Standard bases for local rings of branches and their modules of differentials

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

“…In this way, if we consider B i as the union of Standard Bases for O i and Q i for all i ∈ I, then the same proof of the above theorem shows that r i=0 B i is a Standard Basis for O. It is important to emphasize that B i , for i ∈ I, can be computed as described in [13].…”

“…We can obtain σ i by the algorithm presented in [13] that allows to compute the set of values of any finitely generated O i -module. If n = 2, that is, for a plane curve Q =…”

“…Natural generalizations consist in considering algebroid curves, space irreducible curves or plane curves with several branches. The authors of [4], [5] and [13] present distinct methods to obtain the generators of the semigroup of values of an irreducible algebroid space curve: by the infinitely near points associated to any resolution (see [4] and [5]) and by Standard Bases for the local ring of the curve (see [13]).…”

The analytic classification of plane branches

“…The methods we presented here, together with the algorithms developed in [6], show that the analytic classification of plane branches may be effectively performed. With those algorithms, the sets Λ and the conditions on the coefficients that determine them, for a fixed equisingularity class, may be computed.…”

“…In addition to the tools used by Zariski in [11], we introduce in the game two computational techniques. The first is a SAGBI algorithm, due to Robbiano and Sweedler [9], which we adapted in [5,6] to describe distinguished bases of the local rings of plane branches and of their modules of Kähler differentials, as well, allowing us to compute the set of invariants Λ. The second technique is the algorithm of Complete Transversal due to Bruce, Kirk and du Plessis [1] that permits to determine all normal forms of map germs under Mather's group actions, but does not, in general, predict the final result.…”