Comment by hyperhello
1 day ago
Okay. So what does this overturn, intuitively? Can we no longer assume that functions are differentiable at certain points, or something?
1 day ago
Okay. So what does this overturn, intuitively? Can we no longer assume that functions are differentiable at certain points, or something?
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.
Thanks, I was curious for the motivation behind the conjecture, but it wasn’t mentioned in the Wikipedia article.
Sounds like 3 more iterations on K3/Fable, and we'll able to ask it to break computational hardness assumptions like DLP?
Possibly, but coming up with algorithms is difficult!
And people have been searching for faster DLP algorithms for 50 years
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.
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.
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's not immediately intuitive what it means for something to be globally and locally invertible. After all, it is obvious that it is both in the 1D case.
You can get the inverse of the Jacobian at any point, but you cannot describe the inverse of the Jacobian through a polynomial, which is a function. You need a more complex object to describe the inverse, because the global inverse is not a function due to the potential of overlapping values.
The determinant of a polynomial mapping is a polynomial, which is the subject of the conjecture. To get the Jacobian determinant, all you need to do is compute partial derivatives of polynomials, and add, subtract, and multiply them together. All of these operations map polynomials to polynomials.
The crux of the assumption is that if a polynomial mapping is invertible everywhere (Jacobian nonzero everywhere), its Jacobian must be a constant. Why? Because the only polynomials which are zero nowhere are constants.
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.
There was no particular reason to think it was true. It's easy to find examples using exponentials or trig functions where it's not true. But it would be neat if it was true, and nobody found an example where it wasn't true in 75 years, so it was tempting...
It was really more of a roadblock. If you had an example of where it was false, you could give examples of other things, so various questions required resolving the Jacobian conjecture.