Does every polynomial phi∈R[x_1,x_2,x_3,x_4] satisfying det bigl(operatornameHessphi(x)bigr)=-1 and having Lorentzian Hessian, namely operatornameinertiabigl(operatornameHessphi(x)bigr)=(1,3) for all x∈R^4, have a gradient mapping nabla phi:R^4→R^4 that is a polynomial automorphism?

Hessian conjecture in Lorentzian signature

The problem asks whether the stated four-dimensional Lorentzian Hessian condition forces the gradient map to be a polynomial automorphism. No source in the scan reports a proof or counterexample for this exact restricted problem.

Known results

The general Hessian conjecture is known in dimensions at most two and has counterexamples in dimensions at least five (Meng and Yang, 2026). These counterexamples are not reported to satisfy the required global inertia condition $\operatorname{inertia}(\operatorname{Hess}\phi)=(1,3)$ in dimension four.

August 2026 preprint

Hanwen Liu posted a preprint titled “On the Hessian Conjecture in Lorentzian Signature: Constant Pivots and Hesse Systems.” The scan confirms direct engagement with the problem but gives no result, proof, or verification details, so any claimed progress remains unconfirmed.

Current status (as of August 2026): The exact Lorentzian-signature problem remains unresolved; a directly titled preprint is available, but its mathematical conclusion could not be verified from the retrieved material.

Hessian conjecture in Lorentzian signature

Progress summary

Primary source

Additional references

Solutions 0