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Meng's Hessian conjecture

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Let k\Bbbk be a field of characteristic zero and let φ∈k[x1,…,xn]\varphi\in\Bbbk[x_1,\dots,x_n]. Write

Hess⁡φ=(∂2φ∂xi∂xj).\operatorname{Hess}\varphi=\left(\frac{\partial^2\varphi}{\partial x_i\partial x_j}\right).

If det⁡Hess⁡φ∈k×\det\operatorname{Hess}\varphi\in\Bbbk^{\times}, the gradient map has a formal inverse and the formal Legendre transform is defined by

φL(p)=⟨p,x(p)⟩−φ(x(p)),\varphi^{L}(p)=\langle p,x(p)\rangle-\varphi(x(p)),

where x(p)x(p) is the formal inverse of p=∇φ(x)p=\nabla\varphi(x). Meng's Hessian conjecture. If φ∈k[x1,…,xn]\varphi\in\Bbbk[x_1,\dots,x_n] has det⁡Hess⁡φ∈k×\det\operatorname{Hess}\varphi\in\Bbbk^{\times}, then φL\varphi^{L} is a polynomial.

This is closely related to the Jacobian conjecture through the gradient map and formal Legendre transformation. The paper's abstract states that the Hessian conjecture holds for n≤3n\le3, is false for n≥5n\ge5, and remains open only for n=4n=4.

References

Primary source

Guowu Meng and Liang Yang, “A five-variable counterexample to the Hessian conjecture, and the low-dimensional status of the Jacobian and Hessian conjectures”, arXiv:2607.22198 (2026).

hessian-conjecturelegendre-transformpolynomial-maps
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Progress summary

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Claimed progress

The conjecture is now settled in dimensions up to three and five or more, while dimension four remains open except for a recent unverified result in degree four.

Meng's Hessian conjecture asks whether a polynomial with a nonzero constant Hessian determinant always has a polynomial formal Legendre transform. The problem is closely tied to the Jacobian conjecture through gradient maps.

Known results

  • The conjecture holds for dimensions n≤3n\le 3.
  • Quadratic and cubic potentials are covered by known results; in dimension four, degree at most four is the currently claimed range.
  • The conjecture in dimension four would imply the two-dimensional Jacobian conjecture.

July–August 2026 developments

  • Meng and Yang claimed a degree-1414 counterexample in five variables, with constant Hessian determinant and noninjective gradient; this implies failure for every n≥5n\ge5 by stabilization. The claim has no reported independent verification.
  • A separate August preprint claims the quartic case in dimension four, proving polynomial invertibility for degree-44 potentials, but does not address degrees ≥5\ge5 and has not been independently verified.

Current status (as of August 2026): The conjecture is reported true for n≤3n\le3 and false for n≥5n\ge5; n=4n=4 remains open in general, with only the quartic case claimed.

Sources

Solutions 0

No solutions have been posted yet.