Eulerian operators and the Jacobian Conjecture, III

Abstract

Let k be a field of characteristic zero and F: k n → k n a polynomial map with det JFϵk ∗ and F(0)=0. Using the Euler operator it is shown that if the k-subalgebra of M n ( k[ k 1,…, x n ]) 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+ΣF i( ∂ ∂F i ) is Eulerian.