An effective solution of the problem of analytic classification of plane branches is given. A finite stratification of any given equisingularity class of plane branches is determined and normal forms for each stratum are exhibited in such a way that two branches in normal form are easily recognized to be analytically equivalent or not. In this way, we solve the main problems proposed by O. Zariski (Le problème des modules pour les branches planes.

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We perform the analytic classification of plane branches of multiplicity less or equal than four. This is achieved by computing a Standard basis for the modules of Kähler differentials of such branches by means of the algorithm we developed in [9] and then applying the classification method for plane branches presented in [10].

We introduce the semiring of values Γ with respect to the tropical operations associated to an algebroid curve. As a set, Γ determines and is determined by the well known semigroup of values S and we prove that Γ is always finitely generated in contrast to S. In particular, for a plane curve, we present a straightforward way to obtain Γ in terms of the semiring of each branch of the curve and the mutual intersection multiplicity of its branches. In the analytical case, this allows us to connect directly the results of Zariski and Waldi that characterize the topological type of the curve.

We introduce the concept of good Saito basis for a plane curve and we explore it to obtain a formula for the minimal Tjurina number in a topological class. In particular, we give a lower bound for the Tjurina number in terms of the Milnor number that allow us to present a positive answer for a question of Dimca and Greuel.