Images of higher-order differential operators of polynomial algebras

We investigate images of higher-order differential operators of polynomial algebras over a field k. We show that, when char k > 0, the image of the set of differential operators {xi(i) - tau(i) vertical bar i = 1, 2,..., n} of the polynomial algebra k[xi(1),...,xi(n), z(1),..., z(n)] is a Mathieu subspace, where tau(i) is an element of k[partial derivative(z1),...,partial derivative(zn)] for i = 1,2,...,n. We also show that, when char k = 0, the same conclusion holds for n = 1. The problem concerning images of differential operators arose from the study of the Jacobian conjecture.