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The Jacobian Conjecture

An 85-year-old problem about when a polynomial map must be invertible, and the July 2026 map in C^3 that appears to break it: locally invertible everywhere, yet many-to-one globally.

Part 0

A Counterexample

A Counterexample to the Jacobian Conjecture

For 85 years a nonzero constant Jacobian was believed to force a polynomial map to be invertible. A July 2026 map in C^3 breaks that: watch several paths between two distinct inputs collapse onto one shared output.

8 min · Jacobian conjecture