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 , 2.1). 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 .
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
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 .
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).