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jacobian-conjecture

Revision 19 · 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/Problem__jacobian-conjecture.md).”
Revision 19 · 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/Problem__jacobian-conjecture.md).”

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Revision 19 as rendered

Main article: jc-n

This atlas problem is the Jacobian conjecture in the dimensions where it fell. Because a counterexample in dimension nn pads to one in dimension n+1n+1, “JCₙ for every n≥3n\ge3” holds exactly when JCₙ holds for every nn, so this problem is the all-dimensions Jacobian conjecture. It was refuted by Alpöge's map of ℂ³: a map of degree 7 with Jacobian determinant −2-2 that sends three points to one.

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.

The plane case survives as a separate problem, the Jacobian conjecture in the plane. The map, its relatives and its consequences are laid out in Charting the unknown.

Sources

Algebraic geometry · Mathematics