Parametrization simple irreducible plane curve singularities in arbitrary characteristic

We study the classification of plane curve singularities in arbitrary characteristic. We first give a bound for the determinacy of a plane curve singularity with respect to pararametrization equivalence in terms of its conductor. We then define the notion of parametrization modality for plane curve singularities. Finally, we revisit Mehmood-Pfister's classification of parametrization simple plane curve singularities which are irreducible by giving a concrete list of normal forms of equations and parametrizations. In characteristic zero, the classification of parametrization simple irreducible plane curve singularities was achieved by Bruce and Gaffney.