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# A necessary and sufficient condtion for a curve to have an $A_k$ singularity.

Hi Does any one know of a necessary and sufficient condition for a curve to have a singularity of type A_k.

More precisely, a curve f=0 has a singularity of type A_k at a point, if there exist local coordinates (x,y), where the function can be written as

f(x,y)=x^2+y^{k+1}=0.

If you understand what I am talking about, you need not read the rest. But in case you don't follow the question, let me elaborate a bit.

A necessary and sufficient condition for a curve to have an A_1 node is the following

df:= (f_x, f_y) = 0

Hessian(f) = non degenerate.

This is essentially the morse lemma.

I know conditions for A_2, A_3, ...... until A_6 node. I was wondering if anyone knew something for a general k.