Kaj Hansen

@BalarkaSen, this probably has a generalization to rational functions instead of just polynomials

haha, I know. It makes me crack up every time

the $\aleph_0$ point of view

hahahaha "n-point of view"

Let's query nlab

It always amazes me just how far generalizations can be taken

It can probably be generalized further :P

ahhh, there we go

well, \sum_i x_i^k

p_k here is $\sum_k x_1^k

huh, mathjax rendering doesn't work for me anymore

$p_k(x_1,\ldots,x_n) = (-1)^{k-1}ke_k(x_1,\ldots,x_n)+\sum_{i=1}^{k-1}(-1)^{k-1+i} e_{k - i} (x_1, \ldots, x_n) p_i(x_1, \ldots, x_n)$

nlab <3

@BalarkaSen en.wikipedia.org/wiki/Newton%27s_identities

most people don't know of it

Newton made incredible strides in the study of algebra

He actually did

because he did an enormous number

I'm trying to see if Newton ever calculated the polynomial expression for x^6 + y^6 + z^6 in terms of elementary symmmetrics

Hey there Alessandro

Hey Balarka. Have you started in on IUTeich yet? ;)

I've filled my atomic bomb quota for this month

"and pals" haha

Very cool. I know of Guilleman & Pollack

Huh. Hopefully I can find a lecture series. I like watching those in conjunction w/ a textbook

I really should. I've been curious on looking into homology too. That the jordan curve theorem needs so much machinery intrigues me; I'd love to be able to understand its proof

I'm not sure that question even makes sense how I've asked it

Homotopy equivalence feels like a stronger result. Are there spaces that are homeomorphic but not homotopy equivalent @BalarkaSen ?

I've never actually studied much at all w.r.t. them though

And I always catch myself and have this reminder of "ahhh, so that's what homotopies are good for"

Where I will think of homeomorphisms and linear transformations and such as not being instantaneous. I.e. not just immediately expanding x4 in the case of the matrix $4I$, but instead doing it in a smooth motion.

I have this really bad habit

It's kinda funny

In the one, you lose a vertex, in the other, a face. But same euler characteristic in the end

Like, poking out a single point and cutting out a closed disk are equivalent

Kind of interesting looking at homeomorphism classes in this way

Sorry bout that

Derp, forgot the middle vertex @BalarkaSen

Everything is the same, but minus one face

Remove an interior triangle, and you have an annulus

Imagine a triangulation of a disk with tons of triangles @AlessandroCodenotti

My spatial intelligence is really bad though, lol

I should really look at this sort of stuff again. Gauss-Bonnet and all that was really fascinating.

too much effort

haha! You're right :P

A circular disk I meant

I'd have to triangulate a circle to figure it out though; don't have those memorized off the top of my head

Whatever that of a circle is, minus 1

In particular, if we're cutting out holes, each hole removed reduces the characteristic by $1$ since that's one less face in the triangulation (but same # of edges and vertices)