@jerryjliu0: This is awesome. Instead of doing work this morning, I spent some time brushing up on math to provide a handwavy overvi…

This is awesome. Instead of doing work this morning, I spent some time brushing up on math to provide a handwavy overview to anyone else who like me doesn’t have a math background and is rusty:

At a distance, the conjecture is basically that if a n-dimensional polynomial function (e.g. F(x, y, z) map to some output (x’, y’, z’)) meets some special conditions, then 1) it is invertible, and 2) the inverse is polynomial.

Disproving the conjecture requires finding a n-dimensional polynomial function that meets these special conditions, but is either 1) not invertible at all, or 2) is invertible but not polynomial. (2) is actually not a possibility due to some other math theorem - if there is an inverse, the inverse has to be polynomial.

Invertibility just means the function can be “undone” at every single input point - if input (x1, y1, z1) maps to output (x2, y2, z2), then output (x2, y2, z2) can map to input (x1, y1, z1). A function is not invertible if multiple input points (x, y, z)’s map to the same output (x, y, z)

Here Fable 5 presumably did some search and found a function that was not invertible.

That function is this:
F(x, y, z) = ( z(1+xy)^3 + y^2(1+xy)(4+3xy), y+3x(1+xy)^2z + 3xy^2(4+3xy), 2x-3x^2y-x^3z )

This function meets the special conditions (jacobian determinant is a non-zero constant for every point), but has multiple input points that map to the same output:

Inputs: (0, 0, -1/4), (1, -3/2, 13/2), and (-1, 3/2, 13/2) all map to the same output: (-1/4, 0, 0)

The Jacobian is the multivariable version of a derivative, so it is a function of a specific input point. When the Jacobian determinant is non-zero at a specific input point, there is local invertibility around that point. This conjecture was stronger in that if it was non-zero (and constant, due to being over the complex numbers, which I don’t quite get and I ran out of time to understand), then the function is globally invertible.

What’s interesting here is that finding counterexamples seem like the type of math work that agentic models excel at. It involves generating and crawling a large search space, using priors to narrow it down, and a bit of brute force and raw compute power. Machines can process a wide set of possibilities faster than humans, which is helpful for finding counterexamples.

levent (@alpoge): hello there the jacobian conjecture is false thanx to my close friend akhil for asking about it and my other close friend fable for working during the world cup final

((1+xy)^3 z + y^2 (1+xy) (4+3xy), y + 3 x (1+xy)^2 z + 3 x y^2 (4+3xy), 2 x - 3 x^2 y - x^3 z): \C^3\to \C^3,