openproblem.space·the wiki Sign in to edit
Difference between revisions

hessian-conjecture

(page created)
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).”

Status and infobox

Relations

Source +32 −0

+---
+title: The Hessian conjecture in dimension n
+short: HCₙ — constant Hessian determinant makes the gradient invertible
+symbol: HC
+kind: family
+status: refuted
+tier: V1
+year: 2003
+authors: G. Meng (2003; Appl. Math. Lett. 2006)
+statement: Let $\operatorname{char}k=0$. $\mathrm{HC}_n$: if $\varphi\in k[x_1,\dots,x_n]$ has $\det\operatorname{Hess}\varphi\in k^{\times}$, then the gradient map $\nabla\varphi$ is a polynomial automorphism (equivalently, the formal Legendre transform of $\varphi$ is a polynomial).
+index: n
+cells: 1 = proved V3 | 2 = proved V3 | 3 = proved V3 | 4 = open | 5 = refuted V1 witnessed
+stable: yes
+---
+The Hessian conjecture is the Jacobian conjecture for gradient maps. Since $J(\nabla\varphi)=\operatorname{Hess}\varphi$, a gradient map with constant Hessian determinant is a Keller map, so [[special-case-of::Statement:jc-n|HCₙ is the special case of JCₙ for gradient maps]]{{at|n→n}}.<ref name="meng">G. Meng, [Legendre transform, Hessian conjecture and tree formula](https://arxiv.org/abs/math-ph/0308035), Appl. Math. Lett. 19 (2006) 503–510; arXiv:math-ph/0308035 (28 August 2003).</ref>
 
+It also sits above the Jacobian conjecture in twice the dimension. From a Keller map $F$ of $k^n$, form $\varphi(x,y)=\langle y,F(x)\rangle$ in $2n$ variables. Its Hessian determinant is $(-1)^n(\det JF)^2$, and $\nabla\varphi$ is injective only if $F$ is. So [[implies::Statement:jc-n|HC₂ₙ implies JCₙ]]{{at|2n→n}}.<ref name="my">G. Meng, L. Yang, [A five-variable counterexample to the Hessian conjecture, and the low-dimensional status of the Jacobian and Hessian conjectures](https://arxiv.org/abs/2607.22198), arXiv:2607.22198 (v1 24 July, v2 27 July 2026), Propositions 2.2 and 2.3.</ref> 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.
+ 
+{{ladder|Statement:hessian-conjecture|Statement:jc-n|n=1..6}}
+ 
+## 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.<ref>M. de Bondt, [Polynomials with constant Hessian determinants in dimension three](https://arxiv.org/abs/1203.6605), J. Pure Appl. Algebra 219 (2015) 3743–3754.</ref>
+- **Dimension 5 is false**, by Meng and Yang's integer polynomial $\Psi$ of degree 14. Its gradient sends $(1,-\tfrac32,0,0,0)$ and $(-1,\tfrac32,0,0,0)$ 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.<ref name="my"/> The preprint is unrefereed. The atlas re-checks the certificate itself, in exact arithmetic, on every view of this page: see [[Statement:meng-yang-psi|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 [[Statement:hessian-hc4|HC₄]].
+ 
+{{checker|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 [[Statement:de-bondt-van-den-essen-symmetric|de Bondt and van den Essen]] ties the all-dimensions Hessian conjecture to the Jacobian conjecture a second way.
+ 
+<references />
+ 

Revision 259 as rendered

The Hessian conjecture is the Jacobian conjecture for gradient maps. Since J(∇φ)=Hess⁡φJ(\nabla\varphi)=\operatorname{Hess}\varphi, 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 FF of knk^n, form φ(x,y)=⟨y,F(x)⟩\varphi(x,y)=\langle y,F(x)\rangle in 2n2n variables. Its Hessian determinant is (−1)n(det⁡JF)2(-1)^n(\det JF)^2, and ∇φ\nabla\varphi is injective only if FF 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.

n = 1n = 2n = 3n = 4n = 5n = 6…padding: Padding with identity coordinates: the statement in dimension n + 1 implies it in dimension n.padding: Padding with identity coordinates: the statement in dimension n + 1 implies it in dimension n.padding: Padding with identity coordinates: the statement in dimension n + 1 implies it in dimension n.padding: Padding with identity coordinates: the statement in dimension n + 1 implies it in dimension n.padding: Padding with identity coordinates: the statement in dimension n + 1 implies it in dimension n.padding: Padding with identity coordinates: the statement in dimension n + 1 implies it in dimension n.padding: Padding with identity coordinates: the statement in dimension n + 1 implies it in dimension n.padding: Padding with identity coordinates: the statement in dimension n + 1 implies it in dimension n.padding: Padding with identity coordinates: the statement in dimension n + 1 implies it in dimension n.padding: Padding with identity coordinates: the statement in dimension n + 1 implies it in dimension n.n→n: 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 maps.n→n: 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 maps.n→n: 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 maps.n→n: 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 maps.n→n: 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 maps.n→n: 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 maps.2n→n: So HC₂ₙ implies JCₙ. In particular HC₄ implies JC₂.2n→n: So HC₂ₙ implies JCₙ. In particular HC₄ implies JC₂.2n→n: So HC₂ₙ implies JCₙ. In particular HC₄ implies JC₂.HCnHessian conjectureHC₁: proved (declared, tier V3)HC₁HC₂: proved (declared, tier V3)HC₂HC₃: proved (declared, tier V3)HC₃HC₄: open: nothing settles itHC₄openHC₅: refuted (declared, tier V1, explicit witness)HC₅HC₆: false by implication, tier V1, explicit witness — paddingHC₆HCₙ, n > 6: false by implication, tier V1, explicit witnessHCJCnJacobian conjectureJC₁: proved (declared, tier V3)JC₁JC₂: open: nothing settles itJC₂openJC₃: refuted (declared, tier V0, explicit witness)JC₃JC₄: false by implication, tier V0, explicit witness — paddingJC₄JC₅: false by implication, tier V0, explicit witness — padding → paddingJC₅JC₆: false by implication, tier V0, explicit witness — padding → padding → paddingJC₆JCₙ, n > 6: false by implication, tier V0, explicit witnessJC
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\Psi of degree 14. Its gradient sends (1,−32,0,0,0)(1,-\tfrac32,0,0,0) and (−1,32,0,0,0)(-1,\tfrac32,0,0,0) 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 −4-4).
  • 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
  • ✓ two distinct points (1, -3/2, 0, 0, 0), (-1, 3/2, 0, 0, 0)
  • ✓ 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.

  1. ↑ G. Meng, Legendre transform, Hessian conjecture and tree formula, Appl. Math. Lett. 19 (2006) 503–510; arXiv:math-ph/0308035 (28 August 2003).
  2. ↑ a b G. Meng, L. Yang, A five-variable counterexample to the Hessian conjecture, and the low-dimensional status of the Jacobian and Hessian conjectures, arXiv:2607.22198 (v1 24 July, v2 27 July 2026), Propositions 2.2 and 2.3.
  3. ↑ M. de Bondt, Polynomials with constant Hessian determinants in dimension three, J. Pure Appl. Algebra 219 (2015) 3743–3754.