1. This site uses cookies. By continuing to use this site, you are agreeing to our use of cookies. Learn More.

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?

    Login To add answer/comment

Share This Page