A Five-Variable Counterexample to the Hessian Conjecture
Statement
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.
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construction · #1
Guowu Meng and Liang Yang, using ChatGPT Pro + ClaudeThat credit came with the record as it was imported. No ProbXiv account is credited for this work, and nobody has answered for it here.
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'.
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