Sylvester矩阵、子结式、辗转相除法的三者关系(第四部分)-EW帮帮网

4.执行辗转相除法第四步

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

$\left| S^{'} \right| = \begin{matrix} \begin{matrix} F_{1} \\ F_{1} \\ F_{2} \\ F_{2} \\ F_{3} \\ F_{3} \\ F_{4} \\ F_{4} \\ F_{4} \\ F_{4} \\ F_{4} \\ 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 & 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 & 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} \\ 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} = \begin{matrix} \begin{matrix} F_{1} \\ F_{1} \\ F_{2} \\ F_{2} \\ F_{3} \\ F_{3} \\ F_{4} \\ F_{4} \\ F_{4} \\ F_{4} \\ F_{4} \\ F_{5} \\ F_{5} \\ F_{5} \\ F_{5} \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 & 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} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & f_{3} & f_{2} & f_{1} & f_{0} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & f_{3} & f_{2} & f_{1} & f_{0} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & f_{3} & f_{2} & f_{1} & f_{0} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & f_{3} & f_{2} & f_{1} & f_{0} \end{matrix} \right| \end{matrix}$

对应子结式 $S_{3}$ ：

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

5.执行辗转相除法第五步

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

$\left| S^{'} \right| = \begin{matrix} \begin{matrix} F_{1} \\ F_{1} \\ F_{2} \\ F_{2} \\ F_{3} \\ F_{3} \\ F_{4} \\ F_{4} \\ F_{5} \\ F_{5} \\ F_{5} \\ F_{5} \\ 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 & 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 & f_{3} & f_{2} & f_{1} & f_{0} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & f_{3} & f_{2} & f_{1} & f_{0} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & f_{3} & f_{2} & f_{1} & f_{0} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & f_{3} & f_{2} & f_{1} & f_{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} = \begin{matrix} \begin{matrix} F_{1} \\ F_{1} \\ F_{2} \\ F_{2} \\ F_{3} \\ F_{3} \\ F_{4} \\ F_{4} \\ F_{5} \\ F_{5} \\ F_{5} \\ F_{5} \\ F_{6} \\ F_{6} \\ F_{6} \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 & 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 & f_{3} & f_{2} & f_{1} & f_{0} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & f_{3} & f_{2} & f_{1} & f_{0} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & f_{3} & f_{2} & f_{1} & f_{0} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & f_{3} & f_{2} & f_{1} & f_{0} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & g_{2} & g_{1} & g_{0} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & g_{2} & g_{1} & g_{0} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & g_{2} & g_{1} & g_{0} \end{matrix} \right| \end{matrix}$

对应子结式 $S_{2}$ ：

$S_{2} = ( - 1)^{(m - 2)(l - 2)}detpol\begin{pmatrix} \begin{matrix} F_{1} \\ F_{1} \\ F_{1} \\ F_{1} \\ F_{1} \\ F_{1} \\ F_{0} \\ F_{0} \\ 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 & 0 \\ 0 & b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{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 & b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{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 & 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 & 0 \\ 0 & a_{8} & a_{7} & a_{6} & a_{5} & a_{4} & a_{3} & a_{2} & a_{1} & a_{0} & 0 & 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 & 0 & 0 & a_{8} & a_{7} & a_{6} & a_{5} & a_{4} & a_{3} & a_{2} & a_{1} & a_{0} & 0 \\ 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 - 2)(l - 2)}detpol\begin{pmatrix} \begin{matrix} F_{1} \\ F_{1} \\ F_{2} \\ F_{2} \\ F_{3} \\ F_{3} \\ F_{4} \\ F_{4} \\ F_{5} \\ F_{5} \\ F_{6} \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 & b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{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 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{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 & d_{5} & d_{4} & d_{3} & d_{2} & d_{1} & d_{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 & e_{4} & e_{3} & e_{2} & e_{1} & e_{0} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & f_{3} & f_{2} & f_{1} & f_{0} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & f_{3} & f_{2} & f_{1} & f_{0} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & g_{2} & g_{1} & g_{0} \end{matrix} \right| \end{pmatrix}$

Sylvester矩阵、子结式、辗转相除法的三者关系(第四部分)

4.执行辗转相除法第四步

5.执行辗转相除法第五步

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