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

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

2011-06-04

arxiv.org/abs/1106.0792v1

Mathematics

Commutative Algebra

Scientific paper

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, 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 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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