Problem detail · source-aware
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The Hessian conjecture $HC_n$ asks whether every polynomial $f$ with $\det \mathrm{Hess}(f) \in \mathbb{C}^\times$ has a polynomial gradient inverse. It is known for $n \le 3$, false for $n \ge 5$, and open exactly in dimension four, where it implies the plane Jacobian conjecture. Proved for every quartic polynomial in dimension four: the quartic case reduces to $f = P(x_1,x_2,x_3) + x_4 Q(x_1,x_2,x_3) + a x_4^2$ with $\deg Q \le 2$, and every constant-Hessian polynomial of this form has a polynomial gradient inverse.
The source statement is reproduced for indexing with attribution. Mathematical correctness requires domain-expert or mechanical review. VibeMath has not independently audited statement fidelity, correctness, priority, or novelty.
The acknowledgement credits four systems - GPT-5.6 Sol, GPT-5.6 Luna, Claude Fable 5 and DeepSeek V4 Pro - with exploring candidate arguments, adversarial proof review, algebraic checking and editorial assistance, with the author independently reviewing all arguments. No individual step is attributed, so the lower tier applies.
Provider: OpenAI, Anthropic, DeepSeek · Prompt public: unknown · Independence: unknown
Checked by this site on 17 August 2026 against the paper's LaTeX (arXiv:2608.14217): the acknowledgement is verbatim as quoted, and the introduction's status summary (known for n <= 3 by Dillen and de Bondt, false for n >= 5, open for n = 4, HC_4 implies the plane Jacobian conjecture) matches the literature, including the n >= 5 counterexample recorded as this catalog's sibling entry. Sole-author preprint, days old, not checked here, no independent review.
Correctness: unknown · statement fidelity: unaudited · peer review: none
VibeMathed reports this item as partial. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.
Source-reported tools: argument.
Independent: unknown · difference confidence: 0
VibeMath has not independently audited the mathematical statement, proof, or novelty claim.