>"Anon, you can't just do that."
>You slowly pace across the room and then walk back to the board
>You tense your arms behind your back, partly in frustration
>Partly because it's how you compose yourself when in deep thought
>Explaining these things on Earth was much easier
>Ah, to go back to simpler times would be most welcomed
>But you know that's never going to happen
"Of course I can Twilight. If you had read my paper you'd know I can. We've been doing it for hundreds of years on Earth and it's been of no consequence because it's entirely logical."
>Her face contorts in confusion
>Getting ponies up to speed on your field has been a nightmare, almost entirely due to unfortunate nomenclature
>'imaginary' number, whoever thought that name up was a genius
>You pull an arm from behind your back, chalk in hand
"The object at hand is this polynomial,"
 
P(z)=z^5+z^4+z^3+z^2+z+1
 
>You quickly underscore it with the chalk in an attempt to drive the obvious home
 
and the associated problem."
 
>You point at the text below it
 
Find the sum of reciprocal roots of P and find the sum of reciprocal squares of roots of P.
 
>And your arm retreats back to its hideout behind your back
>You step away from the board and return to meticulously pacing a trench into the floor
>"Oh... Well okay Anon, I'll try it again."
>Her quill levitates up and she begins, once more, furiously working
"Excellent."
 
>Honestly you'd rather be outside
>Gazing out the window Ponyville is in its usual state, everyone blissfully happy, going about their business
>But then again, this is your business
>So perhaps you should be stuck inside
>You gaze down at your watch
>10 minutes have passed
>"I think I got it!"
>The purple horse appears confident
>"Or got the first half of it, at least..."
>Her voice trails off
>Mostly confident
"I'm assuming you did the first part. And the answer is―"
>"Negative One!"
"Yes. Now let me see your work."
>She levitates the paper over to you
>No... this is all...
"This is incorrect."
 
>"Are you sure Anon? Because the only root of P is z equals -1."
>She snatches the paper back with her magic
"Yes, I'm absolutely sure that -1 isn't the only root of P, the other roots are complex. And having the right answer isn't all of the problem, you have to get it the right way. Mathematics is a process, it's a thing you do. If you don't do it right then you aren't doing it at all."
>"But wouldn't those involve /imaginary/ numbers?"
>Back to the start
>Fuck
"Since it's clear you didn't read the paper I submitted to the Equestrian Mathematical Society, I'll restate the core result."
>You pull down the second blackboard and erase the...
>...light cone
>How does Cheerilee know this shit if ponies can't even work with polynomials?
>Never mind that right now
>Trusty chalk in hand, you write the crowning theorem of algebra
"The field of complex numbers is algebraically closed."
>She's unmoved by this statement
>Maybe a different approach is needed
"Given any polynomial Q with coefficients in C, we have that Q splits completely over C."
>She winces
>Come on, Celestia sent you here saying Twilight liked this stuff
>An apparent untruth given that Twilight hasn't engrossed herself in any of the material you've reproduced for her
>"Maybe we don't have to do any of this Anon."
 
"Yes, we won't do any of it. So when I die all the information about and from Earth will disappear with me, with nobody to learn or understand what I have to offer you, and then everyone can forget about how Twilight Sparkle didn't want to listen to Anonymous and didn't like reading."
>The room falls silent and you recompose yourself
>Normally you'd hold back but it's been like this for weeks
>Twilight is taken aback but eventually comes to
"In equivalent terms, any polynomial of degree n and with coefficients in C has n roots up to multiplicity. So that means..."
>Twilight puts a hoof to her chin
>"Well, P is of degree 5... So I guess that means P has 5 roots."
"Up to?"
>"Multiplicity."
"Good."
 
>"See, Anon. That's what I don't get. -1 is the only root of P, so what you're saying must be wrong."
>Gahhhhh, why
"If you truly think -1 is the only root of P then check to see if -1 is also a root of all the higher order derivatives of P, until the derivatives hit 0."
>She goes back to scribbling
>The sun is setting now
>Another afternoon, essentially wasted
>She looks up from her work
>"It's not a root of any of them."
"Which means?"
>I... Er, it's not a root of degree five."
>Which means?"
>Um... My answer is wrong?"
"Say it with confidence."
>"My answer is wrong."
"Only when one can accept that their method isn't working can one move on and try another approach. You've made progress, simply by acknowledging failure."
>She frowns
"This isn't a bad thing, you should be happy that we can move on. I'm going to show you the approach I'm hinting at, then I want you to do the second part on your own, tonight."
 
>A smile appears on her face, something you haven't seen in a long time, and her quill levitates into the air
"As I said before, any polynomial with coefficients in C splits completely over C, which means that we can write P as a product of five linear factors,"
>You carefully place it on the board next to the fundamental theorem of algebra
 
P(z)=(z-a)(z-b)(z-c)(z-d)(z-e)
 
"where a, b, c, d, and e are the roots of P, not necessarily distinct. But in case they are disctinct, but I'll leave that for another time."
>You return to trench making
>When the pony equivalent of World War One comes you'll be more than ready
"So what do you think the next step should be?"
>You can see the gears turning away in her head
>At some times she hesitates
>At others she raises a hoof as if to speak, only to shoot her own ideas down and go back to cranking away those gears
>"Well we should find the expressions for the roots, I assume. Find a, b, c, d, e I mean."
"A fair thought, but such a course is specious. It was proven using Galois theory the roots of an arbitrary degree five polynomial need not be all expressible in terms of radicals."
>"Huh?"
"It's fine if you didn't catch that, I've never mentioned it until now. The true magic here is that we don't need to know what any of the roots are to find the answer, we just need to know that none of them are z=0. Since the coefficient of z^0 is 1, we know none of the roots are 0."
>"I don't like where this is going Anon."
>Her wings rustle
>Damn that's cute
"Why not? What we're about to do is wonderful. We're going to sum over something and get an answer with a closed form, even though we don't know the analytic expressions for all the terms we're summing over."
>"I'll take your word for it."
"You don't have to take my word for it when you can see for yourself. Let us continue, we want a sum, what operation will give us a sum?"
>"Logarithms."
>She truly confident this time
"Yes. Taking the logs of both sides yields,"
 
ln(P(z))=ln(z-a)+ln(z-b)+ln(z-c)+ln(z-d)+ln(z-e)
 
this is about half way there. Now we want the roots in the denominator, as we're summing over reciprocal roots."
 
>"Then we can differentiate with respect to... z"
>"Good, good! Now we have,"
 
P'(z)/P(z)=1/(z-a)+1/(z-b)+1/(z-c)+1/(z-d)+1/(z-e)
 
>It's dark out now, but at least progress is being made!
"so if we allow z to ―"
>"Be 0, that's it!"
>Twilight rushes up to the board
>Forcing your hand open with her magic ― whoa, that feels weird ― she takes the chalk from you
"You could just ask for it you know."
>She puts the final steps into place
 
P'(z)=5z^4+4z^3+3z^2+2z+1
 
>"Then,"
 
-P'(0)/P(0)=-1
 
>"leaving us with,"
 
-1=1/a+1/b+1/c+1/d+1/e
 
"Which tells us that, if we take a=-1, 1/b+1/c+1/d+1/e=0. And that's the real kernel the problem leads you to."
 
>She runs forward and gives you a hug
>"Thanks Anon!"
>All your problems melt away
>You allow yourself to take the moment in
"Hey Twi, sorry about earlier. I just don't get out as much as I used to... I guess."
>She looks up at you and ends her embrace
>"It's okay Anon, I haven't been the best student."
>A few minutes pass
>"How about I find 1/a^2+1/b^2+1/c^2+1/d^2+1/e^2 on my own tomorrow and you can go hang out with Applejack?"
"Actually tomorrow I should really work on ―"
>"They'll be serving their cider for the first time this year tomorrow!"
"Alright, I'll take a day off!"