A Five-Variable Counterexample to the Hessian Conjecture
The paper exhibits an explicit integer polynomial in five variables, of total degree 14 with constant Hessian determinant 128, whose gradient is not injective. Its formal Legendre transform is therefore not a polynomial, so the Hessian conjecture is false. The counterexample comes from a one-variable Schur descent applied to the six-variable doubling of Alpöge's 2026 Jacobian counterexample.
- Result
- Disproved
- Status
- Resolved
- AI contribution
- AI-assisted
- Method
- Construction
- Field
- Affine Algebraic Geometry, Polynomial Maps
- Posed by
- Hessian conjecture literature; after de Bondt and Alpöge
- Year posed
- —
- Years open
- —
- Solved
- 2026-07-24
- Model
- ChatGPT Pro + Claude
- Vendor
- —
- Collaborators
- Guowu Meng, Liang Yang
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 30 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What the AI did
The disclosure credits the second author with finding the Schur descent of Section 6 with the assistance of ChatGPT Pro, and states that the exact-arithmetic verifications throughout were carried out with assistance from Claude. The descent is the step that produces the counterexample; the surrounding framework is the authors'.
Verification
No independent review, but the result is an explicit polynomial with a stated Hessian determinant, so it is checkable by direct computation. Preprint, not refereed.