Standard Bases for fractional ideals of the local ring of an algebroid curve

Abstract

In this paper we present an algorithm to compute a Standard Basis for a fractional ideal I of the local ring O of an n-space algebroid curve with several branches. This allows us to determine the semimodule of values of I. When I = O, we may obtain a (finite) set of generators of the semiring of values of the curve, which determines its classical semigroup. In the complex context, identifying the Kähler differential module Ω O/C of a plane curve with a fractional ideal of O and applying our algorithm, we can compute the set of values of Ω O/C , which is an important analytic invariant associated to the curve.

The Analytic Classification of Plane Curves

“…As we mentioned in Section 2, for each fixed value semiring Γ there are only finitely many A-invariants Λ. Consequently, the same is true for Λ G that can be computed by the algorithm presented in Proposition 16, [CH2].…”

“…In [CH2], algorithms are presented in order to compute a finite set of generators of ν(I) for any fractional ideal of O. In particular, for I = O we obtain ν(O) = Γ.…”

“…In [CH2] (Proposition 19) the authors presented an algorithm to compute Λ and they showed that this set is characterized by their points in the box [0, κ 1 ] × . .…”

“…The conditions imposed to have the semiring Γ are easily determined by the nonzero coefficients of the terms with characteristic exponents and some conditions to get the intersection multiplicity of each pair of branches. On the other hand, the algebraic conditions related with Λ G can be obtained applying the algorithm presented in [CH2].…”

“…The above theorem provides a method to decide if two plane curves are or not analytically equivalent. In fact, considering an associated multigerm to a plane curve C with r irreducible components we compute their A-invariants Γ and Λ G using the appropriate group G and the algorithms in [CH2] for instance. Applying the results in this section we obtain an A-normal…”