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In this paper, we present a solution to the problem of the analytic classification of germs of plane curves with several irreducible components. Our algebraic approach follows precursive ideas of Oscar Zariski and as a subproduct allows us to recover some particular cases found in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a distinguished expression for a given 1-form with respect to a polynomial f ∈ ℂ { x } [ y ] , called Weierstrass form. We will use this form and the properties of plane analytical curves to give new characterizations of nondicritical generalized curve foliations. [ABSTRACT FROM AUTHOR]

The value semigroup Γ and the value set Λ of 1-forms are, respectively, a topological and an analytical invariant of a plane branch. Given a plane branch C with value semigroup Γ there are a finitely number of distinct possible sets Λ according to the analytic class of C . In this work we show that the value set of 1-forms Λ determines the value semigroup Γ and we present an effective method to recover Γ by Λ . In particular, this allows us to decide if a subset of N is a value set of 1-forms for a plane branch. [ABSTRACT FROM AUTHOR]

The value semigroup of a k -semiroot C k of a plane branch C allows us to recover part of the value semigroup Γ = 〈 v 0 , ... , v g 〉 of C , that is, it is related to topological invariants of C. In this paper we consider the set of values of differentials Λ k of C k , that is an analytical invariant, and we show how it determine part of the set of values of differentials Λ of C. As a consequence, in a fixed topological class, we relate the Tjurina number τ of C with the Tjurina number of C k. In particular, we show that τ ≤ μ − 3 n g − 2 4 μ g − 1 where n g = g c d (v 0 , ... , v g − 1) , μ and μ g − 1 denote the Milnor number of C and C g − 1 respectively. If n g = 2 , we have that τ = μ − μ g − 1 for any curve in the topological class determined by Γ that is a generalization of a result obtained by Luengo and Pfister. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we present the algorithms used by the second author in Hernandes (2011) for the implementation in MAPLE of the results of Hefez and Hernandes (2011) that allow to perform the analytic classification of irreducible germs of singular complex plane curves. [Copyright &y& Elsevier]

Abstract: We perform the analytic classification of plane branches of multiplicity less than or equal to four. This is achieved by computing a Standard Basis for the modules of Kähler differentials of such branches by means of the algorithm developed in [Hefez, A., Hernandes, M.E., 2007a. Standard bases for local rings of branches and their module of differentials. J. Symbolic Comput. 42, 178–191] and then applying the classification method for plane branches given in [Hefez, A., Hernandes, M.E., 2007b. The analytic classification of plane branches. arXiv:0707.4502]. [Copyright &y& Elsevier]

In contrast to the Milnor number, there was no known formula relating the Tjurina number of a reducible curve to the Tjurina numbers of its components. In this work we exhibit such a formula relating Tjurina number of a complete intersection algebroid (or analytic) curve over an algebraically closed field of characteristic zero (or over C ) to the Tjurina numbers of its components, involving the intersection indices among the components and numerical analytic invariants extracted from modules of Kähler differentials on unions of branches of the curve. [ABSTRACT FROM AUTHOR]