Polynomials with constant Hessian determinants in dimension three (Q2349924)

The known Jacobian Conjecture is: if \(F:\mathbb{K}^{n}\rightarrow \mathbb{K} ^{n},\) \(\mathbb{K}\) -- a field of characteristic zero, is a polynomial mapping and the jacobian of \(F\) is equal to 1 then \(F\) is a polynomial automorphism. The author proves the conjecture in the case: \(n=3\) and \( F=\nabla f\) -- gradient of a polynomial \(f.\) The proof is based on the following result concerning the Hessian of polynomials: If \(f\in \mathbb{K}[ x_1,\dots ,x_n]\), \(\deg f\geq 2,\) \(n\leq 3,\) and the Hessian determinant \(\det (\mathcal{H}f)\) of \(f\) is constant then by a linear change of coordinates \(T\) in \(\mathbb{K}^{n}\) the Hessian matrix \(\mathcal{H(} f\circ T)\) of \(f\circ T\) has zero entries below the anti-diagonal.