Index degree and the Multiseries theorem | Algebraic Calculus Two | Wild Egg Maths

The Binomial theorem extends to the Multinomial theorem, which then extends to the Multiseries theorem which deals with powers of (ongoing) polyseries (or power series). In order to formulate this in a careful way, we will adopt a polyseries point of view, and we will need the crucial notion of "index degree" of a monomial in several variables c_1,c_2, etc.

This notion of index degree plays also a vital role in our exciting new solution to the age-old problem of solving general polynomial equations -- which is now laid out in the Playlist (40 videos)

This approach to the arithmetic of polyseries, or power series, reflects our fundamental understanding that it does NOT make sense to "do an infinite number of things". We are not allowed to circumvent this essential logical truth by judicious use of phraseology: "now take the limit..." or "now go to infinity", or other such confusions.

By being honest and restricting ourselves to what we can actually, concretely do, we open the door to a lot more interesting and powerful mathematics -- as we see here in the Algebraic Calculus courses.

Video Contents:
00:00 Introduction
00:58 Index degree & the Multiseries theorem
04:30 Graded binomial theorem & graded trinomial theorem
11:46 Graded quadnomial theorem
13:51 Multiseries theorem
18:26 Coefficients of multiseries theorem
23:56 An explicit formula

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[draconyster]

Thanks! It's kind of weird but I listen to these videos frequently in the background when doing random stuff around the house, frequently without watching the actual video part. I still feel like I absorb at least half the information.

One suggestion I have would be for you to appear on the Mindscape podcast by Sean Carrol. It would be really cool to hear you both discuss some of the onthological stuff.

[accountname]

I'm not sure why, maybe it's just the multinomials hanging around, but this makes me think of walking through simplicial complexes. This is very beautiful Norm.

[deanrubine]

Hi Norman. In our adventures we learned that the sum in the theorem can be viewed in a few different ways. It's most apparently a sum over the the partitions of natural number m into n parts, inside the sum over m. The inner sum is also something like the sum over subdigon types with m faces and n-1 edges, the outer sum over faces. In one of my follow-up papers to W&R that I occasionally ask you to read, I raise the general multiseries in an unbounded number of variables to the n th power, what we write as Sum[m>=0] Cm t^m over boldface natural vectors m -- it ends up being an inner sum over vector partitions with n parts.

[txikitofandango]

I would love to see the Wildberger treatment of a standard college or high school level statistics class. As a college tutor, I see students struggle with statistics because no one ever tells them what is going on, where all the formulas are derived, what on earth a normal curve is supposed to be, etc. I'm tempted to undertake such a construction myself, because I can see the broadest outlines of it, starting from the binomial theorem, simple notions of probability, etc. Another construct that is poorly explained in these classes is the fictitious "population mean", a value which we postulate must occur within a certain interval to a particular degree of confidence. I would love any ideas on this

[jaanuskiipli]

The result is kind of directly linked to the "solving polynomial equations" lecture series, where the solution x is in form of an ongoing polyseries and if we substiyute it back into original polynomial equation then we have to raise the polyseries x to various powers up to the degree of original equation

[theoremus]

Thank you for demonstrating the Multiseries. It is a combinatorial slug fest.