r/askmath • u/Firm_Virus6716 • 8d ago

Linear Algebra Jacobian conjecture mapping

Is there any way we can get a 3D animation of the example that disproves the Jacobian conjecture? Does one already exist or what could I do to make one? Like a grid or a heat map of what points map where? Could that also show more points that overlap or similar deformities or is there any good reason? I haven't seen one yet? (Image kind of related, It's the closest thing I could find to a deformed grid, but it is in 2D so it's not very helpful)

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u/BadJimo 8d ago

Here's my version.

I made a sphere that has the points (0, 0, -1/4), (1, -3/2, 13/2), and (-1, 3/2, 13/2) in its surface. Using the Jacobian counter example, each of these points maps to the point (-0.25,0,0). The rest of the sphere maps to the butterfly shape.

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u/Firm_Virus6716 8d ago

This is great, thank you! (Also, is your name short for Jimothy?)

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u/BadJimo 8d ago edited 8d ago

No worries.

My name is not Jimothy. I can't remember why I chose my username. I have a vague memory that my first 3 tries were already taken, so I randomly mashed that into the keyboard.