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.