A digestion of the Jacobian conjecture counterexample

vanderZwan an hour ago | parent | next [-]

> While this is an extremely quick verification, the construction presented in this fashion appears like a massive miracle. The polynomial {F} has degree seven, so a priori the Jacobian {\mathrm{det} DF} ought to be a polynomial in three variables of degree as large as {3 \times 6 = 18}, so the fact that all non-constant coefficients of this polynomial vanish looks like a massive cancellation involving {\binom{18+3}{3}-1 = 1329} coefficients, which is much larger than the {\binom{7+3}{3} = 120} degrees of freedom for a generic degree seven polynomial of three variables. So finding such a polynomial looks highly unlikely to be located by brute force.

Sounds like the most interesting part would be learning what approaches the LLM did use to see if that's reusable elsewhere. I'm guessing that's what the rest of the article is about? Because I also couldn't follow the maths any more.

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tptacek 2 hours ago | parent | prev | next [-]

The introduction to this piece was easy to follow, but as soon as he got into recapitulating it with algebra he lost me (because I'm bad at math). But he includes the GPT5 prompts for his conversation, which are easier to follow:

https://chatgpt.com/share/6a5fdc7a-d6f8-83e8-bbea-8deb42cfed...

	▲	minimaxir 2 hours ago \| parent [-]
		Also note the timestamp: he started working on this thread a few hours after the tweet.

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hyperhello 2 hours ago | parent | prev | next [-]

Okay. So what does this overturn, intuitively? Can we no longer assume that functions are differentiable at certain points, or something?

	▲	mswphd 8 minutes ago \| parent \| next [-]
		it doesn't overturn much. For example, here is a post from 2004 https://www.math.columbia.edu/~woit/wordpress/?p=105 it is about a purported (though incorrect) positive proof of the Jacobian conjecture in 2 dimemnsions. It is true in 1 dimension. The Fable proof is that it is false in >= 3 dimensions. 2 dimensions is still open. Anyway, in that post it says > It now seems that a proof has been found by Carolyn Dean of the University of Michigan, for the case of polynomials in two complex variables (for more variables, many people believe it is not even true) so the resolution of this is a "surprise" in that it is a very long open with many failed proof attempts. But the direction it resolved was not surprising.
	▲	sfpotter 39 minutes ago \| parent \| prev [-]
		No. This is about polynomials. The assumption that the Jacobian is nowhere zero is what is doing so much of the work. This means the Jacobian must in fact be constant. But obviously there are many mappings whose Jacobians are not constant.

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drivebyhooting 37 minutes ago | parent | prev | next [-]

Can we audit the CoT and work the AI did to generate such a remarkable cancellation?

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ChrisArchitect 2 hours ago | parent | prev | next [-]

Claude Fable produced a counterexample to the Jacobian Conjecture

https://news.ycombinator.com/item?id=48973869

Human mathematicians are being outcounterexampled

https://news.ycombinator.com/item?id=48983382

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zzzeek 38 minutes ago | parent | prev [-]

reading through this I eventually realized a situation similar to my experience of it is what my dog sees if I attempt to explain Python programming to him.