http://www.math.tifr.res.in/~publ/studies/SM_03-Ideals-of-Differentiable-Functions.pdf 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

"hannah what is the attachment"

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

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

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!

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 \(\mathcal{E}_k\).

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

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

Please bear in mind this is graduate-level maths most of the time. It's okay if it goes over your head, this level of math is super niche and abstract anyway.

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