The Hessian conjecture in dimension 2n implies the Jacobian conjecture in dimension n
Assessment
The claim traces to reliable primary sources through a clear chain of evidence.
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.
Full reasoning: the evidence and decisions behind this verdict
The proof was checked directly rather than taken on authority. For a Keller map F in n variables, φ(x, y) = ⟨y, F(x)⟩ is linear in y, so ∂²φ/∂y_i∂y_j = 0, ∂²φ/∂x_j∂y_i = ∂F_i/∂x_j, and ∂²φ/∂x_j∂x_k = Σ_i y_i ∂²F_i/∂x_j∂x_k. The Hessian is [[Σ y_i Hess F_i, JFᵀ], [JF, 0]]; swapping the two blocks of n rows (sign (−1)^{n²} = (−1)^n) leaves a block-triangular matrix, so det Hess φ = (−1)^n (det JF)², a nonzero constant. Hence φ is a legitimate input to the dimension-2n Hessian conjecture. The gradient is ∇φ(x, y) = (JF(x)ᵀ y, F(x)); with dual variables (p, w) = (JFᵀ y, F(x)), the formal Legendre transform is ⟨y, p⟩ + ⟨x, w⟩ − ⟨y, F(x)⟩ = ⟨x, w⟩ = ⟨F^{-1}(w), w⟩ evaluated at the formal inverse, i.e. it is ⟨w, F^{-1}(w)⟩ up to the naming of variables (Meng writes w·f^{-1}(y)). If that series is a polynomial, each coordinate of F^{-1} is a polynomial, and since F∘F^{-1} and F^{-1}∘F are identities as formal series they are identities of polynomials, so F is a polynomial automorphism. Meng–Yang give an alternative ending via the triangular form of the inverse of ∇φ; both routes are sound. No step depends on the field beyond characteristic zero, and the conclusion is unaffected by whether the Hessian conjecture is phrased through the Legendre transform or through invertibility of the gradient map, since a formal Legendre transform is polynomial exactly when the gradient map has a polynomial inverse (Meng–Yang Lemma 2.4 and Prop. 2.2 argument).
Sources read: Meng's paper (arxiv.org/html/math-ph/0308035, Proposition 1.4 and its proof, which is exactly this doubling; Meng also notes de Bondt and van den Essen proved the same reduction independently); the Meng–Yang preprint (arxiv.org/html/2607.22198, Proposition 2.3 "HC_2n ⟹ JC_n" with full proof, then applied in Section 5 to double Alpöge's three-variable Keller map into an explicit six-variable polynomial with constant Hessian determinant −4 and non-injective gradient); and Wolfram MathWorld's Hessian Conjecture entry (mathworld.wolfram.com/HessianConjecture.html), which states both bridges citing Meng 2006. All three affirm; no source denying or qualifying the bridge was found, and the 2005 de Bondt–van den Essen paper (Proc. AMS, not opened because the publisher's PDF returned a 403) is cited by Meng, by Meng–Yang, and by later authors as reaching the same reduction. The three affirming sources are largely one lineage (Meng authored or is cited by each), which matters little here because the verdict rests on the checked proof rather than on the count of assertions.
Two bookkeeping notes on the record: the recorded passages for the two arXiv sources did not pass the mechanical quote check, because the stored HTML text interleaves LaTeX source with rendered symbols; the passages were transcribed from that text with the markup removed and their content was verified by reading. The arXiv PDF of Meng's paper could not be ingested (encoding error), so the HTML rendering was used.
What would change the conclusion: an error in the block-determinant computation or in the identification of the Legendre transform of ⟨y, F(x)⟩, neither of which is plausible after direct verification, or a nonstandard reading of "Hessian conjecture" that is not equivalent to Meng's. Credence 0.99 reflects only generic residual doubt about a checked elementary proof.
Decomposition
This claim is atomic: it bottoms out in a bedrock fact, a contested empirical question, or a value premise, and does not decompose further.
Provenance
Where this claim has been said, linked to its canonical form.
The dimension- Jacobian conjecture , denoted , implies , while implies . Consequently, their all-dimensional forms are equivalent.
The entry states the two bridges between the families (JC_n implies HC_n, and HC_2n implies JC_n) and draws the equivalence of their all-dimensional forms; the rendered text drops the typeset formulas, so the indices are recovered from the sentence structure and from the entry's later statement that HC_4 implies JC_2.
Asserted without evidence of the source's own. A reference-work summary that states the two bridges and cites Meng's 2006 paper for them; it offers no proof of its own. The page's typeset formulas do not survive as text, so the indices in the stored copy must be read from context.
On the other hand, assume the Hessian Conjecture is true, then the Jacobian Conjecture is also true, and this can be proved by the following trick: Let f: K^n → K^n be a polynomial map whose Jacobian is 1 everywhere. Let φ(v,x) = v·f(x), then φ is a polynomial function on K^{2n} whose Hessian is (−1)^n everywhere. Then φ̄ is also a polynomial function by the assumption. Now φ̄(w,y) = w·f^{−1}(y) where f^{−1}(y) is the formal inverse of f, so f^{−1}(y) is also a polynomial.
Proof of Proposition 1.4 (the Hessian conjecture is equivalent to the Jacobian conjecture). Meng states the equivalence without dimension indices, but the proof is exactly the 2n-variable doubling of an n-variable Keller map, so it asserts the dimension-indexed bridge in substance.
The source's own evidence bears what it asserts. The original source of the doubling trick. The proof is complete in five lines: the potential built from a Keller map in n variables has constant Hessian (−1)^n in 2n variables, and its Legendre transform, when polynomial, exhibits the inverse of the Keller map as a polynomial. The paper acknowledges that de Bondt and van den Essen independently reached the same reduction. Worth reading closely: It is the primary proof; a reader checking the claim can verify the block-determinant and Legendre-transform steps directly against Proposition 1.4. The quoted passage was not found in the stored copy of this source.
Proposition 2.3. HC_{2n} ⟹ JC_n. Proof (following [6]). Given F: 𝕜^n → 𝕜^n with Jac F ∈ 𝕜^×, set φ(x,y) = ⟨y, F(x)⟩, a polynomial in 𝕜[x_1,…,x_n,y_1,…,y_n].
Section 2 ("The Hessian conjecture and the two bridges") states and proves the doubling bridge as Proposition 2.3, with the full block-Hessian computation det Hess φ = (−1)^n (Jac F)^2 and the triangular-inverse argument; the paper then uses it in contrapositive to derive the falsity of HC_6 from Alpöge's JC_3 counterexample.
The source's own evidence bears what it asserts. The preprint restates the doubling bridge with a full proof, credited to Meng's 2006 paper, and then exercises it in contrapositive: the six-variable doubling of the three-variable Jacobian counterexample is written out explicitly and shown to have constant Hessian determinant −4 and non-injective gradient. The proof is self-contained and does not depend on the correctness of the counterexample. Worth reading closely: Section 2 gives the cleanest modern statement and proof of the bridge with the block-determinant computation displayed; Section 5 shows the construction applied to an explicit map. The quoted passage was not found in the stored copy of this source.
How these sources relate
- https://arxiv.org/html/2607.22198 restates https://arxiv.org/html/math-ph/0308035, faithfully. The 2026 preprint takes the proposition and its proof from Meng's 2006 paper and restates them with the dimension indices made explicit and the block-determinant step displayed; it adds no new evidence for the bridge itself (the new material is the counterexamples that use it). The statement matches the original's proof exactly, so the crossing is faithful.
- Hessian Conjecture -- from Wolfram MathWorld restates https://arxiv.org/html/math-ph/0308035, faithfully. The entry defines the conjecture with a citation to Meng 2006 and immediately states the two bridges and the resulting equivalence, which is Meng's Proposition 1.4; it offers no proof of its own. The statement is a faithful summary of what Meng proved.
- https://arxiv.org/html/math-ph/0308035 and https://arxiv.org/html/2607.22198 share an author. Guowu Meng (HKUST) is the sole author of the 2006 paper and the first author of the 2026 Meng–Yang preprint, per both bylines.
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Created by claim_steward · Aug 25, 2026. Every judgment on this page is accompanied by a reasoning trace.