The homogeneous polynomials whose set of values is closed under multiplication
因 \[(x^2+xy+y^2)(z^2+zw+w^2)=(xz-yw)^2+(xz-yw)[wx+y(z+w)]+[wx+y(z+w)]^2\] 因此, 形如 \(x^2+xy+y^2\) 的数相乘, 所得的积仍为同样的形式. 这恒等式是如何想出来的? 秘密在于行列式, 把 \(x^2+xy+y^2\) 看成行列式 \begin{vmatrix} x& y\cr -y & x+y \end{vmatrix} Let \(f(x_1,x_2,\dotsc,x_n)\) be a homogeneous polynomial. Let \[S=\{f(a_1,a_2,\dotsc,a_n)\mid a_1,a_2,\dotsc,a_n \in\Bbb Z\}.\] …
The homogeneous polynomials whose set of values is closed under multiplication Read More