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 $\sum^{d-1}_{i=1} JF(α_i)$ is invertible for all $i$, which is trivially the case for invertible quadratic maps. More generally, we prove that for affine lines $L = \{β+ μγ| μ\in C\} \subseteq C^n$ ($γ\ne 0$), $F|_L$ is linearly rectifiable, if and only if $\sum^{d-1}_{i=1} JF(α_i) \cdot γ\ne 0$ for all $α_i \in L$. 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 $\sum^{n}_{i=1} JF(α_i)$ is invertible for all $α_i \in C^n$, 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 $\{JH(α) | α\in C^n\}$ consists of nilpotent matrices.