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关于$x_1,x_2,y_1,y_2,z_1,z_2,z_3$的方程组\[\left\{\begin{aligned} & \sum_{i=1}^2\sum_{j=1}^2 a_{i j k} x_i y_j=0 &k\in\{1,2,3\}\\ & \sum_{i=1}^2\sum_{k=1}^3 a_{i j k} x_i z_k=0 &j\in\{1,2\}\\ & \sum_{j=1}^2\sum_{k=1}^3 a_{i j k} y_j z_k=0&i\in\{1,2\}\end{aligned}\right.\]有非零解的充要条件，这里说是
\begin{align*}
0&=
a_{111}^2 a_{122} a_{212} a_{223}^2
+ a_{111}^2 a_{222}^2 a_{123} a_{213}
- a_{111} a_{221} a_{122}^2 a_{213}^2
\\
&
- a_{111} a_{221} a_{212}^2 a_{123}^2
- a_{121}^2 a_{112} a_{222} a_{213}^2
- a_{121}^2 a_{212}^2 a_{113} a_{223}
\\
&
+ a_{121} a_{211} a_{112}^2 a_{223}^2
+ a_{121} a_{211} a_{222}^2 a_{113}^2
- a_{211}^2 a_{112} a_{222} a_{123}^2
\\
&
- a_{211}^2 a_{122}^2 a_{113} a_{223}
+ a_{221}^2 a_{112}^2 a_{123} a_{213}
+ a_{221}^2 a_{122} a_{212} a_{113}^2
\\
&
{}
- a_{111}^2 a_{122} a_{222} a_{213} a_{223}
- a_{111}^2 a_{212} a_{222} a_{123} a_{223}
- a_{111} a_{121} a_{112} a_{212} a_{223}^2
\\
&
+ a_{111} a_{121} a_{122} a_{222} a_{213}^2
+ a_{111} a_{121} a_{212}^2 a_{123} a_{223}
- a_{111} a_{121} a_{222}^2 a_{113} a_{213}
\\
&
- a_{111} a_{211} a_{112} a_{122} a_{223}^2
+ a_{111} a_{211} a_{122}^2 a_{213} a_{223}
+ a_{111} a_{211} a_{212} a_{222} a_{123}^2
\\
&
- a_{111} a_{211} a_{222}^2 a_{113} a_{123}
+ a_{121}^2 a_{112} a_{212} a_{213} a_{223}
+ a_{121}^2 a_{212} a_{222} a_{113} a_{213}
\\
&
- a_{121} a_{221} a_{112}^2 a_{213} a_{223}
+ a_{121} a_{221} a_{112} a_{122} a_{213}^2
+ a_{121} a_{221} a_{212}^2 a_{113} a_{123}
\\
&
- a_{121} a_{221} a_{212} a_{222} a_{113}^2
+ a_{211}^2 a_{112} a_{122} a_{123} a_{223}
+ a_{211}^2 a_{122} a_{222} a_{113} a_{123}
\\
&
- a_{211} a_{221} a_{112}^2 a_{123} a_{223}
+ a_{211} a_{221} a_{112} a_{212} a_{123}^2
+ a_{211} a_{221} a_{122}^2 a_{113} a_{213}
\\
&
- a_{211} a_{221} a_{122} a_{222} a_{113}^2
- a_{221}^2 a_{112} a_{122} a_{113} a_{213}
- a_{221}^2 a_{112} a_{212} a_{113} a_{123}
\\
&
+ a_{111} a_{121} a_{112} a_{222} a_{213} a_{223}
- a_{111} a_{121} a_{122} a_{212} a_{213} a_{223}
\\
&
+ a_{111} a_{121} a_{212} a_{222} a_{113} a_{223}
- a_{111} a_{121} a_{212} a_{222} a_{123} a_{213}
\\
&
+ a_{111} a_{221} a_{112} a_{122} a_{213} a_{223}
+ a_{111} a_{221} a_{212} a_{222} a_{113} a_{123}
\\
&
- a_{121} a_{211} a_{112} a_{122} a_{213} a_{223}
- a_{121} a_{211} a_{212} a_{222} a_{113} a_{123}
\\
&
+ a_{211} a_{221} a_{112} a_{122} a_{113} a_{223}
- a_{211} a_{221} a_{112} a_{122} a_{123} a_{213}
\\
&
+ a_{211} a_{221} a_{112} a_{222} a_{113} a_{123}
- a_{211} a_{221} a_{122} a_{212} a_{113} a_{123}
\\
&
+ a_{111} a_{211} a_{112} a_{222} a_{123} a_{223}
- a_{111} a_{211} a_{122} a_{212} a_{123} a_{223}
\\
&
+ a_{111} a_{211} a_{122} a_{222} a_{113} a_{223}
- a_{111} a_{211} a_{122} a_{222} a_{123} a_{213}
\\
&
+ a_{111} a_{221} a_{112} a_{212} a_{123} a_{223}
+ a_{111} a_{221} a_{122} a_{222} a_{113} a_{213}
\\
&
- a_{121} a_{211} a_{112} a_{212} a_{123} a_{223}
- a_{121} a_{211} a_{122} a_{222} a_{113} a_{213}
\\
&
+ a_{121} a_{221} a_{112} a_{212} a_{113} a_{223}
- a_{121} a_{221} a_{112} a_{212} a_{123} a_{213}
\\
&
+ a_{121} a_{221} a_{112} a_{222} a_{113} a_{213}
- a_{121} a_{221} a_{122} a_{212} a_{113} a_{213}
\\
&
{}
-2 a_{111} a_{221} a_{112} a_{222} a_{123} a_{213}
-2 a_{111} a_{221} a_{122} a_{212} a_{113} a_{223}
\\
&
+2 a_{111} a_{221} a_{122} a_{212} a_{123} a_{213}
-2 a_{121} a_{211} a_{112} a_{222} a_{113} a_{223}
\\
&
+2 a_{121} a_{211} a_{112} a_{222} a_{123} a_{213}
+2 a_{121} a_{211} a_{122} a_{212} a_{113} a_{223}.
\end{align*}
如何证明 |
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