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Jacobian conjecture - mklabgr - 07-22-2026
Jacobian conjecture
Summary
The Jacobian Conjecture is one of the most famous problems in algebraic geometry, first proposed by Ott-Heinrich Keller in 1939. It asks whether every polynomial map from an n-dimensional space to itself with a nonzero constant Jacobian determinant must always have a polynomial inverse. Intuitively, the condition means the map is locally reversible everywhere, and the conjecture asks whether this guarantees global reversibility as well. Although the statement is simple enough to understand with basic multivariable calculus, proving or disproving it has challenged mathematicians for decades.
Numerous published proofs have later been found to contain subtle errors, making it a notorious open problem and one of Stephen Smale’s influential mathematical challenges. Researchers have established many special cases and reduction theorems, showing that solving a few restricted versions would settle the general problem, but a complete solution has remained elusive. The conjecture is also closely connected to several other deep problems in algebra and geometry, including the Dixmier Conjecture, highlighting its central role in modern mathematics.
Quote:On July 19, 2026, mathematician and Anthropic employee Levent Alpöge presented an explicit counterexample in three-dimensional space, discovered using Claude Fable 5, Anthropic's large language model, which disproves the conjecture for $N>2$
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