Nilpotent symmetric Jacobian matrices and the Jacobian conjecture

Let image be a polynomial map such that the Jacobian image of H is nilpotent and symmetric. The symmetric dependence problem, SDP(n), asks whether the rows of the matrix image are dependent over image. We show that if SDP(r) has an affirmative answer for all rless-than-or-equals, slantn, then the Jacobian conjecture holds for all image of the form F=x+H with image nilpotent and symmetric. As a consequence, we deduce the main result of (J. Pure Appl. Algebra, 189/1–3, 123–133), which asserts that the Jacobian conjecture holds for all polynomial maps of the form F=x+H, with image nilpotent, symmetric and homogeneous, and nless-than-or-equals, slant4.