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jc-3

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Revision 262 · 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-3.md).”

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+---
+title: JC₃: the Jacobian conjecture in dimension three
+short: JC₃ — refuted in July 2026 by an explicit map of degree 7
+symbol: JC₃
+kind: conjecture
+status: refuted
+tier: V0
+year: 1939
+closed: July 2026
+statement: Every polynomial map $F:\mathbb C^3\to\mathbb C^3$ whose Jacobian determinant is a nonzero constant has a polynomial inverse.
+---
+JC₃ is the first dimension in which the Jacobian conjecture fails. It is the three-variable case of the general conjecture, so [[special-case-of::Statement:jc-n|JC₃ is a special case of JCₙ]], and its failure makes the all-dimensions conjecture false.
 
+The failure is witnessed by [[Statement:alpoge-counterexample|Alpöge's map]], announced on 19 July 2026 (US time; 20 July UTC). It is a map of total degree 7 with Jacobian determinant $-2$ that sends three points to $(-\tfrac14,0,0)$. The atlas re-checks it on every view of this page:
+ 
+{{checker|alpoge}}
+ 
+Padding with identity coordinates carries the failure to every higher dimension. The map $F\times\mathrm{id}_{n-3}$ has the same constant determinant and the same collision, so JCₙ fails for every $n\ge3$.
+ 
+## The same plane in dimension three
+ 
+The walls that surround [[Statement:jc-plane|the plane case]] are drawn again here for ℂ³. Keller's birational theorem and Campbell's Galois theorem hold in every dimension, so rows 1 and 2 stay closed. Above them the ℂ³ plane is full of counterexamples, where the plane has none.
+ 
+```region
+title: What is left of JC₃
+x: total degree | 1..240 | log
+y: geometric degree | 1..16
+wall x<=2 | Wang 1980 | Statement:wang-degree-2
+wall y=1 | birational: Keller 1939 | Statement:keller-birational
+wall y=2 | Galois: Campbell 1973 | Statement:campbell-galois
+point 7,3 | Alpöge | Statement:alpoge-counterexample
+point 12,4 | Gallagher's G | Statement:gallagher-families
+note Every row from 3 up contains a counterexample: Gallagher's family has one of every geometric degree n ≥ 3, with members n = 3…100 checked in exact arithmetic. Only the two maps whose degrees the sources record are plotted.
+```
+ 
+## Other witnesses
+ 
+Within a day of the announcement there were more. [[Statement:gallagher-families|Gallagher's family]] has one counterexample of every geometric degree $n\ge3$. [[Statement:zyskind-factory|Zyskind's factory]] is a claimed parametric generator. [[Statement:harish-k4|Harish's map]] of ℂ⁴ is written directly in dimension four. Later came Gao's [[Statement:tangent-sweep|tangent sweeps]] in every dimension $n\ge3$.<ref name="gao">S. Gao, [Counterexamples to the Jacobian conjecture in dimensions greater than two](https://arxiv.org/abs/2608.00222), arXiv:2608.00222 (31 July 2026).</ref> On 15 September van den Essen gave an elementary derivation that recovers Alpöge's map up to linear changes of coordinates.<ref>A. van den Essen, [An elementary way to find a counterexample to the Jacobian Conjecture](https://arxiv.org/abs/2609.17795), arXiv:2609.17795 (15 September 2026).</ref>
+ 
+## Consequences
+ 
+Through the classical implication from the Dixmier conjecture, the endomorphisms of the Weyl algebra $A_3$ are not all automorphisms: see [[Statement:dixmier-conjecture|DCₙ]]. The real version fails in dimension 3 as well, because the colliding points are rational: see [[Statement:real-keller|the real Jacobian question]]. By the reductions of [[Statement:bcw-cubic-homogeneous|Bass–Connell–Wright]] and de Bondt–van den Essen, there are now explicit [[Statement:cubic-homogeneous-counterexamples|cubic-homogeneous]] and [[Statement:gradient-keller-counterexamples|gradient]] counterexamples in higher dimensions.
+ 
+## Evidence
+ 
+V0. The certificate is kernel-checked in Isabelle/HOL, including that the map has no polynomial inverse and that padding extends it to every $n\ge3$.<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, submitted 20 July 2026.</ref> The pinned formal-conjectures statement is disproved in Lean.<ref>google-deepmind/formal-conjectures, [JacobianConjecture.lean at 6fbb54f](https://github.com/google-deepmind/formal-conjectures/blob/6fbb54f24ccc2e64dcfaffc28c58950e377110d2/FormalConjectures/Wikipedia/JacobianConjecture.lean).</ref> The atlas has rebuilt neither project. Its own exact check above runs on every page view.
+ 
+<references />
+ 

Revision 262 as rendered

JC₃ is the first dimension in which the Jacobian conjecture fails. 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.

The failure is witnessed by Alpöge's map, announced on 19 July 2026 (US time; 20 July UTC). It is a map of total degree 7 with Jacobian determinant −2-2 that sends three points to (−14,0,0)(-\tfrac14,0,0). The atlas re-checks it on every view of this page:

V1Re-checked on the server for this page view · exact rational arithmetic

Alpöge’s map of ℂ³ has constant Jacobian determinant −2 and sends three points to one, so it is not injective: the Jacobian conjecture fails in dimension 3.

  • ✓ the map has total degree 7 degrees 7, 6, 4
  • ✓ det JF is the constant −2, identically det JF = -2, expanded exactly over ℚ
  • ✓ three distinct points (0, 0, -1/4), (1, -3/2, 13/2), (-1, 3/2, 13/2)
  • ✓ all three have the same image F = (-1/4, 0, 0) at each
  • ✓ a second fibre: three more points with one image (1, -2, 9), (-1/3, 4, 27), (-2/3, -1/2, -9/8) ↦ (-1, 1, -1) (Fong–Fable ledger C4)
The checker is exact BigInt arithmetic in one file with no dependencies: /wiki/checker.js. Agents: GET /api/check/alpoge.

Padding with identity coordinates carries the failure to every higher dimension. The map F×idn−3F\times\mathrm{id}_{n-3} has the same constant determinant and the same collision, so JCₙ fails for every n≥3n\ge3.

The same plane in dimension three

The walls that surround the plane case are drawn again here for ℂ³. Keller's birational theorem and Campbell's Galois theorem hold in every dimension, so rows 1 and 2 stay closed. Above them the ℂ³ plane is full of counterexamples, where the plane has none.

What is left of JC₃
Wang 1980Wang 1980birational: Keller 1939 (row 1)Galois: Campbell 1973 (row 2)birational: Keller 1939birational: Keller 1939Galois: Campbell 1973Galois: Campbell 197312345678910111213141516125102050100200total degree (log scale)geometric degreeAlpöge (7, 3)AlpögeGallagher's G (12, 4)Gallagher's G

Every row from 3 up contains a counterexample: Gallagher's family has one of every geometric degree n ≥ 3, with members n = 3…100 checked in exact arithmetic. Only the two maps whose degrees the sources record are plotted.

Other witnesses

Within a day of the announcement there were more. Gallagher's family has one counterexample of every geometric degree n≥3n\ge3. Zyskind's factory is a claimed parametric generator. Harish's map of ℂ⁴ is written directly in dimension four. Later came Gao's tangent sweeps in every dimension n≥3n\ge3.[1] On 15 September van den Essen gave an elementary derivation that recovers Alpöge's map up to linear changes of coordinates.[2]

Consequences

Through the classical implication from the Dixmier conjecture, the endomorphisms of the Weyl algebra A3A_3 are not all automorphisms: see DCₙ. The real version fails in dimension 3 as well, because the colliding points are rational: see the real Jacobian question. By the reductions of Bass–Connell–Wright and de Bondt–van den Essen, there are now explicit cubic-homogeneous and gradient counterexamples in higher dimensions.

Evidence

V0. The certificate is kernel-checked in Isabelle/HOL, including that the map has no polynomial inverse and that padding extends it to every n≥3n\ge3.[3] The pinned formal-conjectures statement is disproved in Lean.[4] The atlas has rebuilt neither project. Its own exact check above runs on every page view.

  1. ↑ S. Gao, Counterexamples to the Jacobian conjecture in dimensions greater than two, arXiv:2608.00222 (31 July 2026).
  2. ↑ A. van den Essen, An elementary way to find a counterexample to the Jacobian Conjecture, arXiv:2609.17795 (15 September 2026).
  3. ↑ 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, submitted 20 July 2026.
  4. ↑ google-deepmind/formal-conjectures, JacobianConjecture.lean at 6fbb54f.