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Jacobian Conjecture for Baby

muchmirul.github.io

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Re: Jacobian Conjecture for Baby

#2

Post by neuralkoi »

This was an excellent write-up and taught me more in 20 minutes than my linear algebra teacher did in 6 months.

The material however, got fairly dense. After chapter 10, it is no longer "baby" material in my opinion.

Re: Jacobian Conjecture for Baby

#5

Post by y-curious »

I followed very loosely and still got it. Very cool, the kind of math that gets close to feeling like "real life". I was really hoping that your mentions of "solving it by hand" and "you can prove it to yourself by hand" at the end were going to come to some clever maneuver I could do with my own hand (like the physics magnetism hand rule). Still, awesome.

Re: Jacobian Conjecture for Baby

#6

Post by CamperBob2 »

samlinnfer wrote: post #4
NooneAtAll3 wrote: post #3

beginning reeked of... maybe not ai, but bad writing but the further into the text, the better the prose by the end it was great --- good to know that conjecture has been false for real numbers for a while now

It got more and more AI towards the end.

Also more and more use of an extremely annoying effect: text-heavy animations that reset themselves for no reason while you're trying to read them. Step 11, for instance: no value whatsoever is added by erasing and redrawing that figure.

Still, overall a very nice piece of pedagogy. If you didn't grok determinants before, this will probably help.

Re: Jacobian Conjecture for Baby

#7

Post by johnfn »

This is really good - miles ahead of almost all other math writing I have read. One particular thing I really liked was the explanation of the determinant. When I was in college I spent a very long time trying to get some intuitive explanation for a determinant of a matrix and the best I could get is that you just multiplied and subtracted some numbers and poof there you go. I KNEW there had to be some more comprehensible explanation and it drove me crazy that people just parroted textbooks without any real understanding.

This reminds me of that silly video everyone was shown in middle school about inverting a sphere :-)

I am very curious how it was written (will there be a write up to the write up?!). It feels likely AI assisted, but this seems to me to be the best possible use of AI -- crunching out a bunch of visualizations of complex ideas so that they are easy to understand just by looking. The writing also feels a little AI written but definitely feels very human-assisted.

Re: Jacobian Conjecture for Baby

#10

Post by dash2 »

NooneAtAll3 wrote: post #3

beginning reeked of... maybe not ai, but bad writing but the further into the text, the better the prose by the end it was great --- good to know that conjecture has been false for real numbers for a while now

one thing I didn't understand. If the conjecture is false for real numbers, why isn't it false for complex numbers too? Why can't you just take a real number counterexample and use it for complex numbers?

I guess the answer is that a polynomial map whose Jacobian determinant is constant non-zero over the reals may not also be constant non-zero over all the complex numbers.

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