Sylvester矩阵、子结式、辗转相除法的三者关系(第二部分)

书中玉2024-05-29 22:21

【三者的关系】

首先，辗转相除法可以通过Sylvester矩阵进行，过程如下（以 m = 8 、 l = 7 m = 8、l = 7 m=8、l=7为例子）。

首先调整矩阵中 a a a系数到最后面几行，如下所示：

S = ( a 8 a 7 a 6 a 5 a 4 a 3 a 2 a 1 a 0 0 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 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 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 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 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 0 0 a 8 a 7 a 6 a 5 a 4 a 3 a 2 a 1 a 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 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 ) ∼ S ′ = ( 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 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 a 8 a 7 a 6 a 5 a 4 a 3 a 2 a 1 a 0 0 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 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 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 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 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 0 0 a 8 a 7 a 6 a 5 a 4 a 3 a 2 a 1 a 0 ) S = \begin{pmatrix} a_{8} & a_{7} & a_{6} & a_{5} & a_{4} & a_{3} & a_{2} & a_{1} & a_{0} & 0 & 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 \\ 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 & 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 & 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 & 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 & 0 & 0 & a_{8} & a_{7} & a_{6} & a_{5} & a_{4} & a_{3} & a_{2} & a_{1} & a_{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} & 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{pmatrix}\sim S^{'} = \begin{pmatrix} 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} & 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} \\ a_{8} & a_{7} & a_{6} & a_{5} & a_{4} & a_{3} & a_{2} & a_{1} & a_{0} & 0 & 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 \\ 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 & 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 & 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 & 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 & 0 & 0 & a_{8} & a_{7} & a_{6} & a_{5} & a_{4} & a_{3} & a_{2} & a_{1} & a_{0} \end{pmatrix} S= a8000000b70000000a7a800000b6b7000000a6a7a80000b5b6b700000a5a6a7a8000b4b5b6b70000a4a5a6a7a800b3b4b5b6b7000a3a4a5a6a7a80b2b3b4b5b6b700a2a3a4a5a6a7a8b1b2b3b4b5b6b70a1a2a3a4a5a6a7b0b1b2b3b4b5b6b7a0a1a2a3a4a5a60b0b1b2b3b4b5b60a0a1a2a3a4a500b0b1b2b3b4b500a0a1a2a3a4000b0b1b2b3b4000a0a1a2a30000b0b1b2b30000a0a1a200000b0b1b200000a0a1000000b0b1000000a00000000b0 ∼S′= b70000000a8000000b6b7000000a7a800000b5b6b700000a6a7a80000b4b5b6b70000a5a6a7a8000b3b4b5b6b7000a4a5a6a7a800b2b3b4b5b6b700a3a4a5a6a7a80b1b2b3b4b5b6b70a2a3a4a5a6a7a8b0b1b2b3b4b5b6b7a1a2a3a4a5a6a70b0b1b2b3b4b5b6a0a1a2a3a4a5a600b0b1b2b3b4b50a0a1a2a3a4a5000b0b1b2b3b400a0a1a2a3a40000b0b1b2b3000a0a1a2a300000b0b1b20000a0a1a2000000b0b100000a0a10000000b0000000a0

1.执行辗转相除法第一步

F 8 = Q 8 , 7 × F 7 + F 6 deg ⁡ ( F 8 ) = 8 deg ⁡ ( F 7 ) = 7 deg ⁡ ( F 6 ) = 6 F_{8} = Q_{8,7} \times F_{7} + F_{6}\ \ \ \ \ \ \ \ \ \ \deg\left( F_{8} \right) = 8\ \ \ \ \ \ \deg\left( F_{7} \right) = 7\ \ \ \ \ \ \deg\left( F_{6} \right) = 6 F8=Q8,7×F7+F6 deg(F8)=8 deg(F7)=7 deg(F6)=6

( − 1 ) 8 × 7 ∣ S ∣ = F 7 F 7 F 7 F 7 F 7 F 7 F 7 F 7 F 8 F 8 F 8 F 8 F 8 F 8 F 8 ∣ 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 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 a 8 a 7 a 6 a 5 a 4 a 3 a 2 a 1 a 0 0 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 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 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 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 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 0 0 a 8 a 7 a 6 a 5 a 4 a 3 a 2 a 1 a 0 ∣ = F 7 F 7 F 7 F 7 F 7 F 7 F 7 F 7 F 6 F 6 F 6 F 6 F 6 F 6 F 6 ∣ 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 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 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 ∣ ( - 1)^{8 \times 7}|S| = \begin{matrix} \begin{matrix} F_{7} \\ F_{7} \\ F_{7} \\ F_{7} \\ F_{7} \\ F_{7} \\ F_{7} \\ F_{7} \\ F_{8} \\ F_{8} \\ F_{8} \\ F_{8} \\ F_{8} \\ F_{8} \\ F_{8} \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 & 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} \\ a_{8} & a_{7} & a_{6} & a_{5} & a_{4} & a_{3} & a_{2} & a_{1} & a_{0} & 0 & 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 \\ 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 & 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 & 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 & 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 & 0 & 0 & a_{8} & a_{7} & a_{6} & a_{5} & a_{4} & a_{3} & a_{2} & a_{1} & a_{0} \end{matrix} \right| \end{matrix} = \begin{matrix} \begin{matrix} F_{7} \\ F_{7} \\ F_{7} \\ F_{7} \\ F_{7} \\ F_{7} \\ F_{7} \\ F_{7} \\ F_{6} \\ F_{6} \\ F_{6} \\ F_{6} \\ 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 & 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} \\ 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} \end{matrix} \right| \end{matrix} (−1)8×7∣S∣=F7F7F7F7F7F7F7F7F8F8F8F8F8F8F8 b70000000a8000000b6b7000000a7a800000b5b6b700000a6a7a80000b4b5b6b70000a5a6a7a8000b3b4b5b6b7000a4a5a6a7a800b2b3b4b5b6b700a3a4a5a6a7a80b1b2b3b4b5b6b70a2a3a4a5a6a7a8b0b1b2b3b4b5b6b7a1a2a3a4a5a6a70b0b1b2b3b4b5b6a0a1a2a3a4a5a600b0b1b2b3b4b50a0a1a2a3a4a5000b0b1b2b3b400a0a1a2a3a40000b0b1b2b3000a0a1a2a300000b0b1b20000a0a1a2000000b0b100000a0a10000000b0000000a0 =F7F7F7F7F7F7F7F7F6F6F6F6F6F6F6 b700000000000000b6b70000000000000b5b6b700000c6000000b4b5b6b70000c5c600000b3b4b5b6b7000c4c5c60000b2b3b4b5b6b700c3c4c5c6000b1b2b3b4b5b6b70c2c3c4c5c600b0b1b2b3b4b5b6b7c1c2c3c4c5c600b0b1b2b3b4b5b6c0c1c2c3c4c5c600b0b1b2b3b4b50c0c1c2c3c4c5000b0b1b2b3b400c0c1c2c3c40000b0b1b2b3000c0c1c2c300000b0b1b20000c0c1c2000000b0b100000c0c10000000b0000000c0

对应子结式 S 6 S_{6} S6：

S 6 = ( − 1 ) 2 × 1 d e t p o l ( F 7 F 7 F 8 ( b 7 b 6 b 5 b 4 b 3 b 2 b 1 b 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 ) ) = ( − 1 ) 2 × 1 d e t p o l ( F 7 F 7 F 6 ( b 7 b 6 b 5 b 4 b 3 b 2 b 1 b 0 0 0 b 7 b 6 b 5 b 4 b 3 b 2 b 1 b 0 0 0 c 6 c 5 c 4 c 3 c 2 c 1 c 0 ) ) S_{6} = ( - 1)^{2 \times 1}detpol\begin{pmatrix} \begin{matrix} F_{7} \\ F_{7} \\ F_{8} \end{matrix} & \begin{pmatrix} b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{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} \end{pmatrix} \end{pmatrix} = ( - 1)^{2 \times 1}detpol\begin{pmatrix} \begin{matrix} F_{7} \\ F_{7} \\ F_{6} \end{matrix} & \begin{pmatrix} b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} & 0 \\ 0 & b_{7} & b_{6} & b_{5} & b_{4} & b_{3} & b_{2} & b_{1} & b_{0} \\ 0 & 0 & c_{6} & c_{5} & c_{4} & c_{3} & c_{2} & c_{1} & c_{0} \end{pmatrix} \end{pmatrix} S6=(−1)2×1detpol F7F7F8 b70a8b6b7a7b5b6a6b4b5a5b3b4a4b2b3a3b1b2a2b0b1a10b0a0 =(−1)2×1detpol F7F7F6 b700b6b70b5b6c6b4b5c5b3b4c4b2b3c3b1b2c2b0b1c10b0c0

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