The Jacobian conjecture in dimension n

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• equivalent to Jacobian conjecture, n ≥ 3 refuted · V0

  Because a counterexample in dimension $n$ pads to one in dimension $n+1$, “JCₙ for every $n\ge3$” holds exactly when JCₙ holds for every $n$, so this problem is the all-dimensions Jacobian conjecture.

• related to The cubic reduction of Bass–Connell–Wright and Yagzhev proved · V3

  So the reduction constrains no single JCₙ, and it says nothing about the plane.

• generalizes Campbell's theorem: Galois Keller maps are automorphisms proved · V3

  Campbell's theorem says that if this extension is Galois, the map is invertible, so the Jacobian conjecture holds for every Keller map whose function field extension is Galois.

• equivalent to The Jacobian conjecture for cubic-homogeneous maps refuted · V3

  So the cubic-homogeneous case over all dimensions is equivalent to the Jacobian conjecture over all dimensions. The equivalence raises the dimension, so it says nothing about any single $N$.

• implied by The Dixmier conjecture in rank n refuted · V3

  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.

• reduction of The Dixmier conjecture in rank n refuted · V3

  So DCₙ reduces to the Jacobian conjecture in twice the dimension: a proof of JC₂ₙ would prove DCₙ. See the stable equivalence.

• related to Drużkowski's reduction to cubic-linear maps proved · V3

  Like its parent, the reduction raises the dimension, so it bears on the all-dimensions conjecture only, not on any fixed dimension.

• related to The formal inverse and its tree formula proved · V3

  For homogeneous $H$ of degree $d$ in $n$ variables, invertibility is the vanishing of the tree sums of every size m with (d−1)m + 1 > d^{n−1}, and through the cubic reduction the all-dimensions conjecture is this statement for cubic H. The approach was pursued by Wright, by Singer, and by Bisi and coauthors, through random planar trees and $d$-Catalan trees.

• implied by The Gaussian moments conjecture refuted · V0

  Derksen, van den Essen and Zhao proved that the Gaussian moments conjecture in every dimension implies the Jacobian conjecture, and that a non-invertible cubic-homogeneous Keller map in $r$ variables makes GMC($2r$) false. The operators $\partial_i-x_i$ commute, have order one and constant leading coefficients, and by Gaussian integration by parts their images span $\ker\mathbb E$.

• generalizes Graded Keller maps proved · V4

  So the Jacobian conjecture holds for every Keller map graded with one-sign weights, in every dimension.

• refuted by Harish's map of ℂ⁴

  It is the first counterexample written directly in dimension four, rather than obtained by padding a map of ℂ³. It is a Keller map that is not injective, so it refutes JC₄ directly, without stabilizing a three-dimensional map.

• generalizes The Hessian conjecture in dimension n refuted · V1

  Since $J(\nabla\varphi)=\operatorname{Hess}\varphi$, a gradient map with constant Hessian determinant is a Keller map, so HCₙ is the special case of JCₙ for gradient maps.

• implied by The Hessian conjecture in dimension n refuted · V1

  So HC₂ₙ implies JCₙ. In particular HC₄ implies JC₂.

• implied by The image conjecture refuted · V3

  He showed that the Jacobian conjecture, and hence the Dixmier conjecture, are each equivalent to special cases of it. So the image conjecture implies the Jacobian conjecture in every dimension.

• generalizes JC₃: the Jacobian conjecture in dimension three refuted · V0

  It is the three-variable case of the general conjecture, so JC₃ is a special case of JCₙ, and its failure makes the all-dimensions conjecture false.

• generalizes JC₂: the Jacobian conjecture in the plane open

  The two-variable statement is the oldest form of the problem: Kraus stated it in 1884, with a proof that fails at its last step. It is the case $n=2$ of the general conjecture, and after July 2026 it is the only dimension of JCₙ that is still open: every counterexample found that month needs at least three variables.

• generalizes Keller's birational theorem proved · V3

  The proof used homology. In modern terms, the conjecture holds for Keller maps of field degree 1, so Keller's theorem is the Jacobian conjecture restricted to birational maps.

• related to Xavier's dichotomy for the Keller locus proved · V4

  The theorem recasts the Jacobian conjecture as the claim that every component of every bounded-degree Keller locus lies in the automorphism group.

• implied by The Mathieu conjecture refuted · V3

  Mathieu proved that his conjecture, for all compact connected Lie groups, implies the Jacobian conjecture in every dimension. The July 2026 counterexample therefore refutes it as stated.

• reduction of The Poisson conjecture in rank n refuted · V3

  A symplectic map has Jacobian 1, so PCₙ reduces to the Jacobian conjecture in dimension 2n.

• implied by The Poisson conjecture in rank n refuted · V3

  The cotangent lift $(x,\xi)\mapsto(F(x),JF(x)^{-T}\xi)$ of a Keller map is symplectic, and it is injective only if $F$ is, so PCₙ implies JCₙ directly.

• related to For a Keller map, invertible, injective and proper are the same proved · V3

  So the Jacobian conjecture says every Keller map of ℂⁿ is proper, and every counterexample in dimension three and up is a Keller map that is not proper.

• reduction of The real Jacobian question in dimension n refuted · V0

  So the real question reduces to the Jacobian conjecture in the same dimension.

• generalizes Wang's theorem: quadratic Keller maps are invertible proved · V3

  Wang's theorem appears in a paper on a Jacobian criterion for separability. It says the Jacobian conjecture holds, in every dimension, for maps of degree at most two.

• equivalent to Zhao's vanishing conjecture refuted · V3

  The result is that the vanishing conjecture in every dimension is equivalent to the Jacobian conjecture in every dimension. The equivalence does not preserve the dimension, because it passes through the cubic-homogeneous and symmetric reductions.