Mathematics & AI — Algebraic Geometry Meets Machine Reasoning

AI and a Legend Cracked an 85-Year Math Mystery

The Jacobian conjecture, posed in 1939 and untouched for nearly nine decades, has been knocked down in three dimensions — a chat model found the counterexample, and Fields Medalist Terence Tao publicly walked through it, conversation by conversation.

• The conjecture claims that a polynomial map with a constant, non-zero Jacobian determinant must have a polynomial inverse. A single counterexample disproves it.
• A counterexample has now been constructed in three variables, written with polynomials of degree seven whose Jacobian happens to flatten to a constant — a cancellation that looks, at first glance, like a miracle.
• The example was produced with help from Anthropic's "Fable" model, then independently scrutinized, including by Tao, who published his own working through of it on his blog and shared his chat-with-the-AI transcript.

At its core the conjecture is an innocent-sounding question about polynomials. Imagine a rule that turns three numbers into three other numbers, built only from addition, multiplication and powers. The Jacobian of that rule is a single number summarizing how volumes stretch locally; if it never changes and never vanishes, the rule is locally reversible everywhere. The Jacobian conjecture asked whether that local reversibility is enough to guarantee a global one — whether a smooth, nowhere-tangling map must also be untangleable all the way.

For one variable the answer is trivially yes. For two variables it has resisted every attack for 85 years. For three variables, the answer is now no. The constructed map has a Jacobian that is a fixed non-zero number, yet it folds space in a way that no polynomial rule can ever undo. The reason it looked unfindable is precisely the shape of the cancellation required: making the Jacobian collapse from what should have been a huge polynomial of degree up to sixteen down to a single constant demands that dozens of independent coefficients vanish simultaneously, while the map itself only has a handful of free knobs to turn.

What made this episode remarkable was not just the result but the workflow. A mathematician posed the problem to a language model, received a candidate construction, and then spent the working hours proving it by hand. Tao's public notebook — literally the chat log in which he questioned the model step by step — reframed the question from "did the AI solve it?" to "can the best human minds, working with an AI sounding board, now reach problems they could not reach alone?" In a field where progress is measured in lifetimes, the pace has shifted.

The conjecture's fate is still partial: two dimensions remain open, and the counterexample was retroactively explained in more geometric terms, softening the sense that it had appeared from nowhere. But the episode carries a sharper lesson. Large language models are being used not to replace proof, but to compress the search space of human attention — to surface objects and constructions that a mathematician would not have reached by their own reading list. The proof, in the end, is still human. The lead is increasingly not.

Knowledge takeaway: The Jacobian conjecture asserts that a constant, non-zero Jacobian guarantees a polynomial inverse; a three-variable counterexample shows it is false in three and higher dimensions; the two-dimensional case remains open, while AI-assisted exploration is changing how open problems get tackled.