algebra / Affine Algebraic Geometry, Polynomial Maps
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 $\mathrm{HC}_5$ is false. The counterexample comes from a one-variable Schur descent applied to the six-variable doubling of Alpöge's 2026 Jacobian counterexample.
Temporal state
Append-only history
The pinned no-CAS checker reproduced the degree, monomial count, exact rational gradient collision, and deterministic exact Hessian evaluations. VibeMathed's external label remains unreviewed; this is a separate local replay record.
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 $\mathrm{HC}_5$ is false. The counterexample comes from a one-variable Schur descent applied to the six-variable doubling of Alpöge's 2026 Jacobian counterexample.
Research memory
The published polynomial Ψ=A²+13A+2B expands to 42 monomials of degree 14, has the reported rational gradient collision, and returned Hessian determinant 128 at all 40 deterministic exact test points in the no-CAS checker.
This is an artifact replay of the authors' checker at pinned commit 7b23ada. The finite exact evaluations do not replace the paper's symbolic determinant proof.
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 $\mathrm{HC}_5$ is false. The counterexample comes from a one-variable Schur descent applied to the six-variable doubling of Alpöge's 2026 Jacobian counterexample.
Run the pinned no-CAS Hessian checker
success · MFN independent verifier
All checks passed, including the exact rational collision and deterministic Bareiss determinant evaluations.
Evidence graph
parent of · claim · counterexample
parent of · event · reproduction
related to · problem · open