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

Let $F: C^n rightarrow C^m$ be a polynomial map with $degF=d geq 2$. We prove that $F$ is invertible if $m = n$ and $sum^{d-1}_{i=1} JF(alpha_i)$ is invertible for all $i$, which is trivially the case for invertible quadratic maps. More generally, weprove that for affine lines $L = {beta + mu gamma | mu in C} subseteq C^n$ ($gamma e 0$), $F|_L$ is linearly rectifiable, if and only if $sum^{d-1}_{i=1} JF(alpha_i) cdot gamma e 0$ 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 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.