  For K a field of characteristic 0 and d any integer number greater than or
equal to 2, we prove the invertibility of polynomial endomorphisms of the
affine space of dimension d over K of the form F=Id+H, where each coordinate of
H is the cube of a linear form and the cube of the Jacobian matrix of H is
equal to zero. Our proof uses the inversion algorithm for polynomial maps
presented in our previous paper. Our current result leads us to formulate a
conjecture relating the nilpotency degree of the Jacobian matrix of H with the
number of necessary steps in the inversion algorithm.
