HC₄: the Hessian conjecture in four variables

After July 2026, HC₄ and JC₂ are the two statements of the Jacobian and Hessian families still unsettled.^[1] HC₄ is the four-variable case of the family, so HC₄ is a special case of the Hessian conjecture in dimension n.

A plane Keller map $F$ gives $\varphi(x,y)=\langle y,F(x)\rangle$ in four variables, with Hessian determinant $(\det JF)^2$. If $\nabla\varphi$ is injective, so is $F$. Therefore HC₄ implies the plane Jacobian conjecture JC₂. The converse is not known, so HC₄ is a stronger sufficient condition for JC₂, not a restatement of it.^[1]

Evidence so far:

• Ni proves HC₄ for potentials of degree 4 (unrefereed).^[2] With Wang's theorem for degree at most 3, this covers every potential of degree at most 4.
• Hubbers (1994) proved the Jacobian conjecture for cubic-homogeneous Keller maps in dimension 4. That covers quartic potentials with no cubic part.
• de Bondt and van den Essen proved the case $x+H$ with $JH$ symmetric and nilpotent for $n\le4$, in any degree.

1. ↑ a b G. Meng, L. Yang, A five-variable counterexample to the Hessian conjecture, and the low-dimensional status of the Jacobian and Hessian conjectures, arXiv:2607.22198: “Exactly two statements of the two families remain unsettled, JC₂ and HC₄, linked by HC₄ ⇒ JC₂.”
2. ↑ Z. Ni, The Quartic Hessian Conjecture in Dimension Four, arXiv:2608.14217 (14 August 2026), Theorem 1.2. Unrefereed.