A discriminant criterion for an equivalence of an analytic family of plane curve singularities and its applications

In this paper, we generalize Zariski's discriminant criterion under which an analytic family of plane curve singularities are equivalent.Furthermore, we give a necessary condition of local irreducibility of plane curves with singularities.Key words.topological equivalence of plane curve singularities, the standard Puiseux expansion, the Milnor number.