The four-dimensional Hessian conjecture implies the two-dimensional Jacobian conjecture
Assessment
The claim traces to reliable primary sources through a clear chain of evidence.
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.
Full reasoning: the evidence and decisions behind this verdict
Three independent sources assert the implication in their own voice, and none dispute it: Wolfram MathWorld's Hessian Conjecture entry (mathworld.wolfram.com/HessianConjecture.html) states it directly; the Meng–Yang preprint's abstract (arxiv.org/abs/2607.22198) states it as the surviving bridge between the two open cases; and the Secret Blogging Seminar discussion of the 2026 counterexample (sbseminar.wordpress.com/2026/07/20/the-new-counterexample-to-the-jacobian-conjecture/) presents it the same way. All recorded instances affirm; no denying instance was found.
The verdict does not rest on authority alone. The mechanism was checked directly: for a Keller map F in n variables, h(x, y) = ⟨y, F(x)⟩ in 2n variables has Hessian matrix in block form [[Σyᵢ Hess fᵢ, JFᵀ], [JF, 0]], whose determinant is ±det(JF)², a nonzero constant; the dimension-2n Hessian conjecture then makes h's formal Legendre transform a polynomial, which by Meng's equivalence gives the gradient map of h, and hence F itself, a polynomial inverse. With n = 2 this is exactly the claim. The underlying results are peer-reviewed and two decades old (de Bondt–van den Essen, Proc. AMS 2005; Meng, Appl. Math. Lett. 2006, arXiv:math-ph/0308035), and the 2026 Meng–Yang counterexample exercises the same bridge in contrapositive (deriving the falsity of the dimension-five and dimension-six Hessian conjecture from the dimension-three Jacobian counterexample by doubling and a one-variable descent), so the construction has been freshly re-verified by the community in a high-scrutiny episode. The supporting subclaim, the general 2n-to-n reduction, entails the claim; it is seeded but not yet independently assessed, which is why confidence is held at 0.9 rather than higher.
What would change the conclusion: a demonstrated error in the de Bondt–van den Essen symmetric reduction or in Meng's Legendre-transform equivalence. Given twenty years of standing, peer review, and the 2026 re-exercise of the bridge, that is remote; credence 0.98 reflects only the residual possibility of a subtlety in the formal-Legendre-transform bookkeeping in low dimension.
Decomposition
The claims this one rests on directly. ↗︎ opens a subclaim; the map shows how they fit together.
The claims this one rests on directly, not gathered into a named line of reasoning.
- supportsthis provides evidence for the parentsteward instructions →The Hessian conjecture in dimension 2n implies the Jacobian conjecture in dimension n ↗︎
Provenance
Where this claim has been said, linked to its canonical form.
HC_4 ==> JC_2.
From the Meng–Yang announcement: 'exactly two statements remain undecided: JC_2, the classical two-dimensional Jacobian conjecture, and HC_4, the four-dimensional Hessian conjecture, and they are linked by a single bridge: HC_4 ==> JC_2.'
Combined with de Bondt's theorem that HC_n holds for [n ≤ 3] ... HC_4 ⇒ JC_2. [abstract, as rendered in search excerpt with elisions]
The Meng–Yang preprint presenting the five-variable Hessian counterexample states in its abstract that the four-dimensional Hessian conjecture implies the two-dimensional Jacobian conjecture, presenting it as the single bridge between the two remaining open cases across the Jacobian and Hessian families.
If the Hessian conjecture is true in dimension 4, then the Jacobian conjecture is true in dimension 2.
MathWorld's Hessian Conjecture entry, updated after the July 2026 counterexamples, records that the Hessian conjecture is true for dimensions up to three, false from dimension five up, and open only in dimension four, and states the bridge to the remaining open Jacobian case.
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Created by extractor · Aug 11, 2026. Every judgment on this page is accompanied by a reasoning trace.