I.3.19a

5/24/2021

Well folks, what can I say. I'm a little stumped, and I've about had it for now with this section. This section has 21 exercises in total, making it the longest section in this chapter (The next section for instance has less than half as many). I've done all all of them but 3.14, 3.15bd, 3.17, 3.18, 3.20, 3.21. These 5.5 exercises can be done another day. It's not like I haven't gone back to previous sections in the past. If I have to struggle with integrally closed domains in the future, we'll cross that bridge when we get to it. So this is le season finale.

Sometimes, lazy evaluation is useful... Sometimes. In my past of flaily self-learning math attempts, I had an interesting revelation: Hey, instead of going through everything chronologically, why don't I start from the chapter I want to be in and figure things out from there. So I started from here, and proceeded to fill in knowledge as needed. I learned absolutely nothing and wasted a week. So it goes.

On the other hand, the reason I know enough topology and algebra to barely scrape through these exercises isn't via a thorough reading of them. It was a very irresponsible, section-skippy, exercise-exclusionary, lazy reading, sometimes with nightcore playing the background (THIS IS MY MELODY AND IT'S JUST A RAVERS FANTASY ♪ CAUSE I KNOW IF YOU'RE IN LOVE WITH ME TONIIIIIIIIGHT ♪ WE'RE RAVING THROUGH THE NIIIIIIGHT ♪ WHAT THE FUCKING FUCK ♪ IS A "SEPARABLE EXTENSION" ♪ SOMEONE FUCKING HELP ME FUCKING MOTHERFUCKING MOTHERFUCKER ♪)

Breathe in. *Inhales* Find your balance........ Yes...... Breathe out. *Exhales* Think of the ocean..... all from the little wavelets *Inhale* to all the big wide waves *Exhale*..... Both are important..... Both serve their own purpose in this world...... Find your middle ground.... Do you see it? Yes....... Touch it..... Feel it...... Stay calm.... *Inhale* Don't panic... *Exhale*.... Now... Let's check the performance meter:

NOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO

The point of this blog was to build a habit of doing math. 3 sections in, I'd like to reveal to you, reader, that I have not built a "habit" at all. There is still no set time for when I do math. I do it at sporadic, random times, on whim rather than discipline. I generally put it off as long as I can before thinking "God fucking damn it, the blog needs to be updated" and I then stumble my way through an exercise. The accountability is working, in that the job gets done (slowly). But what I've learned over the X months since starting this blog is that "discipline" is completely lost on me. It has been X months and I have not developed a shred of discipline. My brain is officially too broken for consistency. I can go on brainpickings.org and go "ooooo" at the schedules of Benjamin Franklin or Mozart or whatever the fuck, but I now know that that's all useless information for me. I can no longer adapt my brain to my circumstances, I have to adapt my circumstances to my brain. One is usually morally obliged to avoid the latter, because that involves shaking things up around oneself rather than just within oneself, but I'm amoral so its okay. Basically, I'd like to stop being a pussy and pull the trigger on moving locations/jobs. "Nice blog, holeinmyheart, where can I subscribe"... Subscribe to this *unzips* (<— not funny).

BTW: this exercise is easy. I mean it's weird that we suddenly brought in calculus here, but it's fine. I decided to look up the Jacobian matrix Wikipedia page for a bit of refresher, and I noticed that the unsolved problem of part (b) was linked there, as its own page: Jacobian Conjecture. Very interesting. So I scrolled down and saw this:

It follows from the multivariable chain rule that if F has a polynomial inverse function G : kN kN, then JF has a polynomial reciprocal, so is a nonzero constant. The Jacobian conjecture is the following partial converse:

........which is basically the solution to this exercise.

Let me elaborate. Let's look at the Jacobian matrix formulation of the multivariable chain rule

Let I denote the identity matrix and id be the identity morphism. Then

 I = Jid = Jϕ-1∘ϕ = (Jϕ-1 ∘ ϕ) ⋅ Jϕ

So, taking det of both sides,

 1 = det(Jϕ-1 ∘ ϕ) ⋅ det(Jϕ)

Now note that ϕ-1 has polynomial coordinates (because it a morphism An An), so Jϕ-1 has polynomials in all its components. Composed with ϕ (itself having polynomial coordinates), Jϕ-1 ϕ has polynomial components, thus making the determinant of Jϕ-1 ϕ itself polynomial. It can't be 0, otherwise the above equality would give 1 = 0. But it does have to be constant, because only constant polynomial functions have multiplicative inverses. Thus, det(Jϕ) is also a nonzero constant. DONE.

Seeya in the next section!