The Hessian conjecture is false in dimension five
Assessment
The claim traces to reliable primary sources through a clear chain of evidence.
The Hessian conjecture in dimension five asserts that any five-variable polynomial with nonzero constant Hessian determinant has a polynomial formal Legendre transform, which would force its gradient map to be injective. In July 2026, Guowu Meng and Liang Yang refuted this by exhibiting an explicit witness: an integer polynomial in five variables, of total degree 14, with constant Hessian determinant 128 and a non-injective gradient map (arxiv.org/abs/2607.22198). A polynomial with these properties cannot have a polynomial Legendre transform, so the conjecture fails.
The witness was not found in isolation. It descends, by a six-variable doubling followed by a one-variable partial Legendre transform, from the three-variable counterexample to the Jacobian conjecture announced and independently verified in July 2026, and its correctness rests on that map's verified properties of constant Jacobian determinant and non-injectivity together with algebraic identities the authors checked in exact rational arithmetic. No error report or credible dissent has appeared, and standard references now record the result as settled.
This result, combined with earlier work showing the conjecture holds in dimensions up to three and the propagation of the counterexample to every higher dimension, leaves dimension four as the only open case of the Hessian conjecture, where a proof would also settle the two-dimensional Jacobian conjecture.
Full reasoning: the evidence and decisions behind this verdict
The verdict rests on a short, checkable chain. The primary source, Meng and Yang's preprint (arxiv.org/abs/2607.22198, v2 of July 26, 2026), exhibits the explicit polynomial and states that all defining identities were verified in exact rational arithmetic; the refutation then follows by elementary logic, since a non-injective gradient map is incompatible with a polynomial Legendre transform, which is what the conjecture asserts. The load-bearing premise is the witness polynomial's two properties, constant Hessian determinant 128 and non-injective gradient: these are finite algebraic identities on published integer coefficients, decidable by any computer algebra system.
Corroboration is substantial. The Secret Blogging Seminar's coverage (sbseminar.wordpress.com/2026/07/20/the-new-counterexample-to-the-jacobian-conjecture/) states the five-variable counterexample is self-contained and checkable by hand, resting only on Alpöge's map having a constant Jacobian and a two-point collision; both properties are covered by the verified claim that Alpöge's map has constant Jacobian determinant −2 yet is generically three-to-one. Wolfram MathWorld's Hessian Conjecture entry (mathworld.wolfram.com/HessianConjecture.html) asserts in its own voice that the counterexample refutes the conjecture in dimension five and records the family as true for dimensions up to three, false from five upward, open only at four. An independent Zenodo record (zenodo.org/records/21504303) machine-verifies the failure of the Hessian conjecture along the cascade from the Jacobian counterexample, reaching an explicit witness in dimension 48; that does not itself decide dimension five, but it confirms the reduction network the construction travels through. Notably, the cascade alone only reaches dimensions six and above, so dimension five specifically depends on the Meng–Yang descent; this is why the witness claim carries a requires relation.
All recorded source instances affirm; a targeted search for error reports, withdrawals, or dissent found none. The residual uncertainty is that no formal peer review has concluded and this assessment could not re-run the arithmetic itself, since the full coefficient list was not retrievable through the sources read here; confidence is therefore held just below the level a direct computational replication would warrant. What would change the conclusion: a verified error in the witness's Hessian determinant or in the claimed gradient collision, or a retraction of the Alpöge map's properties, either of which would return the dimension-five case to open.
Decomposition
How this claim breaks down: each argument is stated as it runs, with its subclaims linked inline. ↗︎ opens a subclaim; the map shows how they fit together.
The Hessian conjecture in dimension five asserts that the formal Legendre transform of any five-variable polynomial with nonzero constant Hessian determinant is again a polynomial, which forces the gradient map to be injective. Because Meng and Yang's five-variable degree-14 polynomial has constant Hessian determinant 128 and a non-injective gradient map, its Legendre transform cannot be a polynomial, so the conjecture fails in dimension five.
The inference is airtight: a non-injective gradient map is directly incompatible with what the conjecture asserts, so the conclusion follows from the witness's properties by elementary logic. The argument stands or falls entirely with the witness polynomial having constant Hessian determinant 128 and a non-injective gradient, a finite algebraic fact checked by the authors in exact rational arithmetic and disputed by no one to date.
The five-variable witness is not an isolated computation: it is obtained from the six-variable doubling of the July 2026 Jacobian counterexample by a one-variable Schur descent (a partial Legendre transform), and its correctness rests on the source map's two key properties. Because Alpöge's map has constant Jacobian determinant −2 yet is generically three-to-one, and given that the three-variable Jacobian counterexample was announced in July 2026 and independently verified, the construction feeding the five-variable witness is sound, corroborating the failure of the Hessian conjecture in dimension five.
As corroboration the inference goes through: doubling and one-variable descent preserve the properties that make a counterexample, so a sound source map makes the derived witness sound. Both premises are in good standing, with the verified properties of Alpöge's map carrying most of the weight and the independent verification of the three-variable counterexample reinforcing it. The argument corroborates rather than establishes the claim on its own, since the descent step itself is Meng and Yang's contribution and lives in the witness argument.
Provenance
Where this claim has been said, linked to its canonical form.
we construct an explicit integer polynomial in five variables (total degree 14, constant Hessian determinant 128, non-injective gradient) that refutes the Hessian conjecture HC_5.
Announcement by Guowu Meng and Liang Yang of their arXiv paper 'A five-variable counterexample to the Hessian conjecture, and the low-dimensional status of the Jacobian and Hessian conjectures.'
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 HC_5 is false.
The primary source: Meng and Yang's preprint constructing the witness from the six-variable doubling of Alpöge's 2026 Jacobian counterexample via a one-variable Schur descent, with all defining identities checked in exact rational arithmetic.
Meng and Yang (2026) gave an explicit counterexample to [HC_5]. [...] Its gradient map is therefore not an injection, so its formal Legendre transform is not a polynomial. This refutes [HC_5]. [...] It follows that the Hessian conjecture is true for [n <= 3], false for [n >= 5], and open only for [n = 4].
MathWorld's reference entry on the Hessian conjecture, updated after the July 2026 developments, states in its own voice that the Meng–Yang counterexample refutes the conjecture in dimension five and records the conjecture as false for all dimensions of at least five.
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Created by extractor · Aug 11, 2026. Every judgment on this page is accompanied by a reasoning trace.