Singularities of plane projective curves

Problem:

Determine the type of the singularity at (0,0) of

f(x,y) = y²-2x²⁸y-4x²¹y¹⁷+4x¹⁴y³³-8x⁷y⁴⁹+x⁵⁶+20y⁶⁵+4x⁴⁹y¹⁶ ,

and check whether this is the only singularity of the corresponding complex plane projective curve C .

The algorithm proceeds as follows:

Step 1:	Classify the singularity of f at (0,0) following Arnold's classification scheme, in particular, compute the local Tjurina number of f: T_local(f) = dim_KK[x,y]_<x,y>/< jacob(f), f >
Step 2:	Compute the global Tjurina number of f: T_global(f) = dim_KK[x,y]/< jacob(f), f > If T_global(f) = T_local(f) then there is no further singularity in the affine part of C.
Step 3:	Consider the singularities at infinity (coordinate change).

SINGULAR code