Four-dimensional Hessian conjecture

Determine whether every polynomial potential in four variables with nonzero constant Hessian determinant has a polynomially invertible gradient mapping, thereby resolving the remaining four-dimensional case of the Hessian conjecture.

Background

The paper places the four-dimensional Hessian conjecture within the broader Hessian and Jacobian conjecture framework. Earlier results establish the conjecture in dimensions two and three, while a cited construction provides counterexamples in dimension five and higher, with stabilization extending those counterexamples to all higher dimensions. Consequently, the universal Hessian conjecture remains unresolved only in dimension four.

The paper studies the four-dimensional real case under the additional assumption that the Hessian has Lorentzian signature of index 1. Its results settle several structured classes, including potentials admitting constant pivots, two-layer and certain multi-layer potentials, and potentials with low-dimensional or entirely singular Hesse systems, but do not resolve the full four-dimensional conjecture.

References

Together with the preceding positive results, this leaves the Hessian conjecture open only in dimension four; similarly, the Jacobian conjecture is open only in dimension two, and the four-dimensional Hessian conjecture implies the two-dimensional Jacobian conjecture.

— On the Hessian Conjecture in Lorentzian Signature: Constant Pivots and Hesse Systems  (2608.19112 - Liu, 19 Aug 2026) in Section 1, subsection “Historical Background and Overview”