Polynomials with constant Hessian determinants in dimension three

In the last section, we show that the Jacobian conjecture holds for gradient maps over the reals whose linear part is the identity map, by proving that such gradient maps are translations (i.e. have degree 1) if they satisfy the Keller condition. We do this by showing that this problem is the polynomial case of the main result of [13]. For polynomials in dimension n≤3, we generalize this result to arbitrary fields of characteristic zero.