How Fable AI Disproved the 87-Year-Old Jacobian Conjecture
A digestion of the Jacobian conjecture counterexample
I explore the surprising counterexample to the Jacobian conjecture recently discovered by Fable AI in three dimensions. While the conjecture remains open in two dimensions, this new finding reveals a polynomial map that is locally invertible yet globally non-invertible. I break down the geometric intuition behind this result, showing how simple polynomial multiplication and symmetry arguments explain a phenomenon that initially seemed like a mathematical miracle.
The polynomial F has degree seven, so a priori the Jacobian ought to be a polynomial in three variables of degree as large as 18, so the fact that all non-constant coefficients of this polynomial vanish looks like a massive cancellation involving 1299 coefficients.
- 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.
- 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...
- sashank_1509
I don’t understand math but it was amusing seeing Terrence Tao’s chat with chatGPT. Everything Tao said was constantly followed by praise:
“That’s exactly the right way to think about it.”,
“Yes, you are exactly right.”
“You have gotten to the core issue.”
And non stop praise. Seems like sycophancy is still an issue lol.
- aayushdutt
After reading a quarter of the article I started wondering, is this what non coders feel when vibe coding software?
- hyperhello
Okay. So what does this overturn, intuitively? Can we no longer assume that functions are differentiable at certain points, or something?
- jeremyscanvic
> Also, from the fundamental theorem of algebra, once the Jacobian polynomial {\mathrm{det} DF} is non-zero, it must be constant.
I wouldn't have guessed this is true. I'm wondering what the proof looks like!
- 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.
- 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.