On additive-nilpotency of Jacobian matrices of polynomial maps

Let H:Cn→Cn be a polynomial map, JH the Jacobian matrix of H, and VH the linear subspace of Mn(C) spanned by {JH(α)|α∈Cn}, the set of evaluated Jacobian matrices of H. We describe the dimension of VH 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 is a nilpotent subspace.