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

TL;DR

This paper investigates conditions under which polynomial maps are invertible based on the invertibility of sums of their Jacobian matrices and explores implications for affine lines and polynomial map compositions.

Contribution

It establishes new criteria for invertibility of polynomial maps using sums of Jacobian matrices and characterizes polynomial maps with nilpotent Jacobian matrices.

Findings

Invertibility of polynomial maps linked to invertible sums of Jacobians.

Line restriction properties depend on sums of Jacobians and directional derivatives.

Polynomial maps with invertible Jacobian sums decompose into linear and nilpotent parts.

Abstract

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.