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

jc-n

(page created)
Revision 263 · 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__jc-n.md).”

Status and infobox

Source +44 −0

+---
+title: The Jacobian conjecture in dimension n
+short: JCₙ — constant nonzero Jacobian implies a polynomial inverse
+symbol: JC
+kind: family
+status: refuted
+tier: V0
+year: 1939
+authors: O.-H. Keller (1939); the plane case was stated earlier by L. Kraus (1884)
+statement: Let $k$ be a field of characteristic $0$. $\mathrm{JC}_n$: every polynomial map $F\in k[x_1,\dots,x_n]^n$ with $\det JF\in k^{\times}$ has a polynomial inverse.
+formal: jacobian_conjecture, formal-conjectures at 6fbb54f | https://github.com/google-deepmind/formal-conjectures/blob/6fbb54f24ccc2e64dcfaffc28c58950e377110d2/FormalConjectures/Wikipedia/JacobianConjecture.lean
+index: n
+cells: 1 = proved V3 | 2 = open | 3 = refuted V0 witnessed
+stable: yes
+---
+A polynomial map whose Jacobian determinant is a nonzero constant is invertible near every point, by the inverse function theorem. The Jacobian conjecture asks whether it is then invertible globally, with an inverse that is again polynomial. Such a map is called a *Keller map*. The conjecture is one statement for each dimension $n$, and the dimensions are linked: a counterexample $F$ in dimension $n$ gives the counterexample $F\times\mathrm{id}$ in dimension $n+1$, so $\mathrm{JC}_{n+1}$ implies $\mathrm{JC}_n$.
 
+In July 2026 the conjecture failed in dimension three, and so in every dimension from three up. Dimension one is trivial, since a polynomial with constant nonzero derivative is linear. Dimension two, the plane, is open: see [[Statement:jc-plane|JC₂]].
+ 
+## The ladder
+ 
+The Jacobian conjecture sits in a family of conjectures indexed by dimension, each joined to it by a theorem that shifts the index. The figure is computed from those theorems: only the cells marked on each family's page are declared, and the atlas derives the rest.
+ 
+{{ladder|Statement:poisson-conjecture|Statement:dixmier-conjecture|Statement:jc-n|Statement:hessian-conjecture|n=1..6}}
+ 
+Four cells are open after July 2026, and they form one chain. DC₂ implies JC₂, JC₂ implies DC₁, HC₄ implies JC₂, and PC₁ is JC₂ restated. The two-dimensional Jacobian conjecture is the hinge of all four families.
+ 
+## History
+ 
+L. Kraus stated the two-variable case in 1884, with a proof whose last step fails, as Rodríguez Díaz showed in 2026.<ref name="rd">L. O. Rodríguez Díaz, [On the origin of the Jacobian conjecture](https://arxiv.org/abs/2512.23614), C. R. Math. 364 (2026) 363–370.</ref> O.-H. Keller posed the general question in 1939, for polynomials with integer coefficients and functional determinant 1, and proved the birational case.<ref name="keller">O.-H. Keller, Ganze Cremona-Transformationen, Monatsh. Math. Phys. 47 (1939) 299–306, [doi:10.1007/BF01695502](https://doi.org/10.1007/BF01695502).</ref> Wikipedia and nLab read Keller as working in $n$ variables. Wikipedia adds that van den Essen read the paper as two-dimensional.<ref name="wp">[Jacobian conjecture](https://en.wikipedia.org/wiki/Jacobian_conjecture), Wikipedia, accessed 29 September 2026.</ref> Smale listed the conjecture as the sixteenth of his eighteen problems for the next century.<ref>S. Smale, Mathematical problems for the next century, Math. Intelligencer 20(2) (1998) 7–15, [doi:10.1007/BF03025291](https://doi.org/10.1007/BF03025291).</ref>
+ 
+## Why characteristic zero
+ 
+The hypothesis cannot be dropped. Over an algebraic closure of $\mathbb F_p$ the map $x\mapsto x-x^p$ has derivative $1$ everywhere, yet it sends both $0$ and $1$ to $0$.<ref name="wp"/> In characteristic zero the conjecture is also stable under field extension, so it is enough to work over $\mathbb C$.
+ 
+## Equivalent forms and consequences
+ 
+The conjecture has normal forms that hold across all dimensions at once. These are the reductions to cubic-homogeneous maps ([[Statement:bcw-cubic-homogeneous|Bass–Connell–Wright and Yagzhev]]), to cubic-linear maps ([[Statement:druzkowski-cubic-linear|Drużkowski]]) and to gradient maps ([[Statement:de-bondt-van-den-essen-symmetric|de Bondt–van den Essen]]). It is stably equivalent to the [[Statement:dixmier-conjecture|Dixmier conjecture]], the [[Statement:poisson-conjecture|Poisson conjecture]] and the [[Statement:hessian-conjecture|Hessian conjecture]]. It is implied by the [[Statement:mathieu-conjecture|Mathieu conjecture]], the [[Statement:image-conjecture|image conjecture]] and the [[Statement:gaussian-moments|Gaussian moments conjecture]], and it is equivalent to [[Statement:zhao-vanishing|Zhao's vanishing conjecture]]. Every one of these inherits the July 2026 refutation, at least in some dimension.
+ 
+## Formal status
+ 
+In google-deepmind/formal-conjectures, `JacobianConjectureProp` states the conjecture for all dimensions. The theorem `jacobian_conjecture` proves `answer(False)` over every nontrivial commutative ring, with no `sorry` in its proof body. The disproof was merged on 26 July 2026 (PR #4474), and the open plane case was added as `jacobian_conjecture_two_variables` on 28 July (PR #4635). This page pins the file at commit 6fbb54f (18 September 2026).<ref name="fc">google-deepmind/formal-conjectures, [JacobianConjecture.lean at 6fbb54f](https://github.com/google-deepmind/formal-conjectures/blob/6fbb54f24ccc2e64dcfaffc28c58950e377110d2/FormalConjectures/Wikipedia/JacobianConjecture.lean).</ref> The same file contains a variant $G$ of Alpöge's map with Jacobian determinant $1$. It sends the integer points $(1,0,1)$ and $(0,3,-71)$ to the same image, so it also answers Keller's original integer-coefficient question negatively in dimension 3. The atlas has not rebuilt the project itself. The tier V0 rests on the pinned statement and proof, corroborated by an independent Isabelle/HOL entry.<ref name="afp">A. F. Ramos, D. B. Hulak, R. J. G. B. de Queiroz, [Formal Verification of an Explicit Counterexample to the Jacobian Conjecture](https://isa-afp.org/entries/Jacobian_Counterexample.html), Archive of Formal Proofs, July 2026.</ref>
+ 
+<references />
+ 

Revision 263 as rendered

A polynomial map whose Jacobian determinant is a nonzero constant is invertible near every point, by the inverse function theorem. The Jacobian conjecture asks whether it is then invertible globally, with an inverse that is again polynomial. Such a map is called a Keller map. The conjecture is one statement for each dimension nn, and the dimensions are linked: a counterexample FF in dimension nn gives the counterexample F×idF\times\mathrm{id} in dimension n+1n+1, so JCn+1\mathrm{JC}_{n+1} implies JCn\mathrm{JC}_n.

In July 2026 the conjecture failed in dimension three, and so in every dimension from three up. Dimension one is trivial, since a polynomial with constant nonzero derivative is linear. Dimension two, the plane, is open: see JC₂.

The ladder

The Jacobian conjecture sits in a family of conjectures indexed by dimension, each joined to it by a theorem that shifts the index. The figure is computed from those theorems: only the cells marked on each family's page are declared, and the atlas derives the rest.

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.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→2n: A symplectic map has Jacobian 1, so PCₙ reduces to the Jacobian conjecture in dimension 2n.n→2n: A symplectic map has Jacobian 1, so PCₙ reduces to the Jacobian conjecture in dimension 2n.n→2n: A symplectic map has Jacobian 1, so PCₙ reduces to the Jacobian conjecture in dimension 2n.n→n: Reduction modulo p is where the content lies: PCₙ implies the Dixmier conjecture in the same rank.n→n: Reduction modulo p is where the content lies: PCₙ implies the Dixmier conjecture in the same rank.n→n: Reduction modulo p is where the content lies: PCₙ implies the Dixmier conjecture in the same rank.n→n: Reduction modulo p is where the content lies: PCₙ implies the Dixmier conjecture in the same rank.n→n: Reduction modulo p is where the content lies: PCₙ implies the Dixmier conjecture in the same rank.n→n: Reduction modulo p is where the content lies: PCₙ implies the Dixmier conjecture in the same rank.n→n: The cotangent lift (x,ξ)↦(F(x),JF(x)⁻ᵀξ) of a Keller map is symplectic, and it is injective only if F is, so PCₙ implies JCₙ directly.n→n: The cotangent lift (x,ξ)↦(F(x),JF(x)⁻ᵀξ) of a Keller map is symplectic, and it is injective only if F is, so PCₙ implies JCₙ directly.n→n: The cotangent lift (x,ξ)↦(F(x),JF(x)⁻ᵀξ) of a Keller map is symplectic, and it is injective only if F is, so PCₙ implies JCₙ directly.n→n: The cotangent lift (x,ξ)↦(F(x),JF(x)⁻ᵀξ) of a Keller map is symplectic, and it is injective only if F is, so PCₙ implies JCₙ directly.n→n: The cotangent lift (x,ξ)↦(F(x),JF(x)⁻ᵀξ) of a Keller map is symplectic, and it is injective only if F is, so PCₙ implies JCₙ directly.n→n: The cotangent lift (x,ξ)↦(F(x),JF(x)⁻ᵀξ) of a Keller map is symplectic, and it is injective only if F is, so PCₙ implies JCₙ directly.n→n: In short, DCₙ implies JCₙ in the same dimension, because a non-invertible Keller map pulls back to a non-surjective endomorphism of the Weyl algebra.n→n: In short, DCₙ implies JCₙ in the same dimension, because a non-invertible Keller map pulls back to a non-surjective endomorphism of the Weyl algebra.n→n: In short, DCₙ implies JCₙ in the same dimension, because a non-invertible Keller map pulls back to a non-surjective endomorphism of the Weyl algebra.n→n: In short, DCₙ implies JCₙ in the same dimension, because a non-invertible Keller map pulls back to a non-surjective endomorphism of the Weyl algebra.n→n: In short, DCₙ implies JCₙ in the same dimension, because a non-invertible Keller map pulls back to a non-surjective endomorphism of the Weyl algebra.n→n: In short, DCₙ implies JCₙ in the same dimension, because a non-invertible Keller map pulls back to a non-surjective endomorphism of the Weyl algebra.n→2n: So DCₙ reduces to the Jacobian conjecture in twice the dimension: a proof of JC₂ₙ would prove DCₙ. See the stable equivalence.n→2n: So DCₙ reduces to the Jacobian conjecture in twice the dimension: a proof of JC₂ₙ would prove DCₙ. See the stable equivalence.n→2n: So DCₙ reduces to the Jacobian conjecture in twice the dimension: a proof of JC₂ₙ would prove DCₙ. See the stable equivalence.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.PCnPoisson conjecturePC₁: open: nothing settles itPC₁openPC₂: claimed false (V4, not yet adjudicated or checked; still open)PC₂claimedPC₃: refuted (declared, tier V3, explicit witness)PC₃PC₄: false by implication, tier V3, explicit witness — paddingPC₄PC₅: false by implication, tier V3, explicit witness — padding → paddingPC₅PC₆: false by implication, tier V3, explicit witness — padding → padding → paddingPC₆PCₙ, n > 6: false by implication, tier V3, explicit witnessPCDCnDixmier conjectureDC₁: claimed true (V4, not yet adjudicated or checked; still open)DC₁claimedDC₂: open: nothing settles itDC₂openDC₃: refuted (declared, tier V3, explicit witness)DC₃DC₄: false by implication, tier V3, explicit witness — paddingDC₄DC₅: false by implication, tier V3, explicit witness — padding → paddingDC₅DC₆: false by implication, tier V3, explicit witness — padding → padding → padding → In short, DCₙ implies JCₙ in the same dimension, because a non-invertible Keller map pulls back to a non-surjective endomorphism of the Weyl algebra.DC₆DCₙ, n > 6: false by implication, tier V3, explicit witnessDCJCnJacobian 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 witnessJCHCnHessian 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 witnessHC
Open: PC₁, DC₂, JC₂, HC₄. Refuted with an explicit witness: PCₙ for n ≥ 3; DCₙ for n ≥ 3; JCₙ for n ≥ 3; HCₙ for n ≥ 5. Refuted by implication, no witness exhibited: none. Claimed, not yet checked: PC₂ (a counterexample), DC₁ (a proof). 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

Four cells are open after July 2026, and they form one chain. DC₂ implies JC₂, JC₂ implies DC₁, HC₄ implies JC₂, and PC₁ is JC₂ restated. The two-dimensional Jacobian conjecture is the hinge of all four families.

History

L. Kraus stated the two-variable case in 1884, with a proof whose last step fails, as Rodríguez Díaz showed in 2026.[1] O.-H. Keller posed the general question in 1939, for polynomials with integer coefficients and functional determinant 1, and proved the birational case.[2] Wikipedia and nLab read Keller as working in nn variables. Wikipedia adds that van den Essen read the paper as two-dimensional.[3] Smale listed the conjecture as the sixteenth of his eighteen problems for the next century.[4]

Why characteristic zero

The hypothesis cannot be dropped. Over an algebraic closure of Fp\mathbb F_p the map x↦x−xpx\mapsto x-x^p has derivative 11 everywhere, yet it sends both 00 and 11 to 00.[3] In characteristic zero the conjecture is also stable under field extension, so it is enough to work over C\mathbb C.

Equivalent forms and consequences

The conjecture has normal forms that hold across all dimensions at once. These are the reductions to cubic-homogeneous maps (Bass–Connell–Wright and Yagzhev), to cubic-linear maps (Drużkowski) and to gradient maps (de Bondt–van den Essen). It is stably equivalent to the Dixmier conjecture, the Poisson conjecture and the Hessian conjecture. It is implied by the Mathieu conjecture, the image conjecture and the Gaussian moments conjecture, and it is equivalent to Zhao's vanishing conjecture. Every one of these inherits the July 2026 refutation, at least in some dimension.

Formal status

In google-deepmind/formal-conjectures, JacobianConjectureProp states the conjecture for all dimensions. The theorem jacobian_conjecture proves answer(False) over every nontrivial commutative ring, with no sorry in its proof body. The disproof was merged on 26 July 2026 (PR #4474), and the open plane case was added as jacobian_conjecture_two_variables on 28 July (PR #4635). This page pins the file at commit 6fbb54f (18 September 2026).[5] The same file contains a variant GG of Alpöge's map with Jacobian determinant 11. It sends the integer points (1,0,1)(1,0,1) and (0,3,−71)(0,3,-71) to the same image, so it also answers Keller's original integer-coefficient question negatively in dimension 3. The atlas has not rebuilt the project itself. The tier V0 rests on the pinned statement and proof, corroborated by an independent Isabelle/HOL entry.[6]

  1. ↑ L. O. Rodríguez Díaz, On the origin of the Jacobian conjecture, C. R. Math. 364 (2026) 363–370.
  2. ↑ O.-H. Keller, Ganze Cremona-Transformationen, Monatsh. Math. Phys. 47 (1939) 299–306, doi:10.1007/BF01695502.
  3. ↑ a b Jacobian conjecture, Wikipedia, accessed 29 September 2026.
  4. ↑ S. Smale, Mathematical problems for the next century, Math. Intelligencer 20(2) (1998) 7–15, doi:10.1007/BF03025291.
  5. ↑ google-deepmind/formal-conjectures, JacobianConjecture.lean at 6fbb54f.
  6. ↑ A. F. Ramos, D. B. Hulak, R. J. G. B. de Queiroz, Formal Verification of an Explicit Counterexample to the Jacobian Conjecture, Archive of Formal Proofs, July 2026.