"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

the more i read the more confused i am by the choice of notation

"hannah what is the attachment"

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

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...

the unknowns are the functions hᵢⱼ

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!

or simply g(0,...,0)

oh sorry, g₀(x) should be g₀, a constant

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 $$[0,1]$$

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 $$\mathcal{E}_k$$.

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

they Did meant something along the lines of "integrating wrt $$x_n$$"

"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

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

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

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

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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