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 $h\in \mathbb{C}[[y,z]]_{\ge 3}$

  1. Since the term of degree $2$ is $y^2$, it's not of type $A_n$. right?
  2. 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 $E_n$?
  3. Is there an effective way to determine whether this singularity is simple or not?

The easiest way is to use the Newton polyhedron (after a linear change of variables): once you get nondegenerate principal part, just compare to the list. See Arnold, Gussein-Zade, Varchenko (volume 1) for details.

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Yes, there are effective criteria useful to decide whether a (germ of) hypersurface singularity is simple. They usually involve the Milnor or the Tjurina number and the corank (which is defined as $n$ minus the rank of the Hessian matrix of the singularity).

For instance, $f \in \mathfrak{m}^2 \subset \mathbb{C} \{x_1, \ldots, x_n\}$ is right equivalent to a singularity of type $A_1$ if and only if the Milnor number is $1$, and this is in turn equivalent to the fact that the corank is $0$ (i.e, $f$ is a non-degenerate singularity).

For a nice treatment of this subject you can have a look at the book by Greuel, Lossen and Shustin Introduction to Singulartities and Deformations, see in particular Chapter I, Section 2.4 "Classification of Simple Singularities".

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