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Jacobian Conjecture for Baby

muchmirul.github.io

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Re: Jacobian Conjecture for Baby

#11

Post by ahofmann »

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

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.

Re: Jacobian Conjecture for Baby

#12

Post by thebiblelover7 »

johnfn wrote: post #7

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 comprehen…

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

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

Re: Jacobian Conjecture for Baby

#13

Post by CGamesPlay »

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.

Re: Jacobian Conjecture for Baby

#14

Post by thedevilslawyer »

thebiblelover7 wrote: post #12
johnfn wrote: post #7

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 comprehen…

Seems like it was entirely AI written (claude) from the commits: https://github.com/muchmirul/conjectures/commits/main/

Shows how well prompted AI can significantly improve human understanding.

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

Re: Jacobian Conjecture for Baby

#15

Post by NooneAtAll3 »

dash2 wrote: post #10
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.

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

Re: Jacobian Conjecture for Baby

#16

Post by noio »

neuralkoi wrote: post #2

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.

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.

Re: Jacobian Conjecture for Baby

#19

Post by muchmirulys »

CGamesPlay wrote: post #13

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.

revised. hope it clear any confusion

Re: Jacobian Conjecture for Baby

#20

Post by muchmirulys »

CamperBob2 wrote: post #6
samlinnfer wrote: post #4

Earlier quoted context omitted.

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.

revised. thanks for the feedback
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