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 × F 6 + F 4 deg ⁡ ( F 7 ) = 7 deg ⁡ ( F 6 ) = 6 deg ⁡ ( F 4 ) = 4 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 F7=Q7,6×F6+F4 deg(F7)=7 deg(F6)=6 deg(F4)=4

( − 1 ) 8 × 7 + 7 × 6 ∣ S ∣ = 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 ∣ 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 ∣ = 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 ∣ 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 ∣ ( - 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} (−1)8×7+7×6∣S∣=F7F7F6F6F6F6F6F6F6F7F7F7F7F7F7 b700000000000000b6b70000000000000b5b6c6000000b700000b4b5c5c600000b6b70000b3b4c4c5c60000b5b6b7000b2b3c3c4c5c6000b4b5b6b700b1b2c2c3c4c5c600b3b4b5b6b70b0b1c1c2c3c4c5c60b2b3b4b5b6b70b0c0c1c2c3c4c5c6b1b2b3b4b5b6000c0c1c2c3c4c5b0b1b2b3b4b50000c0c1c2c3c40b0b1b2b3b400000c0c1c2c300b0b1b2b3000000c0c1c2000b0b1b20000000c0c10000b0b100000000c000000b0 =F7F7F6F6F6F6F6F6F6F4F4F4F4F4F4 b700000000000000b6b70000000000000b5b6c6000000000000b4b5c5c600000000000b3b4c4c5c60000000000b2b3c3c4c5c6000d400000b1b2c2c3c4c5c600d3d40000b0b1c1c2c3c4c5c60d2d3d40000b0c0c1c2c3c4c5c6d1d2d3d400000c0c1c2c3c4c5d0d1d2d3d400000c0c1c2c3c40d0d1d2d3d400000c0c1c2c300d0d1d2d3000000c0c1c2000d0d1d20000000c0c10000d0d100000000c000000d0

对应子结式 S 5 、 S 4 S_{5}、S_{4} S5、S4：

S 5 = ( − 1 ) 3 × 2 d e t p o l ( F 7 F 7 F 7 F 8 F 8 ( 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 ) ) = ( − 1 ) 3 × 2 + 2 × 1 d e t p o l ( F 7 F 7 F 6 F 6 F 4 ( 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 ) ) 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} S5=(−1)3×2detpol F7F7F7F8F8 b700a80b6b70a7a8b5b6b7a6a7b4b5b6a5a6b3b4b5a4a5b2b3b4a3a4b1b2b3a2a3b0b1b2a1a20b0b1a0a100b00a0 =(−1)3×2+2×1detpol F7F7F6F6F4 b70000b6b7000b5b6c600b4b5c5c60b3b4c4c50b2b3c3c4d4b1b2c2c3d3b0b1c1c2d20b0c0c1d1000c0d0

S 4 = ( − 1 ) 4 × 3 d e t p o l ( F 7 F 7 F 7 F 7 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 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 ) ) = ( − 1 ) 4 × 3 + 3 × 2 d e t p o l ( F 7 F 7 F 6 F 6 F 6 F 4 F 4 ( 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 ) ) 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} S4=(−1)4×3detpol F7F7F7F7F8F8F8 b7000a800b6b700a7a80b5b6b70a6a7a8b4b5b6b7a5a6a7b3b4b5b6a4a5a6b2b3b4b5a3a4a5b1b2b3b4a2a3a4b0b1b2b3a1a2a30b0b1b2a0a1a200b0b10a0a1000b000a0 =(−1)4×3+3×2detpol F7F7F6F6F6F4F4 b7000000b6b700000b5b6c60000b4b5c5c6000b3b4c4c5c600b2b3c3c4c5d40b1b2c2c3c4d3d4b0b1c1c2c3d2d30b0c0c1c2d1d2000c0c1d0d10000c00d0