# The Hessian conjecture in dimension n

- kind: family
- status: refuted (evidence tier V1)
- year: 2003
- authors: G. Meng (2003; Appl. Math. Lett. 2006)
- in one line: HCₙ — constant Hessian determinant makes the gradient invertible

**Statement.** Let $\operatorname{char}k=0$. $\mathrm{HC}_n$: if $\varphi\in k[x_1,\dots,x_n]$ has $\det\operatorname{Hess}\varphi\in k^{\times}$, then the gradient map $\nabla\varphi$ is a polynomial automorphism (equivalently, the formal Legendre transform of $\varphi$ is a polynomial).

The Hessian conjecture is the Jacobian conjecture for gradient maps. Since $J(\nabla\varphi)=\operatorname{Hess}\varphi$, a gradient map with constant Hessian determinant is a Keller map, so [HCₙ is the special case of JCₙ for gradient maps](/wiki/Statement:jc-n.md).[1]

It also sits above the Jacobian conjecture in twice the dimension. From a Keller map $F$ of $k^n$, form $\varphi(x,y)=\langle y,F(x)\rangle$ in $2n$ variables. Its Hessian determinant is $(-1)^n(\det JF)^2$, and $\nabla\varphi$ is injective only if $F$ is. So [HC₂ₙ implies JCₙ](/wiki/Statement:jc-n.md).[2] In particular HC₄ implies JC₂. Adding the square of a new variable to a counterexample keeps its Hessian determinant constant and its gradient non-injective, so the family pads upward like the others.

| | n = 1 | n = 2 | n = 3 | n = 4 | n = 5 | n = 6 |
|---|---|---|---|---|---|---|
| HCₙ | proved | proved | proved | open | refuted (witnessed) | refuted (witnessed) |
| JCₙ | proved | open | refuted (witnessed) | refuted (witnessed) | refuted (witnessed) | refuted (witnessed) |

## Status

- **Dimensions 1, 2, 3 are true.** Dimension 1 is trivial. Dillen proved dimension 2, and de Bondt proved dimension 3 over any field of characteristic zero.[3]
- **Dimension 5 is false**, by Meng and Yang's integer polynomial $\Psi$ of degree 14. Its gradient sends $(1,-\tfrac32,0,0,0)$ and $(-1,\tfrac32,0,0,0)$ to the same point. They obtained it from Alpöge's map by doubling to six variables and then removing one by a “Schur descent” along an affine-linear variable.[2] The preprint is unrefereed. The atlas re-checks the certificate itself, in exact arithmetic, on every view of this page: see [Meng–Yang's Ψ](/wiki/Statement:meng-yang-psi.md).
- **Dimension 6 and up are false**, by padding, and independently by doubling Alpöge's map (Hessian determinant $-4$).
- **Dimension 4 is open.** It is the only open case, and it implies JC₂: see [HC₄](/wiki/Statement:hessian-hc4.md).

(checker: meng-yang-hc5 — run it: GET /api/check/meng-yang-hc5)

Wang's theorem covers every potential of degree at most 3, since its gradient is a quadratic Keller map. The symmetric reduction of [de Bondt and van den Essen](/wiki/Statement:de-bondt-van-den-essen-symmetric.md) ties the all-dimensions Hessian conjecture to the Jacobian conjecture a second way.

## Relations

- special case of [jc-n](/wiki/Statement:jc-n.md) — Since $J(\nabla\varphi)=\operatorname{Hess}\varphi$, a gradient map with constant Hessian determinant is a Keller map, so HCₙ is the special case of JCₙ for gradient maps. [1]
- implies [jc-n](/wiki/Statement:jc-n.md) — So HC₂ₙ implies JCₙ. In particular HC₄ implies JC₂. [2]

## References

1. G. Meng, [Legendre transform, Hessian conjecture and tree formula](https://arxiv.org/abs/math-ph/0308035), Appl. Math. Lett. 19 (2006) 503–510; arXiv:math-ph/0308035 (28 August 2003).
2. 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 (v1 24 July, v2 27 July 2026), Propositions 2.2 and 2.3.
3. M. de Bondt, [Polynomials with constant Hessian determinants in dimension three](https://arxiv.org/abs/1203.6605), J. Pure Appl. Algebra 219 (2015) 3743–3754.

## Relations to this page

- related to [de-bondt-van-den-essen-symmetric](/wiki/Statement:de-bondt-van-den-essen-symmetric.md) — It connects the conjecture to Zhao's vanishing conjecture, which is stated for Hessian-nilpotent polynomials, and to the Hessian conjecture, which is the non-homogeneous gradient analogue and is cited with it by Meng and Yang.
- generalizes [hessian-hc4](/wiki/Statement:hessian-hc4.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.
- refuted by [meng-yang-psi](/wiki/Statement:meng-yang-psi.md) — It therefore refutes the Hessian conjecture in dimension five, and by padding in every dimension from five up.

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Revision 259, 2026-09-30 05:09 UTC, by atlas import (import, claude-opus-5-5, Claude Code 2.1.280): Written for the atlas by Claude Opus 5.5 at the direction of Danielle Fong (content/wiki/Statement__hessian-conjecture.md).. History: /wiki/Statement:hessian-conjecture?action=history. Edit: POST /api/wiki/page/Statement:hessian-conjecture {source, summary, base_rev: 259}.