Let $F: C^n \rightarrow C^m$ be a polynomial map with $degF=d \geq 2$. Weprove that $F$ is invertible if $m = n$ and $\sum^{d-1}_{i=1} JF(\alpha_i)$ isinvertible for all $i$, which is trivially the case for invertible quadraticmaps. More generally, we prove that for affine lines $L = \{\beta + \mu \gamma| \mu \in C\} \subseteq C^n$ ($\gamma \ne 0$), $F|_L$ is linearly rectifiable,if and only if $\sum^{d-1}_{i=1} JF(\alpha_i) \cdot \gamma \ne 0$ for all$\alpha_i \in L$. This appears to be the case for all affine lines $L$ when $F$is injective and $d \le 3$. We also prove that if $m = n$ and $\sum^{n}_{i=1}JF(\alpha_i)$ is invertible for all $\alpha_i \in C^n$, then $F$ is acomposition of an invertible linear map and an invertible polynomial map $X+H$with linear part $X$, such that the subspace generated by $\{JH(\alpha) |\alpha \in C^n\}$ consists of nilpotent matrices.
展开▼
机译：令$ F：C ^ n \ rightarrow C ^ m $是$ degF = d \ geq 2 $的多项式映射。我们证明，如果$ m = n $且$ \ sum ^ {d-1} _ {i = 1} JF（\ alpha_i）$对于所有$ i $都是可逆的，则$ F $是可逆的，这对于可逆的二次映射来说是微不足道的。更普遍地，我们证明对于仿射线$ L = \ {\ beta + \ mu \ gamma | \ mu \ in C \} \ subseteq C ^ n $（$ \ gamma \ ne 0 $），当且仅当$ \ sum ^ {d-1} _ {i = 1 } JF（\ alpha_i）\ cdot \ gamma \ ne 0 $ for全部$ \ alpha_i \ in L $。当$ F $为内射词而$ d \ le 3 $时，所有仿射行$ L $似乎都是这种情况。我们还证明，如果$ m = n $并且$ \ sum ^ {n} _ {i = 1} JF（\ alpha_i）$对于C ^ n $中的所有$ \ alpha_i \ n是可逆的，则$ F $是一个组合可逆线性映射和可逆多项式映射$ X + H $的线性部分$ X $的乘积，使得由$ \ {JH（\ alpha）| \ alpha \ in C ^ n \} $生成的子空间由幂等组成矩阵。
展开▼
