Explicit Computation of The Generic Component of the Analytic Moduli of a Plane Branch

Let ${\mathcal C}$ be a fixed equisingularity class of irreducible germs of complex analytic plane curves. We compute a basis of the ${\mathbb C}[[x]]$-module of K\"ahler differentials for generic $\Gamma \in {\mathcal C}$, algorithmically, and study its behaviour under blow-up. As a first application, we give an algorithm providing the generic semimodule in an equisingularity class in terms of its multiplicity and its Puiseux characteristic exponents. As another application, we give an alternative proof for a formula of Genzmer, that provides the dimension of the moduli of analytic classes in the equisingularity class of $\Gamma$.