On additive-nilpotency of Jacobian matrices of polynomial maps

Let H:Cⁿ→Cⁿ$H:{\mathbb{C}}^n\to {\mathbb{C}}^n$ be a polynomial map, JH the Jacobian matrix of H  , and _VH$V_H$ the linear subspace of _Mn(C)$M_n(\mathbb{C})$ spanned by {JH(α)|α∈Cⁿ}$\{JH(\alpha )|\alpha \in {\mathbb{C}}^n\}$, the set of evaluated Jacobian matrices of H  . We describe the dimension of _VH$V_H$ through a power-linear Gorni–Zampieri mate of H, and give a numerical algorithm for computing this dimension and determining whether JH   is additive-nilpotent, i.e., whether _VH$V_H$ is a nilpotent subspace.