Jacobian Conjecture for Baby

Posted by porphyra 5 days ago

Comments

Comment by johnfn 20 hours ago

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.

Comment by thebiblelover7 19 hours ago

Seems like it was entirely AI written (claude) from the commits:

https://github.com/muchmirul/conjectures/commits/main/

Comment by muchmirulys 6 hours ago

revised. *ai assisted

Comment by thedevilslawyer 18 hours ago

Shows how well prompted AI can significantly improve human understanding.

It's the slop patterns that seem to turn people off.

Comment by muchmirulys 6 hours ago

revised. *but still ai assisted

Comment by muchmirulys 6 hours ago

thanks, revised version up. please do check and hope it helps

Comment by neuralkoi 4 days ago

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.

Comment by noio 16 hours ago

I felt the same. In a way the 'baby' language in the first 10 chapters (e.g. "straight" instead of "linear") left me unprepared for the later chapters.

Comment by y-curious 20 hours ago

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.

Comment by NooneAtAll3 20 hours ago

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

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good to know that conjecture has been false for real numbers for a while now

Comment by ahofmann 19 hours ago

I think the entire post was created by an AI, and I had a similar negative reaction to it. Nevertheless, it's an excellent post with clear explanations of mathematical concepts. This just goes to show how impossible it is to tell the difference between something created by a human and something created by an LLM.

Comment by dash2 19 hours ago

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.

Comment by NooneAtAll3 17 hours ago

conjecture said "if there is no J(point) = 0, there's no f(point)=f(point)"

counterexample for reals moved point=0 outside reals, but kept point=point inside

you can see the "jacobian is sum of squares" being mentioned - that is sufficient to say there's no negative values in the reals, but doesn't work for the whole complex field

for complex numbers you have to have jacobian be a constant, or you'll get the zero somewhere

Comment by samlinnfer 20 hours ago

It got more and more AI towards the end.

Comment by muchmirulys 6 hours ago

revised for better progression *still ai assisted

Comment by CamperBob2 20 hours ago

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.

Comment by muchmirulys 6 hours ago

revised. thanks for the feedback

Comment by CamperBob2 2 hours ago

Great work, don't get me wrong. Looking forward to more like it!

Comment by CGamesPlay 19 hours ago

Chapter 8 is incomplete. It says the local area factor "is zero at one single point... and the map uses it to wrap the plane around twice... so local factor never zero is not enough on its own".

"X is false, therefore X implies Y is false". But this is faulty reasoning.

Comment by muchmirulys 6 hours ago

revised. hope it clear any confusion

Comment by para_parolu 19 hours ago

Awesome. I was literally asking ai to find blog post like this but couldn’t get anything decent