Mathematics - 2017-05-04 (page 5 of 5)

Danu

22:00

so what I have found is that result I mentioned earlier, which lists KE submanifolds with restricted FS metric

It actually lists a quadric, but which embedding of the quadric idk... Mine is the "diagonal" one $\sum z_i^2=0$...

Sylent Nyte

guys

actually hold on

Ted Shifrin

@Danu: So I can only check the case of the surface in my head (because algebraic geometry tells me I have an isometry to $\Bbb P^1\times\Bbb P^1$). The general case depends, as Mike said, on whether the Kähler metric inherited form $\Bbb P^n$ is $SO(n+1)$-invariant or something.

Daminark

holds

trilolil

@TedShifrin I do indeed believe I am not interpreting the formula correctly. How did you come up with the "1st" and "4th" column? I am a bit confused here...

Ted Shifrin

Oh, the nonsingular quadrics are all projectively equivalent, @Danu. They're all isometric.

Danu

22:01

Ted Shifrin

@trilolil: I was continuing with your example a few lines up.

Sylent Nyte

can i send pictures into this chat?

how so?

Ted Shifrin

You mean like Danu just did?

The upload button next to the send button.

Sylent Nyte

I dont have that button

Danu

Oh, really? How so? I talked to Cristiano Spotti today (he's the one giving the KE lectures here) and he was going on about having to find out which embedding it is...

If you're on phone you don't get the button.

Sylent Nyte

22:02

i am on my computer

Daminark

Oh @Eric are you gonna do complex next year?

Danu

Can't help you then

Sylent Nyte

crap

i just switched to mobile

Danu

In either case, you can copy-paste links.

Ted Shifrin

@SYlent: It takes a certain amount of rep, probably.

Sylent Nyte

22:03

help, how to revert

pls

Danu

[something].png links work

Anyways @Ted, so how so?

Hi Mike

Daminark

Hey @Mike!

Eric

@Daminark, probably, I was planning on spending next year doing some more algebraic electives + finishing off the graduate sequences I already started

Danu

I probably misunderstood what you meant by naturally homogeneous

Sylent Nyte

how to revert back to pc

Danu

22:04

Please explain what you meant :D

Balarka Sen

@TedShifrin So, just to clarify, are you asking me to prove exactly that exponential map is $e^{tX}$ for matrix groups?

Daminark

Ah, so like atop-difftop-complex

nbro

Hey Cedrics!

Ted Shifrin

I need to think more, @Danu. I guess ours is $SO(n+1)$-invariant if we think of $SO(n+1)\subset U(n+1)$, but others might not be. That's important.

trilolil

@TedShifrin I am sorry. But I think you will have to explain how you obtained the index nr4. The way I see it is that you will obtain column nr. 1 in both cases.

Eric

22:05

yeah + algebraic geo + number theory as electives (maybe exchanging one of those for Neves minimal surfaces class) @Daminark

Danu

@TedShifrin Didn't you just say they're all isometric?

nbro

Do you think this proof is correct: texpaste.com/n/ey1lkrlf?

Ted Shifrin

No, @trilolil, dammit. When $i=4$, you get the 1st column of $X-Y$. $i=4$, $n=3$, $i-n=1$.

Daminark

Oh is that a new one opening? Lol maybe Neves is gonna open up enough classes that you don't have to worry about running out

Ted Shifrin

I did, @Danu, but you told me I was wrong. So I need to think about that.

Daminark

22:06

The year afterwards he'll do Norse theory

Danu

I didn't!

Daminark

*Morse

Sylent Nyte

imgur.com/a/uCvUm have a try at this, I got the area of like 11.662 if somebody doesnt mind checking that im correct

Danu

Spotti is more of an analytic existence proof-kind of guy. I don't know if he knows about this at all.

I just assumed he knew more than me :P

Ted Shifrin

Well, that's a safe assumption from a global perspective :)

Eric

22:07

oh snap, he told me he'd teach an undergrad curves and surfaces class our fourth year @Daminark, if he's teaching a morse theory class also ill probably take it

Danu

Norse theory... And then Swedes?

Ted Shifrin

And me, too, on most stuff, probably.

Daminark

Should we just open a Neves department or something?

Ted Shifrin

I won't touch that with a nine-foot pole, @Danu. And the czech is in the mail.

Danu

@TedShifrin :)

Daminark

22:07

LOL @Ted

Sylent Nyte

AHA

Eric

@Danu a class on norse theory mentioning the swedes might be a little anachronistic maybe

Balarka Sen

lol

well I'd much prefer Horse theory

Danu

@Eric That's why I said and then

Daminark

I mean Schlag's gonna do curves and surfaces with us so idk if I'll take it.

Eric

22:09

ah yes fair fair @Danu I see now

Danu

@MikeMiller Thanks for making me look up that Besse ch. 8 is actually on this exact topic. I should probably be able to at least orient myself a bit better by just reading that.

Ted Shifrin

totally different, Demonark.

Danu

It's hard to find any overview in the literature...

Ted Shifrin

@Danu: Good idea. And after I'm done reserving dinner in Paris on-line, I'll get back to pondering.

Daminark

How so? @Ted

Eric

22:10

@Daminark I mean I won't take it, maybe i'll ask him if i can TA so I can get $$$

Danu

Reserving dinners already? Going full fancy mode?

Ted Shifrin

Morse theory vs curves and surfaces? ... or did I miss something?

Balarka Sen

@Ted Is my understanding of your question right? Just reposting in case you missed.

Daminark

Oh Neves is also teaching an undergrad class on curves and surfaces

trilolil

I agree with you that in that case for $X-Y$ you get the first column. But could you then tell me which column you will get for the second formula (X+Y)at the first iteration. The first one as well right?

first iteration (n=3)

sec. formula: i = 1
third forumula: (n+1)-n = 3+1-3 = 1

Eric

22:11

@Ted I think he's teaching two courses, one is a second/third year grad course on morse theory, one is an undergrad curves and surfaces course

Balarka Sen

There's too many people pinging Ted 24/7 :P

Daminark

Which is why I think we should just open a department for everything he's teaching because holy crap

Eric

but this is like two years from now I think

Danu

@BalarkaSen Yeah, it's gotten pretty crazy over the past 6 months or so. I feel a bit bad for contributing to a large extent.

On the other hand, I really like these semi-high level discussions we sometimes have.

nbro

ok

Balarka Sen

22:12

Nah nobody's to blame. Ted is just popular.

Ted Shifrin

@trilolil: Damn, you're still not understanding. Read carefully, man. When $i=4$, we're looking at the $4$th column of our new matrix.

nbro

popular fiction

Ted Shifrin

ohhhh ... got it. Sorry, not following everything.

But it's great that you guys love him so much. I had some students take 5 and 6 classes from me (nuts they were).

nbro

ahah

lol

Danu

I took 4 of one professor, just by accident sorta :P

Balarka Sen

22:13

@nbro are you writing one?

Eric

@Ted I do really like him as a teacher, he is a very nice guy, and also the only person teaching these geometry/geometric analysisy classes in the department and that stuff is just very exciting to me

Ted Shifrin

I get it, @Eric. You're fortunate. And it seems he assigns appropriate homework, etc., even if his test questions were a bit off the deep end (not for Demonark's class).

Eric

yeah, the flipside is that he takes the philosophy that if you're an undergrad in a grad course putting forth a decent effort and following along, you shouldn't have to worry about your grade, so the exam being hard isn't a super big deal

Balarka Sen

Ted double missed my question :( I guess I'm just going to prove what I think I am asked then

Ted Shifrin

LOL

@Balarka: NO!

trilolil

22:17

@TedShifrin I may be a total retard on that one, but would you mind calculating one iteration and writing it? I really believe that both formulas always get the same index:

n=3

first iteration
$X+Y$: i=1
$X-Y$: (3+1)-3 = 1

second iteration
$X+Y$: i=2
$X-Y$: (3+2)-3 = 2

third iteration
$X+Y$: i=3
$X-Y$: (3+3)-3 = 3

Ted Shifrin

I'm asking you to tell me where $g$ ends up at time $t$ when you follow the left-invariant vector field $X$.

Waiting

Time to go.

Ted Shifrin

@trilolil: My reading is that he is defining a new matrix. (Let's stick with $n=3$.) Its first column will be the first column of $X+Y$. Its fourth column will be the first column of $X-Y$. What is so difficult?

Waiting

Also robjohn is missing lately.

Ted Shifrin

Why are you talking about iterating? Isn't he defining a new matrix and these are the column vectors?

If it's about iterating, then I totally am not reading it right. Granted, I assumed I knew what he meant and didn't read that carefully.

nbro

22:19

I don't know Ted but every time I come here it seems that he's the savior and shepherd of all these sheep, metaphorically, clearly.

Ted Shifrin

Sometimes, @nbro, I am very stooopid. And I also make mistakes because too many people are yelling at me :)

Daminark

is relieved that nbro didn't actually figure out that I'm literally a sheep

Ted Shifrin

Demonark: in an old wolf's clothing.

nbro

:D

James Bond

@TedShifrin Sometimes, you sound quite obnoxious. =)

Ted Shifrin

22:21

I am totally frustrated with one person right now, @James, so obnoxious I am.

trilolil

@TedShifrin yes he totally is defining a new matrix. I am thinking in a softwarish way... I see what you did. It's just that he is substracting n every time. You don't seem to take that into consideration, I think.

Benjamin

Are there any logics in which the AND or NOR operations are non-commutative?

James Bond

@TedShifrin You have also just become Yoda.

Ted Shifrin

@trilolil: There is no "every time" ... He's talking about columns 1,2,...,2n of his new matrix. For the first n he does $X+Y$, for the last n he does $X-Y$.

Balarka Sen

@TedShifrin Ah. For some reason I thought it'd be clear if I knew the exponential map.

Daminark

22:22

@Ted shrugs

James Bond

@Benjamin It appears to my naive ears that this question needs a specialist is logic to answer, better to post on the main site.

Ted Shifrin

@Balarka: There's a reason I'm asking you this. It will make you understand the importance of bi-invariance.

(Take that as a hint if you like.)

Benjamin

@JamesBond Will do.

@JamesBond Anyone I might want to tag who could do a good job?

trilolil

@TedShifrin I think I now understand my mistake! the 'i' he is referring to when he writes "for i=1...n" has actually nothing to do with the i in the subscript in $[....]_i$ it refers to $\chi_i$.

Did I understand that correctly?

Daminark

@Balarka defines $e^z$

James Bond

22:25

@Benjamin Naming anyone won't get you to an answer quickly, I think. Also, I don't know anyone.

Ted Shifrin

I think it's in both places, @trilolil, but I need to find your thing again.

trilolil

there you go :)

ok no!

I got it!

he substracts n to get the first, second etc.. column of $\Sigma$ while being able to get the 4th column of $\chi$

James Bond

Every time I take the train, there are bidirectional gates, lol.

Ted Shifrin

Right. As I said, he's first defining the first n columns of $\Chi$, then defining the last n columns of $\Chi$.

trilolil

@TedShifrin are you sure he is defining columns and not rows in some way? As you can see the matrix called "sigmas" has the following size: 2n x n-1

Balarka Sen

22:29

Say $g$ goes after time $t$ to $h$ by the flow. Then $1$ goes at time $t$ to $g^{-1}h$, by left-invariance, doesn't it? Does that not mean $\exp(tX) = g^{-1}h$?

Or am I completely misunderstanding

Ted Shifrin

@trilolil: Part of what he's writing is garbage. He's pretending that he's given you a definition of square root. He has not. That's why I addressed that earlier. But how do you get $\Sigma$ is $2n\times (n-1)$?

Huh @Balarka? I don't think you understand my question.

trilolil

Ted Shifrin

I know what $e^{tX}$ means, Balarka. In terms of it, where is $g$ after time $t$?

trilolil

@TedShifrin as you can see this matrix has 2n rows and n-1 columns. Where "sigmas" is the matrix we are trying to construct

Ted Shifrin

And I want $X$ to mean $X_e\in\mathfrak g$ in the exponential. It's not the vector field everywhere.

@trilolil: It was $n$ columns, not $n-1$. Look carefully.

trilolil

22:33

@TedShifrin right, starts from zero... But you do agree that he is constructing this matrix by calculating the rows and not the columns, right?

Balarka Sen

@Ted Well, you want to describe $h$ (where $g$ lands after $t$) right? If what I said is right, $\exp(tX_0) = g^{-1}h$. So $g\exp(tX_0) = h$ - understanding the exponential map describes $h$.

Ted Shifrin

Ah, ok, @Balarka. Fine. Thanks. Note something interesting. Which side is the exp on?

Danu

this again :P

Balarka Sen

You wanted me to say just that? lol I was confuzzled for 15 minutes because it seemed like a triviality

Ted Shifrin

No, @trilolil, it's columns, and he says the subscript means column.

Balarka Sen

22:35

@TedShifrin Right.

Ted Shifrin

Flow of left-invariant vector field is right action. That's why you need bi-invariance.

Danu

...for what?

Balarka Sen

Ah, that's an interesting perspective.

Danu

invariant objects?

Ted Shifrin

Sorry I've been too busy, @Balarka. And I still need to think about whether all conics in $\Bbb CP^2$ are isometric. Of course they're not. They're projectively equivalent, but not unitarily equivalent.

That was my main point in the question, @Balarka.

@Danu, I think I'm being dopy. Projective equivalence will certainly not give isometries in general.

Balarka Sen

22:37

No worries, I can see you're getting cornered by the whole chat :)

Danu

Right, Spotti said something similar (if they're related by a U(1) they are isometric else not)

Alessandro Codenotti

Hey @Ted, do you have 10 minutes for a short question? (Just joking, I don't actually have one)

Ted Shifrin

But your quadric is about the nicest it can be. Doesn't the $SO(2n)$ act compatibly with $U(2n)$?

smacks @Alessandro ... I need a few martinis, but it's early for that.

Balarka Sen

@Alessandro That made my heart jump up a bit.

Alessandro Codenotti

@TedShifrin it's the right time somewhere

Ted Shifrin

22:39

Good point!

Balarka Sen

(Until I read the stuff in the parenthesis that is.)

Ted Shifrin

@Balarka, it's un-sleep time ...

Balarka Sen

I wonder if I will un-sleep though.

Daminark

@Ted So let's say you have $f:X\to \mathbb{R}^m$ whose intersection number with $Z$ is 0...

Lol jk

Ted Shifrin

Demonark: In $\Bbb R^n$ all intersection numbers are 0.

Danu

22:41

#rekt

Balarka Sen

lol

James Bond

@Daminark Seems lol jk is your pet phrase.

Daminark

Lol yeah I know

Ted Shifrin

flunks Demonark on his final exam

Daminark

B... But...

James Bond

22:42

Mine is simply LOL, LOL.

Daminark

@JamesBond It's gonna be a fight between that and "kek"

Balarka Sen

@Daminark What are you learning in manifolds?

Let's see if I can give you some exercises to stop you from joking.

Daminark

Also referring to things as dank, it's a habit I got even though I never realized for quite some time that it had anything to do with smoking, I just thought it was memetic. That also happened for the word "pregaming", I used it for months before someone was like "I don't think you know what that means..."

Today we did intersection number, though I do have some exercises to keep me busy, I'm just procrastinating really

Balarka Sen

always know your words

James Bond

@BalarkaSen Just do 100 situps and see if he can still laugh.

Balarka Sen

22:46

Why would me doing 100 situps have anything to do with him laughing?

I am confused.

James Bond

So am I.

Daminark

In defense of pregaming someone mentioned that to me in a non-alcoholic context. There was this lab that took my partner and me a while so we decided we'd meet up for lunch before and "pregame the lab", as my partner put it

Balarka Sen

What do we do now that we're all confused?

James Bond

Partner always sounds like romantic partner to me.

Danu

anyways, I'm out

Ted Shifrin

22:47

Not yet, @Danu

Daminark

@JamesBond Clearly you don't know me well enough :P

Ted Shifrin

I was about to make a comment.

Then you can out.

Danu

Okay.

I found the original reference (Koszul in J Can. Math. 1955) proving that these spaces are supposed to have KE metric. Perhaps it will tell me something...

Ted Shifrin

So any two nonsingular quadrics are projectively equivalent, say by $f$. But $f^*\omega = \omega$ only if $f$ is (the restriction of) an isometry of $\Bbb P^N$. The fact that the hyperplane classes correspond isn't strong enough.

Oh, cool.

Bye now.

Daminark

But yeah like, I inferred that pregaming was a generic term for doing things early. So I'd talk about how I was not busy and was gonna try to pregame the homework for next week, etc.

Danu

22:50

@TedShifrin What does that last sentence mean?

Ted Shifrin

The cohomology class of $\omega$ is Poincaré dual to a hyperplane section. The projective equivalence pulls hyperplane class back to hyperplane class. But that's not giving me isometry.

James Bond

That's giving me a headache. =)

Danu

Oh, okay.

Balarka Sen

Mosquitoes are committing cannibalization on me.

Ted Shifrin

Now who's being obnoxious, @James?

James Bond

22:52

=)

Danu

I don't really know the definition of "projective equivalence"

Ted Shifrin

$PGL(N+1)$

Daminark

Wait... @Balarka you're not a mosquito...

Semiclassical

hi chat

Daminark

Hey @Semi!

James Bond

22:55

Hi @Semiclassical LOL. I am still alive.

Danu

@TedShifrin OK. I still don't know why they're all equivalent in that sense, though.

Semiclassical

@BalarkaSen they're eating each other on top of each other? how bizarre.

Danu

Probably look at the defining equation and induce an action of PGL?

Ted Shifrin

Because linear algebra.

Balarka Sen

@Daminark Sure I am.

Ted Shifrin

22:56

Any two nondegenerate quadratic forms are equivalent /$\Bbb C$.

Danu

oh, I know that result :D

Daminark

gasps

Ted Shifrin

So you get a linear change of coordinates ...

Daminark

Also kek @Semi

James Bond

@Daminark Some people can't pronounce that word.

Daminark

22:58

What mispronunciation do you typically hear?

Ted Shifrin

One of the kids I was tutoring yesterday was having trouble pronouncing the word RURAL. It's a bit tough.

Daminark

Yeah that's harder. Did "Rual" happen?

Forever Mozart

chinese?

Danu

lular

Okay, now I'm out.

Forever Mozart

lol exactly

Balarka Sen

22:59

lulz

Daminark

Lel

James Bond

@Daminark Maybe GUPS instead.

Daminark

(Ah the variants of lol)

Ted Shifrin

Bye all. Go un-sleep, @Balarka and Danu.

Daminark

@Ted One last question tho...

Jk see you!

Balarka Sen

23:00

See ya

James Bond

C U has so many ways of being written, LOL.

Akiva Weinberger

(Una) silla

Daminark

That was clever @Akiva

Akiva Weinberger

cya

Balarka Sen

@Akiva Bye.

"Goonight Bill. Goonight Lou. Goonight May. Goonight.
Ta ta. Goonight. Goonight.
Good night, ladies, good night, sweet ladies, good night, good night. "

Akiva Weinberger

23:05

I'm reading a bit on algebraic topology. They proved that every algebraic set has a decomposition into irreducible sets, and that it's unique.

Their proof uses the axiom of choice. Can it not be done without it?

Balarka Sen

What's an algebraic set

Semiclassical

@BalarkaSen I approve.

Akiva Weinberger

@BalarkaSen Closed set in Zariski topology

Balarka Sen

@Semiclassical :)

Akiva Weinberger

Zero set of a polynomial, or the intersection of such

Semiclassical

23:06

I forget, is that the very end up "A Game of Chess"?

Or are there a few lines after.

Balarka Sen

Yup.

Semiclassical

okay. Couldn't remember if it went directly into "The Fire Sermon."

Balarka Sen

Nope, none.

Semiclassical

mmkay.

I found some of the lines from the end of "The Fire Sermon" bouncing around in my head the other day.

Akiva Weinberger

@BalarkaSen Do they eat palak paneer in Kolkata?

James Bond

23:08

I finally got Gelfand and Fomin's Calculus of Variations. Is that a good book, anyone?

Semiclassical

Highbury bore me,
Richmond and Kew undid me.
By Richmond I raised my knees.
My heart is at Moorgate, and my heart
Under my feet.

(Now, let's see how well I remembered that...)

Akiva Weinberger

Just out of curiosity. That's what my dinner looks like.

Balarka Sen

@Semiclassical A few lines from Waste Land keeps coming back to me once in a while.

Semiclassical

Yeah.

Balarka Sen

Under the brown fog of a winter dawn,
A crowd flowed over London Bridge, so many,
I had not thought death had undone so many.
Sighs, short and infrequent, were exhaled,
And each man fixed his eyes before his feet.
Flowed up the hill and down King William Street,
To where Saint Mary Woolnoth kept the hours
With a dead sound on the final stroke of nine.
There I saw one I knew, and stopped him, crying: “Stetson!
“You who were with me in the ships at Mylae!
“That corpse you planted last year in your garden,

This is my favorite passage.

Semiclassical

23:09

Yeah, that's good.

James Bond

When did Balarka and Semi become so poetic?

Akiva Weinberger

@BalarkaSen What's Waste Land?

Balarka Sen

@AkivaWeinberger Yeah, they do

Semiclassical

I've been an Eliot buff for ages.

Balarka Sen

It's a poem by T. S. Eliot.

James Bond

23:10

I only know about Elias Stein, LOL.

Semiclassical

And it's The Waste Land, to be precise.

Balarka Sen

Quite an epic poem, at that.

(Right.)

Akiva Weinberger

"A Game of Chess" is the same author?

Semiclassical

It's part of the same poem.

It's in five parts.

Akiva Weinberger

Oh.

I know Chess: The Musical, but I suppose it's very different.

Balarka Sen

23:11

It's literally a collage made of five parts. That's one bit of the collage.

Akiva Weinberger

starts singing Chess to self

James Bond

I have taken a look at Barry Simon's five-volume Comprehensive Course in Analysis.

Semiclassical

"The Burial of the Dead", "A Game of Chess", "The Fire Sermon", "Death by Water", and...oh, damn, I'm actually forgetting.

Sha Vuklia

@Waiting haha, I'll be doing physics the next coming days, so you won't see me around in this chat a lot, I'm afraid!

Akiva Weinberger

"Death Under Troubled Waters" could be a good, if unsettling, song

Balarka Sen

23:12

What the Thunder Said

Semiclassical

Bah, yes.

Akiva Weinberger

> "Boom" - Thunder

Semiclassical

nope

da

Akiva Weinberger

What?

Balarka Sen

It says Da, thrice.

Akiva Weinberger

23:14

Oh

Semiclassical

poetryfoundation.org/poems-and-poets/poems/detail/47311

Akiva Weinberger

I don't think it sounds very much like a "da"

Semiclassical

Look for "Then spoke the thunder"

Balarka Sen

Datta. Dayadhvam. Damyata.

Semiclassical

That's an allusion to Indian mythology, isn't it?

I mean, the words obviously are.

Balarka Sen

23:14

Yes.

Semiclassical

(Eliot was surprisingly well-versed in such things.)

Akiva Weinberger

beeps in Morse code Ditditditdah dah dah ditditdit

Daminark

@@JamesBond Just do p-adic analysis and forget normal

Semiclassical

@BalarkaSen I also had some of the imagery from "What The Thunder Said" in my head.

Balarka Sen

Yes, it's very vivid.

I can really connect to this poem. I haven't seen anything like this.

Semiclassical

23:20

In this decayed hole among the mountains
In the faint moonlight, the grass is singing
Over the tumbled graves, about the chapel
There is the empty chapel, only the wind’s home.
It has no windows, and the door swings,
Dry bones can harm no one.

(back later)

Akiva Weinberger

How would I show that $Y^2+X^2(X-1)^2$ is irreducible in $\Bbb R[X,Y]$?

I have no idea how I'd approach that.

Hm. In $\Bbb C[X,Y]$, it's $(Y+X(X-1)i)(Y-X(X-1)i)$.

Balarka Sen

@AkivaWeinberger Also, I don't think this can be done without choice. Ideal theoretically, this is the same as saying any ideal of k[X1, ..., Xn] can be decomposed uniquely into prime ideals.

Semiclassical

23:43

back

PVAL-inactive

@Akiva math.stackexchange.com/questions/150917/…

Simplex

Just got out of a calc exam. One of the questions asked if x|x| is differentiable at x=0. Wolfram alpha says it's undefined but I don't see how. Any hints?

Akiva Weinberger

It should be differentiable there.

PVAL-inactive

plug it in to the definition of derivative.

Akiva Weinberger

I don't know what Wolfram Alpha is doing, but it seems to be incorrect here.

@PVAL-inactive So?

Oh!

Simplex

23:50

For some reason it interpreted it as the partial derivative. Not sure if that makes a difference since I'm only in calc 1

Akiva Weinberger

OK, so $Y\pm X(X-1)i$ is irreducible in $\Bbb C$.

@Simplex I don't think it does. Partial derivative is when you have a function with more than one input (like $x\sin(y)$, which takes in a value for $x$ and for $y$)

Which means that that's the only factorization in $\Bbb C$, which means it's irreducible in $\Bbb R$. Yeah? @PVAL

Simplex

@AkivaWeinberger thanks. I thought it should be differentiable

Akiva Weinberger

@Simplex I think so too.

PVAL-inactive

@Akiva The way you stated it isn't quite right.

Akiva Weinberger

By $\Bbb C$ I mean $\Bbb C[X,Y]$. Is there something else I'm missing? @PVAL

Semiclassical

23:53

Doing D[x Abs[x], x] /. x -> 0 in Mathematica gives 0 as the output.

PVAL-inactive

Well if Y^2+X^2(X-1)^2 were reducible in $\Bbb R[X,Y]$

Semiclassical

So I'm a bit surprised if WolframAlpha can't do it.

Akiva Weinberger

@Semi http://www.wolframalpha.com/input/?i=derivative+of+x%7Cx%7C+at+0

Semiclassical

note what it does if you don't ask for the evaluation at zero: wolframalpha.com/input/?i=derivative+of+x%7Cx%7C

Akiva Weinberger

Strange. If you click on the "plaintext" in the input interpretation, it gives you ReplaceAll[D[x Abs[x], x], {x -> 0}]

and that works.

PVAL-inactive

23:54

@Akiva its not hard to show the two factors you have are irreducible over $\Bbb C$

Semiclassical

since it takes the output of that to be 2x^2/|x|, one has $x\to 0$ as a (removable) singularity.

Akiva Weinberger

@PVAL Right. Quadratic formula in $X$.

PVAL-inactive

If your polynomial was reducible over $\Bbb R$ it would factor into real polynomials which have factors the irreducible factors over $\Bbb C$

Akiva Weinberger

But there's only two of them.

PVAL-inactive

but this is impossible unless your real polynomials are unit multiples of those complex factors

because as you said there are only two of them.

Akiva Weinberger

23:57

And no multiple of those is real.

PVAL-inactive

So now all there is to check is that the two complex polynomials are not unit multiples of real polynomials which is easy.

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$Mathematics$ Mathematics