# Meng–Yang's five-variable potential Ψ

- kind: counterexample
- year: 2026
- authors: G. Meng and L. Yang (arXiv:2607.22198, July 2026)
- in one line: Ψ ∈ ℤ[x₁,x₂,y₁,y₂,y₃], det Hess Ψ = 128, gradient 2-to-1

**Statement.** With $u=1+x_1x_2$, $A=y_1u^3+3x_1y_2u^2-x_1^3y_3$ and $B=y_1x_2^2u(4+3x_1x_2)+y_2\big(x_2+3x_1x_2^2(4+3x_1x_2)\big)+y_3(2x_1-3x_1^2x_2)$, the polynomial $\Psi=A^2+13A+2B$ has $\det\operatorname{Hess}\Psi=128$, and $\nabla\Psi(1,-\tfrac32,0,0,0)=\nabla\Psi(-1,\tfrac32,0,0,0)=(0,0,-\tfrac12,0,0)$.

$\Psi$ is a polynomial of degree 14 in five variables with integer coefficients. Its Hessian determinant is the constant 128, and its gradient map is not injective. It therefore [refutes the Hessian conjecture in dimension five, and by padding in every dimension from five up](/wiki/Statement:hessian-conjecture.md). Meng and Yang derived it from [Alpöge's map](/wiki/Statement:alpoge-counterexample.md): doubling gives a six-variable potential with Hessian determinant $-4$, and a Schur descent along an affine-linear variable removes one variable. Under the descent, the three-point collision of the map becomes a two-point collision of the gradient.[1]

The preprint is unrefereed, and the research behind this page found no independent re-verification of the determinant. The atlas now provides one, in exact rational arithmetic and in about ten milliseconds:

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

Two negative controls run with the atlas test suite. Changing $2B$ to $3B$ changes the determinant to 648, so the check fails. Changing $13A$ to $12A$ leaves the determinant at 128 but separates the two gradients. So the coefficient 13 is what makes the collision, and it does not enter the determinant.

The two witness points have rational coordinates, and $\Psi$ has integer coefficients. The failure is therefore also a failure over ℝ and over ℚ.

## Relations

- refutes [hessian-conjecture](/wiki/Statement:hessian-conjecture.md) — It therefore refutes the Hessian conjecture in dimension five, and by padding in every dimension from five up.

## 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, Theorem 6.1.

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