Polynomial maps with invertible sums of Jacobian matrices and of directional Derivatives.
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.
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Sat, 10/22/2011 - 13:22 — passguysuch that the subspace generated by $\{JH(\alpha) |
\alpha \in C^n\}$ consists of nilpotent matrices.
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Sat, 10/22/2011 - 18:59 — johnfish633@gma...Mercedes Benz S500 Alternator\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 $\alpha_i \in C^n$, t
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