Hessian conjecture refuted in five variables, derived from Claude-found Jacobian counterexample
Five days after Levent Alpöge's Claude Fable 5-assisted counterexample to the Jacobian conjecture, Guowu Meng and Liang Yang used "Schur descent" on it to build a five-variable counterexample to the related Hessian conjecture. The Hessian conjecture now holds for n≤3, fails for n≥5, and is open only for n=4.
Key facts
- arXiv 2607.22198, submitted 2026-07-24 (revised 07-27)
- Explicit polynomial in 5 variables, degree 14, constant Hessian determinant 128, with non-injective gradient
- Derived from Alpöge's Jacobian counterexample; the paper itself does not report AI use
Science result
- Field
- mathematics / algebraic geometry / polynomial maps
- Problem
- Hessian conjecture: a polynomial whose Hessian determinant is a nonzero constant has an injective gradient map
- Result
- Counterexample for n=5 (hence all n≥5); status now: true for n≤3, false for n≥5, open for n=4
- AI system
- Claude Fable 5 (indirectly, via the Jacobian counterexample)
- Human role
- Human-led; built by hand on an AI-assisted result
- Verification
- arXiv preprint; explicit and checkable
- Status
- pending
- Why surprising
- An AI-found counterexample spread to a neighbouring conjecture within days.
What happened
Guowu Meng and Liang Yang turned Alpöge's three-variable Jacobian counterexample into a five-variable counterexample to the Hessian conjecture.
Why it matters
It shows how AI-found results feed quickly into human follow-up work. It also leaves one clean open case, n=4.
Changelog
- 2026-09-29: created during a snowball check while verifying the Jacobian entry
Related events
Sources (2)
- paperarXiv 2607.22198: A five-variable counterexample to the Hessian conjecture
- discussionTerence Tao: A digestion of the Jacobian conjecture counterexample
id: 2026-07-24-hessian-conjecture-counterexample · updated 2026-09-29 · open in the interactive timeline