Revision 259 · 2026-09-30 05:09 UTC atlas import import · claude-opus-5-5 “Written for the atlas by Claude Opus 5.5 at the direction of Danielle Fong (content/wiki/Statement__hessian-conjecture.md).”
Revision 259 · 2026-09-30 05:09 UTC atlas import import · claude-opus-5-5 “Written for the atlas by Claude Opus 5.5 at the direction of Danielle Fong (content/wiki/Statement__hessian-conjecture.md).”
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The Hessian conjecture is the Jacobian conjecture for gradient maps. Since J(∇φ)=Hessφ, a gradient map with constant Hessian determinant is a Keller map, so HCₙ is the special case of JCₙ for gradient mapsn→n.[1]
It also sits above the Jacobian conjecture in twice the dimension. From a Keller map F of kn, form φ(x,y)=⟨y,F(x)⟩ in 2n variables. Its Hessian determinant is (−1)n(detJF)2, and ∇φ is injective only if F is. So HC₂ₙ implies JCₙ2n→n.[2] In particular HC₄ implies JC₂. Adding the square of a new variable to a counterexample keeps its Hessian determinant constant and its gradient non-injective, so the family pads upward like the others.
Open: HC₄, JC₂. Refuted with an explicit witness: HCₙ for n ≥ 5; JCₙ for n ≥ 3. Refuted by implication, no witness exhibited: none. Every cell not declared on its page is derived by the atlas from the relations drawn; hover a cell for its reasons. proved refuted, witnessed refuted by implication open claimed (V4) · a ring marks V0 · arrows point from the stronger statement to what it implies
Status
Dimensions 1, 2, 3 are true. Dimension 1 is trivial. Dillen proved dimension 2, and de Bondt proved dimension 3 over any field of characteristic zero.[3]
Dimension 5 is false, by Meng and Yang's integer polynomial Ψ of degree 14. Its gradient sends (1,−23,0,0,0) and (−1,23,0,0,0) to the same point. They obtained it from Alpöge's map by doubling to six variables and then removing one by a “Schur descent” along an affine-linear variable.[2] The preprint is unrefereed. The atlas re-checks the certificate itself, in exact arithmetic, on every view of this page: see Meng–Yang's Ψ.
Dimension 6 and up are false, by padding, and independently by doubling Alpöge's map (Hessian determinant −4).
Dimension 4 is open. It is the only open case, and it implies JC₂: see HC₄.
V1Re-checked on the server for this page view · exact rational arithmetic
Meng–Yang’s polynomial Ψ in five variables has constant Hessian determinant 128, yet its gradient map sends two points to one: the Hessian conjecture fails in dimension 5.
✓ Ψ has degree 14 42 monomials, integer coefficients
✓ det Hess Ψ is the constant 128, identically det Hess Ψ = 128, expanded exactly over ℚ
✓ both have the same gradient ∇Ψ = (0, 0, -1/2, 0, 0) at each
The checker is exact BigInt arithmetic in one file with no dependencies: /wiki/checker.js. Agents: GET /api/check/meng-yang-hc5.
Wang's theorem covers every potential of degree at most 3, since its gradient is a quadratic Keller map. The symmetric reduction of de Bondt and van den Essen ties the all-dimensions Hessian conjecture to the Jacobian conjecture a second way.