The remaining four-variable Hessian conjecture

Let $\Bbbk$ be a field of characteristic zero and let $\varphi\in\Bbbk[x_1,x_2,x_3,x_4]$ satisfy

$$\det\operatorname{Hess}(\varphi)\in\Bbbk^\times.$$

Then the gradient map $p=\nabla\varphi(x)$ has a formal inverse $x=x(p)$. The formal Legendre transform of $\varphi$ is

$$\varphi^L(p)=\langle p,x(p)\rangle-\varphi(x(p)).$$

Four-variable Hessian conjecture ($\mathrm{HC}_4$). Must $\varphi^L$ be a polynomial? Equivalently, must every four-variable gradient polynomial map with constant nonzero Jacobian determinant have a polynomial inverse?

Current status. De Bondt proved $\mathrm{HC}_n$ for $n\leq 3$. A July 2026 preprint of Meng and Yang constructs an explicit five-variable polynomial $\Psi$ of degree $14$ such that

$$\det\operatorname{Hess}(\Psi)=128$$

and $\nabla\Psi$ is not injective, refuting $\mathrm{HC}_5$. Adding independent quadratic variables propagates this counterexample to every $n\geq 5$.

Consequently, $\mathrm{HC}_4$ is the only unresolved member of the dimension-indexed Hessian conjectures. It is linked to the remaining two-dimensional Jacobian conjecture by

$$\mathrm{HC}_4\Longrightarrow\mathrm{JC}_2.$$

The Schur-descent mechanism producing the five-variable counterexample does not iterate to four variables, so the four-variable case appears to require a new idea.