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A.4.6 Invariants of plane curve singularities

The Puiseux pairs of an irreducible and reduced plane curve singularity are probably its most important invariants. They can be computed from its Hamburger-Noether expansion (which is the analogue of the Puiseux expansion in characteristic 0 for fields of arbitrary characteristic).

The library hnoether.lib (see hnoether_lib) uses the algorithm of Antonio Campillo in "Algebroid curves in positive characteristic" SLN 813, 1980. This algorithm has the advantage that it needs least possible field extensions and, moreover, works in any characteristic. This fact can be used to compute the invariants over a field of finite characteristic, say 32003, which will most probably be the same as in characteristic 0.

We compute the Hamburger-Noether expansion of a plane curve singularity given by a polynomial f in two variables. This expansion is given by a matrix, and it allows us to compute a primitive parametrization (up to a given order) for the curve singularity defined by f and numerical invariants such as the

  • characteristic exponents,
  • Puiseux pairs (of a complex model),
  • degree of the conductor,
  • delta invariant,
  • generators of the semigroup.
Besides commands for computing a parametrization and the invariants mentioned above, the library hnoether.lib provides commands for the computation of the Newton polygon of f, the square-free part of f and a procedure to convert one set of invariants to another.

 


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