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Symbolic Hamburger-Noether expressions

We recall the definition of Hamburger-Noether expansions (HNE), resp. symbolic Hamburger-Noether expressions, for the branches of a reduced plane curve singularity. They can be regarded as the analogue of Puiseux expansions when working over a field of positive characteristic (cf. the discussion in [2], 2.1). As being the case for Puiseux expansions, many invariants of plane curve singularities (such as the multiplicity sequence, the -invariant, the intersection multiplicities of the branches, etc.), can be computed directly from the corresponding system of HNE.

Let be a point and local parameters at . Moreover, let the germ be given by a local equation with irreducible decomposition . Finally, let's suppose that is a transversal parameter for , that is, the order of is equal to the order of .

Definition 2.1   A Hamburger-Noether expansion (HNE) of at for the local branch given by (defined over some finite algebraic extension ) is a finite sequence of equations
 (1)

where is a non-negative integer, and , , are positive integers, such that     in

If the local equation of is polynomial in , i.e., , then the last (infinite) row of (1) can be replaced, equivalently, by a (finite) implicit equation

The resulting system is called a symbolic Hamburger-Noether expression (sHNE) for the branch.

Any HNE leads to a primitive parametrization of the branch (setting and mapping , ). It can be computed from a sHNE up to an arbitrary finite degree in .

Remark 2.2   There exist constructive algorithms to compute a system of sHNE's (resp. HNE's up to a given degree) for the branches of a reduced plane curve singularity (cf. [2] for the irreducible case, resp. [34] for the reducible case). A modification of the latter algorithm is implemented in the computer algebra system SINGULAR [26,17].

To perform the algorithm one does not need any knowledge about the irreducible factorization of in . Moreover, in the reducible case, the probably necessary (finite) field extension is not performed a priori, but by successive (primitive) extensions introduced exactly when needed for computations (factorization).

Remark 2.3   Given a Hamburger-Noether expansion (1) for a local branch , the corresponding multiplicity sequence can be easily read off:

where the are defined recursively by setting and, for ,

In particular, we can compute the -invariant of the branch directly from the Hamburger-Noether expansion, since

Moreover, if is the primitive parametrization of obtained from the HNE then the intersection multiplicity of with the plane curve germ given by can be computed as .

Next: Adjoint curves and Brill-Noether Up: Effective Construction of Algebraic Previous: Preliminaries, Notations
Christoph Lossen
2001-03-21