Build. Blocks
Comb. Appl.
HCA Proving
Arrangements
Branches
Classify
Coding
Deformations
Equidim Part
Existence
Finite Groups
Flatness
Genus
Hilbert Series
Membership
Nonnormal Locus
Normalization
Primdec
Puiseux
Plane Curves
Saturation
Solving
Space Curves
Spectrum
Analysis of Singularities of Plane Projective Curves
We know:
  • dim (C[[x,y]] / < fx, fy >) = 2*delta - number of branches + 1,  

where the left-hand side is the Milnor number of f.

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

  1. Test for Ak- and Dk-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.

SINGULAR Example

Sao Carlos, 08/02 http://www.singular.uni-kl.de