The problem
Let F: ℂⁿ → ℂⁿ be a polynomial map whose Jacobian determinant is a nonzero constant. Must F be invertible with polynomial inverse? Keller asked for n = 2 in 1939; the case n = 3 carries the same reputation for truthfulness.
The Jacobian Conjecture (three variables)
AI-resolvedReferences
History & significance
On Smale's 1998 list of mathematical problems for the 21st century. Decades of partial results (injectivity criteria, reduction attempts) all pointed the same way as intuition: surely true. Almost nobody was trying to disprove it.
Connected problems
The resolution (AI-assisted resolution)
Disproved — Levent Alpöge (Harvard), announced 19 July 2026, working with an AI system he credits as a substantial collaborator. The counterexample fits in 216 characters: a concrete polynomial map in three variables with constant Jacobian 1 that admits no polynomial inverse. Experts (Abhishek Saha, Queen Mary) note the single-line answer was verified almost immediately across the community — the surprise is not the checking but the finding. The two-variable case survives and remains open; the result is widely described as the hardest mathematical problem yet significantly cracked by AI.