The Quartic Hessian Conjecture in Dimension Four

The Hessian conjecture $HC_n$ asks whether every polynomial $f$ with $\det \mathrm{Hess}(f) \in \mathbb{C}^\times$ has a polynomial gradient inverse. It is known for $n \le 3$, false for $n \ge 5$, and open exactly in dimension four, where it implies the plane Jacobian conjecture. Proved for every quartic polynomial in dimension four: the quartic case reduces to $f = P(x_1,x_2,x_3) + x_4 Q(x_1,x_2,x_3) + a x_4^2$ with $\deg Q \le 2$, and every constant-Hessian polynomial of this form has a polynomial gradient inverse.