A digestion of the Jacobian conjecture counterexample

tptacek

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...

ChrisArchitect

Related: 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

hyperhello

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

vanderZwan

> 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.

zzzeek

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.

drivebyhooting

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

jmward01

Finding a different way of thinking about a problem often leads to a breakthrough. This is what an ecosystem in nature shows us, that diversity matters in finding hard solutions. I think the great thing here is we are getting a chance to find whole new ways of thinking about problems that were hard. I suspect many old problems will fall because of it and, hopefully, some really new interesting ones will replace them.

aayushdutt

After reading a quarter of the article I started wondering, is this what non coders feel when vibe coding software?