Polynomial maps with invertible sums of Jacobian matrices and directional derivatives

Let F:Cn→CmF:Cn→Cm be a polynomial map with degF=d≥2degF=d≥2. We prove that FF is invertible if m=nm=n and ∑i=1d−1(JF)|αi is invertible for all αi∈Cnαi∈Cn, which is trivially the case for invertible quadratic maps.More generally, we prove that for affine lines L={β+μγ∣μ∈C}⊆CnL={β+μγ∣μ∈C}⊆Cn (γ≠0γ≠0), F∣LF∣L is linearly rectifiable, if and only if ∑i=1d−1(JF)|αi⋅γ≠0 for all αi∈Lαi∈L. This appears to be the case for all affine lines LL when FF is injective and d≤3d≤3.We also prove that if m=nm=n and ∑i=1n(JF)|αi is invertible for all αi∈Cnαi∈Cn, then FF is a composition of an invertible linear map and an invertible polynomial map X+HX+H with linear part XX, such that the subspace generated by {(JH)|α∣α∈Cn}{(JH)|α∣α∈Cn} consists of nilpotent matrices.