A digestion of the Jacobian conjecture counterexample

I was reading another source that claimed this example was inspired by an existing (rational polynomial) example from the literature (created in 1999 by a Russian mathematician Vitushkin).

> The seed is almost certainly Vitushkin's old rational "counterexample."

From https://claude.ai/share/22abed98-d9af-43c5-9881-b19e009a07b0

This is not quite lore laundering, but it seems to be close.

Also note the timestamp: he started working on this thread a few hours after the tweet.

I like how he goes one-on-one with it like Gandalf fighting Yoda on fifteen planes at once for 80% of the transcript, and then we read "OK, I've activated Pro."

I'm not a mathematian but I know enough linear algebra and vector calculus to understand the conjecture. This is my interpretation:

Firstly, the determinant of the Jacobian is measuring if at any point the function is crushing space / flattening out.

If the Jacobian is a nonzero constant everywhere this means that nowhere does the the function flatten out. A small change in X along any line will always produce a non zero change in Y. Not flattening out means that locally you can invert it.

What was conjectured is that this local invertibability property everywhere would mean global invertibility.

Turns out to not be the case.

For a simple case, the falsified conjecture is trivially true in 1D.

Specifically consider f(x) = x^2

This function happens to flatten out right at x=0. At that x coodrinate the function flattens out and folds over on itself. This fold means you can't invert x^2. It's also not locally invertible around x=0.

If a function f(x) has constant derivative evewhere then it would flatten out nowhere and it would be invertible everwhere. It would also be globally invertible.

The Jacobian conjecture was stating that the extension of this property holds in higher dimensions. That if the function had no fold in space then it would be invertible globally.

The counterexample shows that you can create a simple function in 3 variables, where the function demonstratably is invertible evewhere, but is not injective globally (they specifically show 3 points that map to the same output).

What's interesting is this is like if someone showed you a parabola where somehow you got back to the same y coordinate without a kink bending over back to itself.

For the Jacobian determinant to be constant is a massive coincidence, in general it is some complicated and messy polynomial. The conjecture was that this coincidence couldn't happen, except for simple special cases.

So Alpoge and Fable found an example of a function that was believed to be too strange to exist.

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.

Not much.

But it does give credible plausibility to the concept that we might be mistaken about the exact boundaries of hardness for adjacent (but not equivalent) polynomial systems. Most (all?) of which have also stood up to a whole lot of undeniably sharp people poking at them for about as long.

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.

It overturns the Jacobian conjecture (i.e., speculation) for dim >= 3, which we now know was an overgeneralization. Tao characterizes it as "can be viewed as an assertion that local invertibility implies global invertibility". It was already widely suspected to be false. Assuming that it was true was never warranted, so this really doesn't change anything. The significance is that an AI was able to find a relatively simple counterexample. Its "chain of thought" would be very interesting to see.

"problems that were hard"

They are still hard problems - As we say in the UK: "one swallow does not a summer make".

As you well know: birds are not renowned for their arithmetic skills, nor eating encourages the weather!

The difference is your dog will never understand the Python code but you could probably understand this post in a matter of days or weeks if you really wanted to. Can we all please stop acting like this Terry guy is so special?

Some people downvoting you, but I think it is a valuable illustration of IQ gap.

And chances are that humanity at large will be soon trying to follow ai inventions and discoveries not unlike your dog follows your Python code.

I doubt Anthropic will share the details (or at least the full true details). The mystery of the magic makes for much better marketing.

I think a reasonable assumption is that there is an interaction between an LLM, a https://en.wikipedia.org/wiki/Computer_algebra_system tool, a human prompting with deep math expertise, and lots of compute that explains hitting upon the remarkable cancellation.

I am sure he did the minimum effort needed to communicate what he wanted to communicate.

If you are offended by his math gifs and feel that the widely regarded best mathematician of our time should use embedded LaTex or something better, why not offer to upgrade his blog?

Why are you so upset about people oohing and aahing? These may not be the fireworks you like, but don't yuck their yum. We should do more praising of each other for doing work.