# JC₃: the Jacobian conjecture in dimension three - kind: conjecture - status: refuted (evidence tier V0) - year: 1939 - in one line: JC₃ — refuted in July 2026 by an explicit map of degree 7 **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 [JC₃ is a special case of JCₙ](/wiki/Statement:jc-n.md), and its failure makes the all-dimensions conjecture false. The failure is witnessed by [Alpöge's map](/wiki/Statement:alpoge-counterexample.md), 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 — run it: GET /api/check/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 [the plane case](/wiki/Statement:jc-plane.md) 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** — x: total degree (1–240), y: geometric degree (1–16). - excluded: x <= 2 — Wang 1980 (Statement:wang-degree-2) - excluded: y = 1 — birational: Keller 1939 (Statement:keller-birational) - excluded: y = 2 — Galois: Campbell 1973 (Statement:campbell-galois) - witness at (7, 3): Alpöge (Statement:alpoge-counterexample) - witness at (12, 4): Gallagher's G (Statement:gallagher-families) 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](/wiki/Statement:gallagher-families.md) has one counterexample of every geometric degree $n\ge3$. [Zyskind's factory](/wiki/Statement:zyskind-factory.md) is a claimed parametric generator. [Harish's map](/wiki/Statement:harish-k4.md) of ℂ⁴ is written directly in dimension four. Later came Gao's [tangent sweeps](/wiki/Statement:tangent-sweep.md) in every dimension $n\ge3$.[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 $A_3$ are not all automorphisms: see [DCₙ](/wiki/Statement:dixmier-conjecture.md). The real version fails in dimension 3 as well, because the colliding points are rational: see [the real Jacobian question](/wiki/Statement:real-keller.md). By the reductions of [Bass–Connell–Wright](/wiki/Statement:bcw-cubic-homogeneous.md) and de Bondt–van den Essen, there are now explicit [cubic-homogeneous](/wiki/Statement:cubic-homogeneous-counterexamples.md) and [gradient](/wiki/Statement:gradient-keller-counterexamples.md) 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 $n\ge3$.[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. ## Relations - special case of [jc-n](/wiki/Statement:jc-n.md) — 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. ## References 1. S. Gao, [Counterexamples to the Jacobian conjecture in dimensions greater than two](https://arxiv.org/abs/2608.00222), arXiv:2608.00222 (31 July 2026). 2. 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). 3. 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. 4. google-deepmind/formal-conjectures, [JacobianConjecture.lean at 6fbb54f](https://github.com/google-deepmind/formal-conjectures/blob/6fbb54f24ccc2e64dcfaffc28c58950e377110d2/FormalConjectures/Wikipedia/JacobianConjecture.lean). ## Relations to this page - refuted by [alpoge-counterexample](/wiki/Statement:alpoge-counterexample.md) — It therefore refutes the Jacobian conjecture in dimension three, and by padding in every dimension from three up. Alpöge posted it on X at 02:19 UTC on 20 July 2026 (the evening of 19 July in the US), crediting its discovery to Claude Fable 5. - refuted by [gallagher-families](/wiki/Statement:gallagher-families.md) — The day after Alpöge's announcement, Gallagher showed that the map is the first member of a one-parameter family, not an isolated accident. Each member is a Keller map of ℂ³ that is not injective, so each member of the family refutes JC₃ independently of Alpöge's map. - refuted by [tangent-sweep](/wiki/Statement:tangent-sweep.md) — The result is counterexamples in every dimension $n\ge3$, with arbitrarily large geometric degree in each fixed dimension, and five new explicit maps checked in exact rational arithmetic. So the tangent-sweep theorem refutes JC₃ and every higher dimension, by a mechanism rather than by a single certificate. - refuted by [zyskind-factory](/wiki/Statement:zyskind-factory.md) — If the claim holds, every factory output is a Keller map of ℂ³ with at least three preimages over a general point, and so refutes JC₃. --- Revision 262, 2026-09-30 05:09 UTC, by atlas import (import, claude-opus-5-5, Claude Code 2.1.280): Written for the atlas by Claude Opus 5.5 at the direction of Danielle Fong (content/wiki/Statement__jc-3.md).. History: /wiki/Statement:jc-3?action=history. 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