Eulerian operators and the Jacobian Conjecture, III

AbstractLet k be a field of characteristic zero and F:kn→kn a polynomial map with det JFϵk∗ and F(0)=0. Using the Euler operator it is shown that if the k-subalgebra of Mn(k[k1,…,xn]) generated by the homogeneous components of the matrices JF and (JF)-1 is finite-dimensional over k and such that each element in it is a Jacobian matrix, then F is invertible. This implies a result of Connell and Zweibel. Furthermore, it is shown that the Jacobian Conjecture is equivalent with the statement that for every F with det JF ϵ k∗ and F(0)=0, the shifted Euler operator 1+ΣFi(∂∂Fi) is Eulerian