#subvarities

As I continue with my exploration of the paper here, I found Michael Penn's video relatively helpful (we'll go into that later). I am now working through the fourth paragraph in this blog post. The proof states that the degree of the polynomial can be as large as necessary, so things can get very weird. It's easy to imagine x^3, for example, but it is not in any way easy to imagine x^20th. Nevertheless, the nature of the matter is such that these powers, regardless of how weird and wily they get, still have to finagle with some specific laws in math space. First of all, points that create shapes on algebraic sub-varieties, or sets/systems of solutions usually of the polynomial type (in fact, perhaps always of the polynomial type, though my understanding is not yet sufficient to understand the necessity there) must obviously have to be rational to exist. That means that any nature we derive theoretically must map actually somewhere on the intersections of all the component parts (see weird shape that resulted from such an operation above), and that even if we get it wrong, those points we observe do have a natural explanation. Yay. The creatures are not weirder than that, fortunately. Also, all fields operated on have to have an algebraic closure. More or less, this is the anti-flat earther statement of mathematics. Things have to fold in on each other, and there's a certain gravity to the possibilities supported by any given field. Finally, an irreducible subvariety means that it is within a given constraint, however, within that constraint, it cannot be further "gardened down" into seperate varieties. That's the math carrot, the math radish, or the math flower, whether you like it or not. It's not salad time over here, dude.

So far so good. We're getting there. Things I notice as I learn: I, again, don't understand why powers up to insanely complex magnitudes can be so reduced, other than reducing the whole function by a common denominator. But say you couldn't do it. It just strikes me as mind-boggling, but, it is mathematics.

Other things to note; there weren't any exceptionally strong videos on rational points. What didn't I like about the videos I found?

They don't explain the strategy behind the moves in a proof. They just show you how to do it. That doesn't relay a deeper understanding.

As usual, the why of solving the issue at hand isn't addressed either.

Both of these issues are problematic, as our mind needs relevancy and cause to learn best. Yes, we can show the mind how to do something, but if it doesn't know what for, as teachers we're going to get weak retention scores. And that's on us.