Reader Beware.

I planned to do a post about tensor products (what they are, why we should care, what we do with them, etc.) but because I’m not comfortable with all of that quite yet, I’m going to assume you know what tensor products are, and do a few explicit calculations.  So, in short, if you don’t already know what tensor products are, don’t read this post.

Our notation will be as follows: k is a field, R is a commutative ring with 1\neq 0, and \otimes_{R} will denote the tensor product of modules over a ring R.  As usual, R[x] will denote the polynomials in x with coefficients in R.

(Note:  My thanks to Brooke, who pointed out that I kept writing "+" when I meant "\otimes."  I hope I’ve not made this error elsewhere, as tensors are "pretty different" from standard addition.)

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(I’ve decided against giving a proof of Rouché’s theorem until such a time as I find one that doesn’t use algebraic topology or isn’t tedious as hell.)

 

Let’s simply state Rouché’s theorem, and then we’ll talk about how to actually apply Rouché’s theorem.

 

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Sometimes, we can read a whole bunch of math and not get it until a simple picture is drawn.  For example, the notion of upper semi-continuous was not clear to me (at all) until a picture was drawn.  The "idea" of the mean value theorem is similar — drawing the tangent and parallel secant really shows what the theorem is trying to say. 

This post is to say in words, and then show in pictures, why we should have the fundamental group about a basepoint x_{0} of {\mathbb R}^{n} be trivial.  In other words, we’d like to prove \pi_{1}({\mathbb R}^{n}, x_{0}) = 0

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