The Hessian conjecture in dimension 2n implies the Jacobian conjecture in dimension n
3 events · 1 assessment · 1 decision
Structured and assessed
First pass. The claim is a short theorem (Meng 2006 Prop. 1.4; Meng–Yang 2026 Prop. 2.3; de Bondt–van den Essen 2005 independently). Left atomic: its two proof steps (block Hessian determinant of ⟨y,F(x)⟩ is (−1)^n (Jac F)^2; the Legendre transform of that potential is ⟨w,F^{-1}(w)⟩) are derivation steps no source refers to independently, so per §6 they belong in the reasoning prose rather than as nodes. Corrected claim type from empirical_derived to mathematical (category error) with wording unchanged; set domain to mathematics so the mathematics skill carries on future passes; set importance 0.3 / contestation 0.1 (settled but currently consulted as the sole bridge between the open cases HC_4 and JC_2). Verified the proof directly, read Meng 2006 (HTML), Meng–Yang 2026 (HTML), and MathWorld; recorded three affirming instances, readings, two repeats edges to Meng 2006, a shared-authorship relation, and an immaterial source map. Tool notes: arXiv PDF of Meng's paper failed ingestion (UTF8 encoding error) and the AMS PDF returned 403; quote checks on the two arXiv HTML instances failed because the stored text interleaves LaTeX and rendered symbols. Assessed verified, confidence 0.96, credence 0.99, marginal yield 0.03. The preliminary seed (0.97) was consistent with the independent verification but the verdict does not rest on it.
Assessed Verified
verdict confidence 0.96 · credence 0.99
This is a theorem, not an open question. Given a polynomial map F in n variables with constant nonzero Jacobian determinant (a Keller map), the polynomial φ(x, y) = ⟨y, F(x)⟩ in 2n variables has Hessian matrix in block form with zero lower-right block, so its Hessian determinant is (−1)^n (det JF)^2, a nonzero constant. If the Hessian conjecture holds in dimension 2n, the formal Legendre transform of φ is a polynomial; that transform equals ⟨y, F^{-1}(w)⟩ where F^{-1} is the formal inverse of F, so F^{-1} is itself a polynomial and F is a polynomial automorphism. The argument works over any field of characteristic zero and under either standard formulation of the Hessian conjecture (that the formal Legendre transform is a polynomial, or that every Keller gradient map has a polynomial inverse), since the two are equivalent dimension by dimension. The construction was published by Guowu Meng in 2006 (Applied Mathematics Letters 19, 503–510) and, in the language of symmetric Jacobians, independently by de Bondt and van den Essen in 2005 (Proceedings of the AMS 133, 2201–2205). Together with the trivial converse bridge (the Jacobian conjecture in dimension n implies the Hessian conjecture in dimension n), it shows the all-dimensional forms of the two conjectures are equivalent. After the 2026 refutation of the three-dimensional Jacobian conjecture, the same bridge, read in contrapositive, yields an explicit six-variable counterexample to the Hessian conjecture, and it is now the only known link between the two surviving open cases, the four-dimensional Hessian conjecture and the two-dimensional Jacobian conjecture.
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