Two power series $G(x_1, \ldots, x_n)$ and $F(x_1, \ldots, x_n)$ are equivalent over $\mathbb{C}$ if there is an automorphism of the ring $\mathbb{C}[[x_1, \ldots, x_n]]$ given by $x_1 \to \phi(x_1, \ldots, x_n)$ and an invertible element $$ u(x_1, \ldots, x_n) \in \mathbb{C}[[x_1, \ldots, x_n]] $$ such that $$ u(x_1, ...x_n)G(x_1, \ldots, x_n) =F(\phi_1, \ldots, \phi_n) $$ We can use this to say that two singularities given by $(F=0)$ and $(G=0)$ are equivalent if their equations are equivalent on the previous sense (see [1]).

**Example** ([2]): The quartic plane curve
$
(xz+y^2)^2+x^4
$
has an $A_7 =x^2+y^8$ singularity at $(x=y=0)$.

I am quite comfortable with this sort of games for **isolated** singularities..However, for **non isolated singularities**, I do get concerned about things making sense...

What are the technical details and/or traps that I should look ? Is there any warning... counterexample.. reference?

Thanks!

[1] Real Algebraic Threefolds I by Kollar http://arxiv.org/pdf/alg-geom/9712004v1.pdf

[2] Wall, C. T. C. "Geometry of quartic curves." Mathematical Proceedings of the Cambridge Philosophical Society. Vol. 117. No. 3. [Cambridge, Eng.] Cambridge Philosophical Society., 1995.