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

Abstract

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 ${\sum }_{i=1}^{d-1}JF({\alpha }_i)\cdot \gamma \ne 0$ 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 ${\sum }_{i=1}^{n}JF({\alpha }_i)$ 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.

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