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Given a polynomial f(x,y,z), it defines a hypersurface in $\mathbb C^3$. I guess there is a classification of hypersurface singularity like Arnold normal form. I wonder given an explicite example of f, how to determine its singularity type.

For instance is $z_1^3+z_2^3+z_3^3=0$ locally analytic isomorphic to $z_1^3+z_2^3+z_3^3+z_1^6+z_2^6+z_3^6=0$ near zero?

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    $\begingroup$ This is discussed at length in chap. 6 of Singularities of Differentiable Maps by Arnold, Gusein-Zade, Varchenko. In particular Tougeron's theorem on finite determinacy is relevant. $\endgroup$
    – abx
    Commented Apr 3, 2023 at 5:35
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    $\begingroup$ In your example the answer is clearly yes: just put $y_i=z_i(1+z_i^3)^{1/2}$. $\endgroup$
    – abx
    Commented Apr 3, 2023 at 8:57
  • $\begingroup$ Thanks @abx. The reference is helpful. Also I guess you mean $z_i(1+z_i)^\frac{1}{3}$. For this example, I tried formal power sereis solution and don't know how to show the convergence. It turns out it is easy here. Thanks a lot. $\endgroup$
    – xin fu
    Commented Apr 3, 2023 at 16:09
  • $\begingroup$ You are right of course, sorry for the typo. $\endgroup$
    – abx
    Commented Apr 3, 2023 at 18:10

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