The Hessian conjecture in dimension n

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 mapsn→n.^[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 JCₙ2n→n.^[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.

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.
• 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.

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 ties the all-dimensions Hessian conjecture to the Jacobian conjecture a second way.

1. ↑ G. Meng, Legendre transform, Hessian conjecture and tree formula, Appl. Math. Lett. 19 (2006) 503–510; arXiv:math-ph/0308035 (28 August 2003).
2. ↑ 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 (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, J. Pure Appl. Algebra 219 (2015) 3743–3754.