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"p-adic" is one of those things I try to read about every now and then without much success

Exercise is a very pleasant thing to do. In mathematics, and in real life. Often, I fail to do either of them though 😂!

math.tifr.res.in/~publ/studies here you can have a look it feels like someone managed to convert a latex PDF file into epub and them back into PDF

if you ever got books from archive.org into a kindle you know what I'm talking about

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"hannah what is the attachment"

oh sure i wanted to say they just gave me this wall of formulas in PAGE 2

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I came here thinking "when will the rabbit hole of papers end??" and this is not even a paper

tfw you go so deep into a chain of papers you end up in a proper book

and that is actually very powerful in singularity theory... no doubt that condition emerges so often...

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So, basically, finite means that the system of equations

\(x_i=h_{i,1}(x)f_1(x)+\dots+h_{i,n}(x)f_n(x)\) for \(i=1,\dots,n\)

has a solution!

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By definition \(f\) is finite if \(\mathcal{E}_n\) is finitely generated as a \(\mathcal{E}_p\)-module via \(f^*\). So, what does this mean?

Well, any germ of smooth function can be written in the form
\(g(x)=g_0(x)+x_1g_1(x)+\dots+x_ng_n(x)\) (this is a consequence of Hadamard's lemma, but the proof bsaically boils down to derivating the function \(g(tx)\) with respect to \(t\) and then integrating wrt \(t\) in the interval \([0,1]\)

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Ok so let \(f\colon U \subseteq \mathbb{R}^n \longrightarrow \mathbb{R}^p\) with \(n \leq p\). I will denote the germ of smooth functions on \((\mathbb{R}^k,0)\) by \(\mathcal{E}_k\).

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I've been struggling with that paper all day

In particular, trying to understand a step which involved using finiteness (now i will say what that means)

And I just realisef what finiteness **actually** means

i read the proof

they Did meant something along the lines of "integrating wrt \(x_n\)"

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"where \(\pi_{U,x}\) is the map sending a function onto its germ at \(x\)"

hnnnnnnnng i guess? sure i get the idea but it feels like you're pushing the envelope in that definition but hey i only read about sheaf theory once!

so for context: we have two sheaves of functions, \(D_f\) and \(S_f\), and the formula relating them is

\(D_f=(\partial x_n)^{-1}S_f\)

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it took me a while to take in the fact we're taking the "inverse" of a vector field

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\(\displaystyle\left(\frac{\partial}{\partial x_n}\right)^{-1}\)

so it's going to be one of those days huh

\(R(f,g;x)=a_n^mb_m^n\displaystyle\prod^n_{i=1}\prod^m_{j=1}(\alpha_i-\beta_j)\)

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