2026/07/24 by Guowu Meng, Liang Yang
#math.AC #math.AG
We exhibit an explicit integer polynomial in five variables, of total degree 14 and with constant Hessian determinant 128, whose gradient is not injective. Consequently its formal Legendre transform is not a polynomial, and the Hessian conjecture \HC5 is false. The counterexample is obtained from the six-variable doubling of Alpöge's 2026 Jacobian counterexample by a one-variable Schur descent---a partial Legendre transform in a single variable. Combined with de~Bondt's theorem that \HCn holds for n≤3, with the elementary doubling and stabilization bridges relating the Jacobian conjectures \JCn to the Hessian conjectures \HCn, and with Alpöge's refutation of \JC3, this decides the Hessian conjecture in every dimension except n=4: \HCn is true for n≤3, false for n≥5, and open only at n=4. Exactly two statements of the two families remain unsettled, \JC2 and \HC4, linked by \HC4 ⇒ \JC2. Along the way we record, as a warm-up, an explicit six-variable counterexample to \HC6 with constant Hessian determinant -4 and non-injective gradient. This note adds the five-variable counterexample to, and updates the status recorded in, the first author's earlier educational preprint \citeMengRG2026.