Polynomials with constant Hessian determinant (Q1175207)

The author proves the Jacobian conjecture for polynomial mappings \(F:\mathbb{C}^ 2\to\mathbb{C}^ 2\) with symmetric Jacobian matrix. He uses the fact that, in this case, there exists a polynomial \(P:\mathbb{C}^ 2\to\mathbb{C}\) such that \(F=\text{grad}(P)\) (then \(P\) has constant Hessian determinant), and next, he gives the explicit form of such \(P\)'s. He also indicates two applications of the above results in differential geometry of surfaces.