Polynomial maps with invertible sums of Jacobian matrices and directional derivatives

Abstract

Let F:Cn→Cm be a polynomial map with degF=d≥2. We prove that F is invertible if m=n and ∑i=1d−1(JF)|αi is invertible for all αi∈Cn, which is trivially the case for invertible quadratic maps.More generally, we prove that for affine lines L={β+μγ∣μ∈C}⊆Cn (γ≠0), F∣L is linearly rectifiable, if and only if ∑i=1d−1(JF)|αi⋅γ≠0 for all αi∈L. This appears to be the case for all affine lines L when F is injective and d≤3.We also prove that if m=n and ∑i=1n(JF)|αi is invertible for all αi∈Cn, 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)|α∣α∈Cn} consists of nilpotent matrices.