• dim (C[[x,y]] / < f_x, f_y>) = 2*delta - number of branches + 1,  

To compute the number of (local) branches, we proceed as follows:

1. Test for A_k- and D_k-singularities.

2. Compute the Newton Polygon.

3. If the Newton Polygon is non-degenerate, then the number of branches can be computed combinatorically from the faces.

4. If the Newton Polygon is degenerate and has more than one face, then f can be splitted (modulo analytic equivalence) into a product.

5. If the Newton Polygon is degenerate and has only one face, then we use the Puiseux expansion to compute the number of branches.

Needs:

• Puiseux expansion

• Primary decomposition

• Field extensions

• Newton polygon.