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Applying analytic coordinate changes to singular function germs

Discussion in 'Mathematics' started by Morph, Aug 1, 2020 at 8:03 PM.

  1. Morph

    Morph Guest

    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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