Let $f(z_1,z_2,\dots ,z_n)$ be an analytic function in $\mathbb{C}[[z_1,z_2,\dots ,z_n]]$ whose leading term defines an isolated singularity at the origin. If we have the following types of singularities, then it is called a simple(ADE) singularity.

$A_n:z_1^{n+1}+\sum_2^{n} z_i^2=0$ $(n\ge 1)$

$D_n:z_1^{n-1}+z_1z_2^2+\sum_3^{n} z_i^2=0$ $(n\ge 4)$

$E_6:z_1^4+z_2^3+\sum_3^{n} z_i^2=0$

$E_7:z_1^3z_2+z_2^3+\sum_3^{n} z_i^2=0$

$E_8:z_1^5+z_2^3+\sum_3^{n} z_i^2=0$.

I have many (homogeneous) equations which define hypersurfaces with a singularity(in some pojective space). I would like to know if these singularities are simple.

For example, a equation is (locally) given by:

$(f(y,z)-y)x^2+2g(y,z)x+(y^2+h(y,z))=0$, where $f,g\in \mathbb{C}[[y,z]]_{\ge 2}$ and $h\in \mathbb{C}[[y,z]]_{\ge 3}$

- Since the term of degree $2$ is $y^2$, it's not of type $A_n$. right?
- Why is the singularity of type $D_n$ only if the coefficient of $z^2$ in $g$ does not vanish? What happen if the coefficient of $z^2$ in $g$ vanish? Can't it be of type $E_n$?
- Is there an effective way to determine whether this singularity is simple or not?