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Suppose we are given a function germ \begin{align} f = \sum a_{ijk}x^iy^jz^k \end{align} such that $f\in \mathfrak{m}^2$, where $\mathfrak{m}$ is the ideal in $\mathbb{C}\{x,y,z\}$ of holomorphic functions vanishing at 0. I am currently reading an expository text on the du Val surface singularities in which the author sometimes simplifies a germ like this by using analytic coordinate changes. Stuff like: "If the 2-jet of $f$ is $x^2$, an analytic coordinate change can be used to remove any further appearances of $x$ in $f$" or "if $J_2(f)=x^2 + y^2$ then the existence of an $a_{ijk}\neq 0$ with $i + j <2$ implies that at least one term of the form $z^m$ or $xz^m$ or $yz^m$ appears in $f$, and then an analytic coordinate change can be used to make $f = x^2 + y^2 + z^{n+1}$."

I don't really understand which coordinate changes are applied here. Does anyone have a reference explaining techniques like this in further detail?

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Analytic change of coordinates in $(\mathbb{C}^3,0)$ must take $0$ to $0$. So they are induced by (local) $\mathbb{C}$-automorphisms of $\mathcal{O}_{\mathbb{C}^3,0}=\mathbb{C}\{x,y,z\}$ taking $\mathfrak{m}=(x,y,z)$ to itself. It is not difficult to check that those maps are always of type $$x=a_{1,1}x_1+a_{2,1}y_1+a_{3,1}z_1+\ldots$$ $$y=a_{1,2}x_1+a_{2,2}y_1+a_{3,2}z_1+\ldots$$ $$z=a_{1,3}x_1+a_{2,3}y_1+a_{3,3}z_1+\ldots$$ where $A=(a_{i,j})\in GL_3$ and stand $\ldots$ for 'higher order terms'.

Being very practical, in each case you should express a given function $f(x,y,z)$ in the new coordinates $(x_1,y_1,z_1)$ in a way that simplify its form. Most of the times it reduces to a linear algebra question...

Of course this also works for every dimension $n$.

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  • $\begingroup$ I don't think that your observation helps with any of the particular examples that were presented in the question. $\endgroup$
    – Ben McKay
    Commented Apr 29, 2021 at 8:18

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