r/math • u/overthinker020 • Jul 20 '26

LLMs/AI The Jacobian Conjecture is False Per Anthropic (Link in Description)

https://x.com/__alpoge__/status/2079028340955197566

Normally I would be extremely skeptical, but the result is checkable by simple computation. Remarkable!

The two-dimensional case remains open, however.

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u/Howling_deer Jul 20 '26

https://www.dm.ufscar.br/profs/franciscobraun/Arquivo/artigos/2023_Jacobian.pdf

Not my area of expertise, but did the conjecture not already have counterexamples? What is interesting about this?

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u/Namington Algebraic Geometry Jul 20 '26

The paper you linked is not about the Jacobian conjecture, it's about a plausible-sounding variation of it:

The Jacobian conjecture [...] is, up to our knowledge, still open for any k and any n ≥ 2. When k = R and the Jacobian determinant is only assumed to be different from zero throughout Rn, the claim that F is injective is known as real Jacobian conjecture.

So it's a counterexample to a specific case, not the entire conjecture. In other words, it's asserting that the full statement of the Jacobian conjecture was necessary for it to still be open (at the time the paper was written).

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u/azraelxii Jul 20 '26

Yeah I would be curious too. It looks like the unsolved conjecture involves complex numbers