A digestion of the Jacobian conjecture counterexample
terrytao.wordpress.com
July 21, 2026
12 min read
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Summary
The Jacobian conjecture posits that a polynomial map in complex variables with a non-zero constant Jacobian is invertible with a polynomial inverse. Local invertibility is equivalent to the Jacobian being non-zero, following from the inverse function theorem.
Key Takeaways
- The Jacobian conjecture asserts that if a polynomial map in complex variables has a non-zero constant Jacobian, then it is invertible with a polynomial inverse.
- A counterexample has been found showing that the Jacobian conjecture is false in three dimensions, where a polynomial has a non-zero constant Jacobian but is not invertible.
- The conjecture remains open in two dimensions and is easily established in one dimension.
- The counterexample can be reformulated to show the existence of an affine variety that is locally injective but not globally injective.
Community Sentiment
Positives
- The introduction is easy to follow, and including the GPT-5 prompts makes the complex conversation more accessible to those not deeply versed in math.
- Finding a different way of thinking about a problem often leads to breakthroughs, and this diversity in approaches could lead to solving long-standing mathematical challenges.
Concerns
- There's skepticism about the implications of the findings, with some arguing it doesn't overturn much of the existing understanding of the Jacobian conjecture.
- Doubts linger about the transparency of the AI's workings; the 'magic' behind the remarkable results may never be fully disclosed, which raises concerns about trust in AI-generated solutions.