Sylvester矩阵、子结式、辗转相除法的三者关系(第三部分)

2.执行辗转相除法第二步

$F_{7} = Q_{7,6} \times F_{6} + F_{4}\ \ \ \ \ \ \ \ \ \ \deg\left( F_{7} \right) = 7\ \ \ \ \ \ \deg\left( F_{6} \right) = 6\ \ \ \ \ \ \deg\left( F_{4} \right) = 4$

$1)^{8 \times 7 + 7 \times 6}|S| = \begin{matrix} \begin{matrix} F_{7} \\ F_{7} \\ F_{6} \\ F_{6} \\ F_{6} \\ F_{6} \\ F_{6} \\ F_{6} \\ F_{6} \\ F_{7} \\ F_{7} \\ F_{7} \\ F_{7} \\ F_{7} \\ F_{7} \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} \right| \end{matrix} = \begin{matrix} \begin{matrix} F_{7} \\ F_{7} \\ F_{6} \\ F_{6} \\ F_{6} \\ F_{6} \\ F_{6} \\ F_{6} \\ F_{6} \\ F_{4} \\ 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 & 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 & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} \end{matrix} \right| \end{matrix}$

对应子结式 $S_{5}、S_{4}$ ：

$S_{5} = ( - 1)^{3 \times 2}detpol\begin{pmatrix} \begin{matrix} F_{7} \\ F_{7} \\ F_{7} \\ F_{8} \\ F_{8} \end{matrix} & \begin{pmatrix} b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 & 0 \\ 0 & b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 \\ 0 & 0 & b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} \\ a_{8} & a_{7} & a_{6} & a_{5} & a_{4} & a_{3} & a_{2} & a_{1} & a_{0} & 0 \\ 0 & a_{8} & a_{7} & a_{6} & a_{5} & a_{4} & a_{3} & a_{2} & a_{1} & a_{0} \end{pmatrix} \end{pmatrix} = ( - 1)^{3 \times 2 + 2 \times 1}detpol\begin{pmatrix} \begin{matrix} F_{7} \\ F_{7} \\ F_{6} \\ F_{6} \\ F_{4} \end{matrix} & \begin{pmatrix} b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 & 0 \\ 0 & b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 \\ 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} & 0 \\ 0 & 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} \\ 0 & 0 & 0 & 0 & 0 & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} \end{pmatrix} \end{pmatrix}$

$S_{4} = ( - 1)^{4 \times 3}detpol\begin{pmatrix} \begin{matrix} F_{7} \\ F_{7} \\ F_{7} \\ F_{7} \\ F_{8} \\ F_{8} \\ F_{8} \end{matrix} & \begin{pmatrix} b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 & 0 & 0 \\ 0 & b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 & 0 \\ 0 & 0 & b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 \\ 0 & 0 & 0 & b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} \\ a_{8} & a_{7} & a_{6} & a_{5} & a_{4} & a_{3} & a_{2} & a_{1} & a_{0} & 0 & 0 \\ 0 & a_{8} & a_{7} & a_{6} & a_{5} & a_{4} & a_{3} & a_{2} & a_{1} & a_{0} & 0 \\ 0 & 0 & a_{8} & a_{7} & a_{6} & a_{5} & a_{4} & a_{3} & a_{2} & a_{1} & a_{0} \end{pmatrix} \end{pmatrix} = ( - 1)^{4 \times 3 + 3 \times 2}detpol\begin{pmatrix} \begin{matrix} F_{7} \\ F_{7} \\ F_{6} \\ F_{6} \\ F_{6} \\ F_{4} \\ F_{4} \end{matrix} & \begin{pmatrix} b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 & 0 & 0 \\ 0 & b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 & 0 \\ 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} & 0 & 0 \\ 0 & 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} & 0 \\ 0 & 0 & 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} \\ 0 & 0 & 0 & 0 & 0 & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & d_{4} & d_{3} & d_{2} & d_{1} & d_{0} \end{pmatrix} \end{pmatrix}$