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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.

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