# Applying analytic coordinate changes to singular function germs [closed]

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?

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 $$n$$.