In this paper we solve the problem of analytic classification of plane curves singularities with two branches by presenting their normal forms. This is accomplished by means of a new analytic invariant that relates vectors in the tangent space to the orbits under analytic equivalence in a given equisingularity class to Kähler differentials on the curve. * The first two authors were partially supported by CNPq grantsThe A-equivalence in B is induced by the action of the group A = Aut(C{t 1 })×Aut(C{t 2 })× Aut(C{X, Y }), as follows:Our analysis will be splitted into two cases, namely, whether the two components of (f ) have distinct tangents (the transversal case) or equal tangents. In what follows, we will denote by m i the multiplicity of f i , i = 1, 2.Case 1) Distinct tangents. In this case, by A-equivalence, we may assume that the tangent of the first component is (Y ) and of the second one is (X), so that φ i = (x(t i ), y(t i )), where ord t 1 x(t 1 ) < ord t 1 y(t 1 ) and ord t 2 x(t 2 ) > ord t 2 y(t 2 ).Case 2) Same tangent. In this case, by A-equivalence, we may assume that the common tangent

We describe an algorithm to compute a presentation of the pushforward module f * OX for a finite map germ f : X → (C n+1 , 0), where X is Cohen-Macaulay of dimension n. The algorithm is an improvement of a method by Mond and Pellikaan. We give applications to problems in singularity theory, computed by means of an implementation in the software Singular.Mathematics Subject Classification. Primary 58K05; Secondary 32S10, 14Q99, 13C10 .

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.

No abstract

In this work, we classify parametrized monomial surfaces f:(ℂ2,0)→(ℂ4,0) that are A-finitely determined. We study invariants that can be obtained in terms of invariants of a parametrized curve.