可上三角化的多项式映射

设F=X H:Kn→Kn为特征0的域k上的多项式映射,当F=(x1 h1,…,xn hn),hi(x)=xi (ai1x1 … ainxn)3,i=1,…,n时,称F为三次线性多项式映射.通过矩阵A=aij:i,j=1,…,n]的幂零性质,研究了上述三次线性多项式的上三角化问题,证明在秩为3时A是强幂零的,而在秩为4时不是强幂零的,从而在秩为4时,多项式映射F并不总是可上三角化.为进一步了解强幂零性质,最后讨论了与强幂零性质有紧密联系的一些猜想和性质.

Linearly triangularizable polynomial mapping

Let F=X+H:Kn→Kn be a polynomial map, where K is a field with char k = 0. It has been proved that F is not always linearly triangularizable if F is a so-called cubic linear mapping,i. e. F = (x1+h1,…,xn+hn) , hi(x)=xi+(ai1x1+…+ainxn)3,i=1,…, n, with the Jacobian matrix J(H) being nilpotent, and the matrix A = aji:i,j=1,…,n] has a rank of four. Then, for the strongly nilpotent property, which is equivalent to the linearly triangularizable property, J(H) is nilpotent, which implies that J(H) is strongly nilpotent if n ≤ 3. A similar result has been shown in cubic linear form in the case where rank(A) = 3, but it is not true when rank(A) = 4.Finally, some remarks were given about the properties that may be equivalent to the strongly nilpotent property.