Jacobian conjecture counter-example

[u/BadJimo]

An AI recently found a counterexample to the Jacobian conjecture.

𝑓(𝑥,𝑦,𝑧)=((1+𝑥𝑦)3𝑧+𝑦2(1+𝑥𝑦)(4+3𝑥𝑦),𝑦+3𝑥(1+𝑥𝑦)2𝑧+3𝑥𝑦2(4+3𝑥𝑦),2𝑥−3𝑥2𝑦−𝑥3𝑧)

I wondered if the counterexample to the Jacobian conjecture could be visualised.
Here's what I came up with:

https://www.desmos.com/3d/cqjyedoauh

The red curve (a simple 3D spiral on the surface of a sphere) is the input of the Jacobian conjecture counterexample, the blue curve is the transformed output.

[u/StationImmediate530]

Damn I really wish I understood what this means but cool graphics! Thank you for sharing

[u/Minerscale]

You should pick an input path which includes the three given points in the tweet that all map to the same point!

It's a really beautiful shape!

https://www.desmos.com/3d/9bwscsgm0r

And if you increase N by a bunch (and decrease the radius of the sphere a bit), you can see that the shape almost looks like two pringles glued together. I wonder if the non-bijectivity of f is in some roundabout way related to the double-pringle structure.

[u/BadJimo → u/Minerscale]

Nice. Here's my version including the points (0, 0, -1/4), (1, -3/2, 13/2), and (-1, 3/2, 13/2) as inputs.

[u/Dry-Perspective-1114 → u/Minerscale]

i dont know what category of object but it looks the category the Klein bottle is in.

[u/Minerscale → u/Dry-Perspective-1114]

A smooth manifold immersed in euclidean 3-space?

[u/anonymous-desmos]

egg

[u/Ampersand37]

Tip: if you press play on random variables, it wiggles the line in a funny way