Complex Analysis

[fleet finch]

Can anyone help me understand how they conclude with the implicit map theorem? I don't understand why the g has to be equal to f...

[lost roost]

1. Wait patiently for a helper to come along.
2. Once someone helps you, say thank you and close the thread with:

+close

3. Feel free to nominate the person for helper of the week in #helper-nominations
4. Do not ping the mods, unless someone is breaking the rules.
5. If you're happy with the help you got here, and the server overall, you can contribute financially as well:

[wise sinew]

Is this from a SCV book could you give a reference?

Ok found it trying to read it

[wise sinew]

@fleet finch what does the notation for unbolded D mean here? The notation is this book is very foreign to me. Regardless of choice of y i don't see how implicit function theorem is being used either. The book seems to define it for "functions regular of order 1" and that is when the partials are equal to 0. But I've never heard of the theorem stated this way

Ok nvm ignore that above msg

[fleet finch]

@wise sinew It is a germ of the Discriminant, then unbolded because he fixed the Delta as a domain and is choosing a "represtative" of the germ

[fleet finch]

it is Gunning Rossi

[wise sinew]

Ohhhhhh okay

[wise sinew]

Yea i figured that lol not many SCV books in general just checked a few

Okay i think i just spent an hour understanding the first half and now I'm stuck at the same place you are not sure why g(y)=f(y)

Is there some corollary you've missed from before?

[fleet finch]

could it just be

because P(f(y)) is 0 and then when he uses implicti function theorem

he wants the level curve set of that

something like that?

no, i don't think it

[wise sinew]

Is it some sort of analogue to analytic continuation in SCV or am I being dumb? The way that line is worded reminds me of that

[fleet finch]

i suspect it is just cluncky notation and yes some analytic continuation, he is kinda of saying that they coincide in one point and solve the same equation

something like that

[wise sinew]

I'm sure here's some corrolary before hand if he just stated that f=g so casually

[wise sinew]

I'm thinking its probably something like that

Cuz P(f) restricted to W coincides with P(g)

Maybe just be hopeful and assume its something like that for now but you should definitely ask your prof for a clarification

[fleet finch]

ohhhhh

[wise sinew]

Wait y is the point at which the representative of the discriminant is non zero so would that mean P^-1 would be well defined no

Ok i think I'm gonna take a break from this but i feel like this kinda makes sense

[fleet finch]

ok thanks for your time

[wise sinew]

Does this work or am I thinking about it wrong?

[fleet finch]

i don't see it but i will think about it, might just be me 🙂

discriminant different than 0 means it has distinct roots

i feel like i have to show that g and f are the same root

they are root because g satisfies the implict function theorem (?)

but why g(y)=f(y)?

[wise sinew]

Yes yes

[fleet finch]

it's a mess for me honestly I think I will just ask

[wise sinew]

Ok but how is that even possible to show

[wise sinew]

g and f are both roots of the representative P which has distinct roots so hmm yes i think so

Ok I'm convinced it is analytic continuation im just unfamiliar with the scv vase

I think i might have to look up about the germ of the discriminant to get an idea why y is so important. i feel like maybe i am making some mistake about how it behaves.

I'm sorry I haven't really been able to help maybe there's someone else here who might be more experienced

[wise sinew]

@regal spade it's this problem

It's mainly germs tho

Just guessing it's somewhat related enough that you can make some sense of it

[fleet finch]

okay might be dumb...

might it be that there exist only one continuos solution of implicit theorem, now we know that this only one is ALSO holo. Now he calls this g and notices that f is continuos and holds the same requirements, hence f is holo.

@wise sinew

[regal spade]

Hmm, I just glanced at it and looked over the convo. It's not clear to me either. Might have to open a copy of Gunning Rossi myself to see their formulation of implicit function theorem.

[wise sinew]

I feel like this would only work if P(x)=0 iff x=f no? I think that property y has must be important in some way

[wise sinew]

Oh dear well thanks for having a look anyway

I'll try going through this again today @fleet finch but if you do find a solution could u send it here I wanna know now too

[fleet finch]

unfortunately I think this awful answer is the right one ahah

[wise sinew]

Did your prof say something?

[fleet finch]

he agreed with it

[wise sinew]

Well shit ig that's an unclimatic ending haha