000000 c_{0} c_{1} c_{2} 000 d_{0} d_{1} d_{2} 0000000 c_{0} c_{1} 0000 d_{0} d_{1} 00000000 c_{0} 00000 d_{0}

对应子结式 $S_{5}$ ：

$S_{5} = ( - 1)^{(m - 5)(l - 5)}detpol$

\begin{pmatrix} \begin{matrix} F_{1} \\ F_{1} \\ F_{1} \\ F_{0} \\ F_{0} \end{matrix} & \begin{pmatrix} b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 & 0 \\ 0 & b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 \\ 0 & 0 & b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} \\ a_{8} & a_{7} & a_{6} & a_{5} & a_{4} & a_{3} & a_{2} & a_{1} & a_{0} & 0 \\ 0 & a_{8} & a_{7} & a_{6} & a_{5} & a_{4} & a_{3} & a_{2} & a_{1} & a_{0} \end{pmatrix}

$\begin{pmatrix} \begin{matrix} F_{1} \\ F_{1} \\ F_{1} \\ F_{0} \\ F_{0} \end{matrix} & \begin{pmatrix} b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 & 0 \\ 0 & b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 \\ 0 & 0 & b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} \\ a_{8} & a_{7} & a_{6} & a_{5} & a_{4} & a_{3} & a_{2} & a_{1} & a_{0} & 0 \\ 0 & a_{8} & a_{7} & a_{6} & a_{5} & a_{4} & a_{3} & a_{2} & a_{1} & a_{0} \end{pmatrix}$ \end{pmatrix} = ( - 1)^{(m - 5)(l - 5)}detpol

\begin{pmatrix} \begin{matrix} F_{1} \\ F_{1} \\ F_{2} \\ F_{2} \\ F_{3} \end{matrix} & \begin{pmatrix} b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 & 0 \\ 0 & b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 \\ 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} & 0 \\ 0 & 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} \\ 0 & 0 & 0 & 0 & d_{5} & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} \end{pmatrix}

$\begin{pmatrix} \begin{matrix} F_{1} \\ F_{1} \\ F_{2} \\ F_{2} \\ F_{3} \end{matrix} & \begin{pmatrix} b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 & 0 \\ 0 & b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 \\ 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} & 0 \\ 0 & 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} \\ 0 & 0 & 0 & 0 & d_{5} & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} \end{pmatrix}$ \end{pmatrix}

S_{5} = (- 1)^{(m - 5) (l - 5)} d e tp o l F_{1} F_{1} F_{1} F_{0} F_{0} b_{7} 00 a_{8} 0 b_{6} b_{7} 0 a_{7} a_{8} b_{5} b_{6} b_{7} a_{6} a_{7} b_{4} b_{5} b_{6} a_{5} a_{6} b_{3} b_{4} b_{5} a_{4} a_{5} b_{2} b_{3} b_{4} a_{3} a_{4} b_{1} b_{2} b_{3} a_{2} a_{3} b_{0} b_{1} b_{2} a_{1} a_{2} 0 b_{0} b_{1} a_{0} a_{1} 00 b_{0} 0 a_{0} = (- 1)^{(m - 5) (l - 5)} d e tp o l F_{1} F_{1} F_{2} F_{2} F_{3} b_{7} 0000 b_{6} b_{7} 000 b_{5} b_{6} c_{6} 00 b_{4} b_{5} c_{5} c_{6} 0 b_{3} b_{4} c_{4} c_{5} d_{5} b_{2} b_{3} c_{3} c_{4} d_{4} b_{1} b_{2} c_{2} c_{3} d_{3} b_{0} b_{1} c_{1} c_{2} d_{2} 0 b_{0} c_{0} c_{1} d_{1} 000 c_{0} d_{0}

3.执行辗转相除法第三步

$F_{2} = Q_{3} \times F_{3} + F_{4}\ \ \ \ \ \ \ \ \ \ \deg\left( F_{2} \right) = 6\ \ \ \ \ \ \deg\left( F_{3} \right) = 5\ \ \ \ \ \ \deg\left( F_{4} \right) = 4$

$\left| S^{'} \right| =$

\begin{matrix} \begin{matrix} F_{1} \\ F_{1} \\ F_{2} \\ F_{2} \\ F_{3} \\ F_{3} \\ F_{3} \\ F_{3} \\ F_{3} \\ F_{3} \\ F_{2} \\ F_{2} \\ F_{2} \\ F_{2} \\ F_{2} \end{matrix}

$\begin{matrix} \begin{matrix} F_{1} \\ F_{1} \\ F_{2} \\ F_{2} \\ F_{3} \\ F_{3} \\ F_{3} \\ F_{3} \\ F_{3} \\ F_{3} \\ F_{2} \\ F_{2} \\ F_{2} \\ F_{2} \\ F_{2} \end{matrix}$ & \left|

\begin{matrix} b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & d_{5} & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & d_{5} & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & d_{5} & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & d_{5} & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & d_{5} & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & d_{5} & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} \\ 0 & 0 & 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} \end{matrix}

$\begin{matrix} b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & d_{5} & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & d_{5} & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & d_{5} & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & d_{5} & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & d_{5} & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & d_{5} & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} \\ 0 & 0 & 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} \end{matrix}$ \right| \end{matrix} =

\begin{matrix} \begin{matrix} F_{1} \\ F_{1} \\ F_{2} \\ F_{2} \\ F_{3} \\ F_{3} \\ F_{3} \\ F_{3} \\ F_{3} \\ F_{3} \\ F_{4} \\ F_{4} \\ F_{4} \\ F_{4} \\ F_{4} \end{matrix}

$\begin{matrix} \begin{matrix} F_{1} \\ F_{1} \\ F_{2} \\ F_{2} \\ F_{3} \\ F_{3} \\ F_{3} \\ F_{3} \\ F_{3} \\ F_{3} \\ F_{4} \\ F_{4} \\ F_{4} \\ F_{4} \\ F_{4} \end{matrix}$ & \left|

\begin{matrix} b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & d_{5} & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & d_{5} & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & d_{5} & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & d_{5} & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & d_{5} & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & d_{5} & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} \\ 0 & 0 & 0 & 0 & 0 & 0 & e_{4} & e_{3} & e_{2} & e_{1} & e_{0} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & e_{4} & e_{3} & e_{2} & e_{1} & e_{0} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & e_{4} & e_{3} & e_{2} & e_{1} & e_{0} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & e_{4} & e_{3} & e_{2} & e_{1} & e_{0} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & e_{4} & e_{3} & e_{2} & e_{1} & e_{0} \end{matrix}

$\begin{matrix} b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & d_{5} & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & d_{5} & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & d_{5} & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & d_{5} & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & d_{5} & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & d_{5} & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} \\ 0 & 0 & 0 & 0 & 0 & 0 & e_{4} & e_{3} & e_{2} & e_{1} & e_{0} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & e_{4} & e_{3} & e_{2} & e_{1} & e_{0} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & e_{4} & e_{3} & e_{2} & e_{1} & e_{0} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & e_{4} & e_{3} & e_{2} & e_{1} & e_{0} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & e_{4} & e_{3} & e_{2} & e_{1} & e_{0} \end{matrix}$ \right| \end{matrix}

S^{^{'}} = F_{1} F_{1} F_{2} F_{2} F_{3} F_{3} F_{3} F_{3} F_{3} F_{3} F_{2} F_{2} F_{2} F_{2} F_{2} b_{7} 00000000000000 b_{6} b_{7} 0000000000000 b_{5} b_{6} c_{6} 000000000000 b_{4} b_{5} c_{5} c_{6} 00000000000 b_{3} b_{4} c_{4} c_{5} d_{5} 00000 c_{6} 0000 b_{2} b_{3} c_{3} c_{4} d_{4} d_{5} 0000 c_{5} c_{6} 000 b_{1} b_{2} c_{2} c_{3} d_{3} d_{4} d_{5} 000 c_{4} c_{5} c_{6} 00 b_{0} b_{1} c_{1} c_{2} d_{2} d_{3} d_{4} d_{5} 00 c_{3} c_{4} c_{5} c_{6} 0 0 b_{0} c_{0} c_{1} d_{1} d_{2} d_{3} d_{4} d_{5} 0 c_{2} c_{3} c_{4} c_{5} c_{6} 000 c_{0} d_{0} d_{1} d_{2} d_{3} d_{4} d_{5} c_{1} c_{2} c_{3} c_{4} c_{5} 00000 d_{0} d_{1} d_{2} d_{3} d_{4} c_{0} c_{1} c_{2} c_{3} c_{4} 000000 d_{0} d_{1} d_{2} d_{3} 0 c_{0} c_{1} c_{2} c_{3} 0000000 d_{0} d_{1} d_{2} 00 c_{0} c_{1} c_{2} 00000000 d_{0} d_{1} 000 c_{0} c_{1} 000000000 d_{0} 0000 c_{0} = F_{1} F_{1} F_{2} F_{2} F_{3} F_{3} F_{3} F_{3} F_{3} F_{3} F_{4} F_{4} F_{4} F_{4} F_{4} b_{7} 00000000000000 b_{6} b_{7} 0000000000000 b_{5} b_{6} c_{6} 000000000000 b_{4} b_{5} c_{5} c_{6} 00000000000 b_{3} b_{4} c_{4} c_{5} d_{5} 0000000000 b_{2} b_{3} c_{3} c_{4} d_{4} d_{5} 000000000 b_{1} b_{2} c_{2} c_{3} d_{3} d_{4} d_{5} 000 e_{4} 0000 b_{0} b_{1} c_{1} c_{2} d_{2} d_{3} d_{4} d_{5} 00 e_{3} e_{4} 000 0 b_{0} c_{0} c_{1} d_{1} d_{2} d_{3} d_{4} d_{5} 0 e_{2} e_{3} e_{4} 00 000 c_{0} d_{0} d_{1} d_{2} d_{3} d_{4} d_{5} e_{1} e_{2} e_{3} e_{4} 0 00000 d_{0} d_{1} d_{2} d_{3} d_{4} e_{0} e_{1} e_{2} e_{3} e_{4} 000000 d_{0} d_{1} d_{2} d_{3} 0 e_{0} e_{1} e_{2} e_{3} 0000000 d_{0} d_{1} d_{2} 00 e_{0} e_{1} e_{2} 00000000 d_{0} d_{1} 000 e_{0} e_{1} 000000000 d_{0} 0000 e_{0}

对应子结式 $S_{4}$ ：

$S_{4} = ( - 1)^{(m - 4)(l - 4)}detpol$

\begin{pmatrix} \begin{matrix} F_{1} \\ F_{1} \\ F_{1} \\ F_{1} \\ F_{0} \\ F_{0} \\ F_{0} \end{matrix} & \begin{pmatrix} b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 & 0 & 0 \\ 0 & b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 & 0 \\ 0 & 0 & b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 \\ 0 & 0 & 0 & b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} \\ a_{8} & a_{7} & a_{6} & a_{5} & a_{4} & a_{3} & a_{2} & a_{1} & a_{0} & 0 & 0 \\ 0 & a_{8} & a_{7} & a_{6} & a_{5} & a_{4} & a_{3} & a_{2} & a_{1} & a_{0} & 0 \\ 0 & 0 & a_{8} & a_{7} & a_{6} & a_{5} & a_{4} & a_{3} & a_{2} & a_{1} & a_{0} \end{pmatrix}

$\begin{pmatrix} \begin{matrix} F_{1} \\ F_{1} \\ F_{1} \\ F_{1} \\ F_{0} \\ F_{0} \\ F_{0} \end{matrix} & \begin{pmatrix} b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 & 0 & 0 \\ 0 & b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 & 0 \\ 0 & 0 & b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 \\ 0 & 0 & 0 & b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} \\ a_{8} & a_{7} & a_{6} & a_{5} & a_{4} & a_{3} & a_{2} & a_{1} & a_{0} & 0 & 0 \\ 0 & a_{8} & a_{7} & a_{6} & a_{5} & a_{4} & a_{3} & a_{2} & a_{1} & a_{0} & 0 \\ 0 & 0 & a_{8} & a_{7} & a_{6} & a_{5} & a_{4} & a_{3} & a_{2} & a_{1} & a_{0} \end{pmatrix}$ \end{pmatrix} = ( - 1)^{(m - 4)(l - 4)}detpol

\begin{pmatrix} \begin{matrix} F_{1} \\ F_{1} \\ F_{2} \\ F_{2} \\ F_{3} \\ F_{3} \\ F_{4} \end{matrix} & \begin{pmatrix} b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 & 0 & 0 \\ 0 & b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 & 0 \\ 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} & 0 & 0 \\ 0 & 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} & 0 \\ 0 & 0 & 0 & 0 & d_{5} & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} & 0 \\ 0 & 0 & 0 & 0 & 0 & d_{5} & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} \\ 0 & 0 & 0 & 0 & 0 & 0 & e_{4} & e_{3} & e_{2} & e_{1} & e_{0} \end{pmatrix}

$\begin{pmatrix} \begin{matrix} F_{1} \\ F_{1} \\ F_{2} \\ F_{2} \\ F_{3} \\ F_{3} \\ F_{4} \end{matrix} & \begin{pmatrix} b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 & 0 & 0 \\ 0 & b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 & 0 \\ 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} & 0 & 0 \\ 0 & 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} & 0 \\ 0 & 0 & 0 & 0 & d_{5} & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} & 0 \\ 0 & 0 & 0 & 0 & 0 & d_{5} & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} \\ 0 & 0 & 0 & 0 & 0 & 0 & e_{4} & e_{3} & e_{2} & e_{1} & e_{0} \end{pmatrix}$ \end{pmatrix}

S_{4} = (- 1)^{(m - 4) (l - 4)} d e tp o l F_{1} F_{1} F_{1} F_{1} F_{0} F_{0} F_{0} b_{7} 000 a_{8} 00 b_{6} b_{7} 00 a_{7} a_{8} 0 b_{5} b_{6} b_{7} 0 a_{6} a_{7} a_{8} b_{4} b_{5} b_{6} b_{7} a_{5} a_{6} a_{7} b_{3} b_{4} b_{5} b_{6} a_{4} a_{5} a_{6} b_{2} b_{3} b_{4} b_{5} a_{3} a_{4} a_{5} b_{1} b_{2} b_{3} b_{4} a_{2} a_{3} a_{4} b_{0} b_{1} b_{2} b_{3} a_{1} a_{2} a_{3} 0 b_{0} b_{1} b_{2} a_{0} a_{1} a_{2} 00 b_{0} b_{1} 0 a_{0} a_{1} 000 b_{0} 00 a_{0} = (- 1)^{(m - 4) (l - 4)} d e tp o l F_{1} F_{1} F_{2} F_{2} F_{3} F_{3} F_{4} b_{7} 000000 b_{6} b_{7} 00000 b_{5} b_{6} c_{6} 0000 b_{4} b_{5} c_{5} c_{6} 000 b_{3} b_{4} c_{4} c_{5} d_{5} 00 b_{2} b_{3} c_{3} c_{4} d_{4} d_{5} 0 b_{1} b_{2} c_{2} c_{3} d_{3} d_{4} e_{4} b_{0} b_{1} c_{1} c_{2} d_{2} d_{3} e_{3} 0 b_{0} c_{0} c_{1} d_{1} d_{2} e_{2} 000 c_{0} d_{0} d_{1} e_{1} 00000 d_{0} e_{0}

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