Polynomial maps with invertible sums of Jacobian matrices and of directional Derivatives

Let $F:C^n\to C^m$ be a polynomial map with $degF=d\ge 2$. We prove that $F$ is invertible if $m=n$ and ${\sum }_{i=1}^{d-1}JF({\alpha }_i)$ is invertible for all $i$, which is trivially the case for invertible quadratic maps. 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 ${\sum }_{i=1}^{d-1}JF({\alpha }_i)\cdot \gamma \mathrm{\ne }0$ 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 $\sum^{n}{i=1} JF(\alpha_i)$isinvertibleforall$\alpha_i \in C^n$,then$F$ is a composition 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.