# HC₄: the Hessian conjecture in four variables

- kind: conjecture
- status: open
- in one line: HC₄ — the one open Hessian case; it implies JC₂

**Statement.** If $\varphi\in\mathbb C[x_1,x_2,x_3,x_4]$ has constant nonzero Hessian determinant, then $\nabla\varphi$ is a polynomial automorphism of $\mathbb C^4$.

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](/wiki/Statement:hessian-conjecture.md).

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](/wiki/Statement:jc-plane.md). 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.

## Relations

- special case of [hessian-conjecture](/wiki/Statement:hessian-conjecture.md) — After July 2026, HC₄ and JC₂ are the two statements of the Jacobian and Hessian families still unsettled. HC₄ is the four-variable case of the family, so HC₄ is a special case of the Hessian conjecture in dimension n. [1]
- implies [jc-plane](/wiki/Statement:jc-plane.md) — Therefore HC₄ implies the plane Jacobian conjecture JC₂.

## References

1. G. Meng, L. Yang, [A five-variable counterexample to the Hessian conjecture, and the low-dimensional status of the Jacobian and Hessian conjectures](https://arxiv.org/abs/2607.22198), 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](https://arxiv.org/abs/2608.14217), arXiv:2608.14217 (14 August 2026), Theorem 1.2. Unrefereed.

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