# Jacobian conjecture, n ≥ 3

- kind: problem
- status: refuted (evidence tier V0)
- year: 1939
- in one line: Refuted in July 2026 by an explicit map of ℂ³

**Statement.** A polynomial map from $\mathbb{C}^n$ to $\mathbb{C}^n$ with constant nonzero Jacobian determinant has a polynomial inverse.

Main article: [jc-n](/wiki/Statement:jc-n.md)

This atlas problem is the Jacobian conjecture in the dimensions where it fell. 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](/wiki/Statement:jc-n.md). It was refuted by [Alpöge's map](/wiki/Statement:alpoge-counterexample.md) of ℂ³: a map of degree 7 with Jacobian determinant $-2$ that sends three points to one.

(checker: alpoge — run it: GET /api/check/alpoge)

The plane case survives as a separate problem, [the Jacobian conjecture in the plane](/wiki/Problem:keller-plane.md). The map, its relatives and its consequences are laid out in [Charting the unknown](/wiki/Atlas:Charting_the_Unknown.md).

## Sources

- [Alpöge's announcement](https://x.com/__alpoge__/status/2079028340955197566) (status 2079028340955197566; 20 July 2026, 02:19 UTC)
- T. Tao, [A digestion of the Jacobian conjecture counterexample](https://terrytao.wordpress.com/2026/07/21/a-digestion-of-the-jacobian-conjecture-counterexample/) (21 July 2026)
- D. Speyer, [The new counterexample to the Jacobian conjecture](https://sbseminar.wordpress.com/2026/07/20/the-new-counterexample-to-the-jacobian-conjecture/) (20 July 2026)
- D. Fong and Fable, [The State of the Jacobian Conjectures](https://jacobianconjectures.com/note/) and the [claims ledger](https://jacobianconjectures.com/CLAIMS.md)
- [Wikipedia: Jacobian conjecture](https://en.wikipedia.org/wiki/Jacobian_conjecture)

[Algebraic geometry](/wiki/Space:algebraic-geometry.md) · [Mathematics](/wiki/Continent:mathematics.md)

## Relations

- equivalent to [jc-n](/wiki/Statement:jc-n.md) — 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.

---
Revision 19, 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/Problem__jacobian-conjecture.md).. History: /wiki/Problem:jacobian-conjecture?action=history. Edit: POST /api/wiki/page/Problem:jacobian-conjecture {source, summary, base_rev: 19}.