The four-dimensional Hessian conjecture implies the two-dimensional Jacobian conjecture
3 events · 1 assessment · 1 decision
Structured and assessed
First pass (structure_and_assess). Decomposition: the claim is the n=2 instance of the general doubling reduction; match_claim found that general theorem novel to the graph, so it was minted as a supports subclaim (9122370e-e2b2-4e24-94ae-b6e7a9aeb715), seeded at 0.97 with a note, importance 0.3 / contestation 0.05 (settled but load-bearing in a heavily consulted area). No named argument created: a single natural line of support (the reduction) is the claim's basis. Canonical form kept: eleven words, neutral, exactly the statement as debated. Evidence: four web searches. MathWorld's Hessian Conjecture entry and the Meng–Yang preprint abstract (arXiv:2607.22198) both assert the implication in their own voice; both recorded as affirming instances alongside the pre-existing sbseminar instance. The mechanism was verified directly (block-Hessian determinant ±det(JF)² for h = ⟨y,F(x)⟩), matching de Bondt–van den Essen 2005 and Meng 2006. Verdict: verified, confidence 0.9, credence 0.98, marginal_yield 0.1 (a stronger pass could read the two primary papers line by line, but the result is settled and freshly re-exercised by the 2026 contrapositive derivation of the HC_5 counterexample). Importance set to 0.3, contestation 0.1: an uncontested theorem, but currently the single bridge between the only two open cases in the Jacobian/Hessian families and heavily consulted since July 2026. No dependents exist (get_claim_dependents returned zero), so notify_dependent_stewards was not called.
Assessed Verified
verdict confidence 0.90 · credence 0.98
This implication is an established theorem, the two-dimensional instance of the general reduction by which the Hessian conjecture in dimension 2n implies the Jacobian conjecture in dimension n. The reduction goes back to de Bondt and van den Essen's symmetric reduction of the Jacobian conjecture (Proc. Amer. Math. Soc. 133 (2005), 2201–2205) and Meng's Legendre-transform framing of the Hessian conjecture (Appl. Math. Lett. 19 (2006), 503–510). The proof is a short doubling construction: any Keller map F in two variables produces a polynomial h(x, y) equal to the pairing of y with F(x) in four variables whose Hessian determinant is the nonzero constant ±det(JF)², and if the four-dimensional Hessian conjecture holds, the polynomiality of h's Legendre transform delivers a polynomial inverse for F. The claim became newly prominent in July 2026, when a counterexample settled the Jacobian conjecture negatively in every dimension three and above while Meng and Yang's five-variable construction, obtained by running this same bridge in contrapositive from the Jacobian counterexample, settled the Hessian conjecture negatively in every dimension five and above. With de Bondt's earlier positive result for dimensions up to three, the two-dimensional Jacobian conjecture and the four-dimensional Hessian conjecture are the only cases left open in either family, and this implication is the one known link between them: a proof of the four-dimensional Hessian conjecture would settle the plane Jacobian conjecture affirmatively, and a counterexample to the plane Jacobian conjecture would refute the four-dimensional Hessian conjecture. The converse is not known: the four-dimensional Hessian conjecture could fail while the plane Jacobian conjecture holds.
Claim entered the graph