A bound for the Milnor number of plane curve singularities

Let $f=0$ be a plane algebraic curve of degree $d>1$ with an isolated singular point at the origin of the complex plane. We show that the Milnor number $\mu_0(f)$ is less than or equal to $(d-1)^2-\left[\frac{d}{2}\right]$, unless $f=0$ is a set of $d$ concurrent lines passing through 0. Then we characterize the curves $f=0$ for which $\mu_0(f)=(d-1)^2-\left[\frac{d}{2}\right]$

Introduction to the local theory of plane algebraic curves

We consider the algebroid plane curves de ned by formal power series of two variables with coe cients in an algebraically closed eld. Using quadratic transformations we prove the local normalization theorem. Then we study the intersection multiplicity of algebroid curves and give an introduction to the Newton diagrams

Formal and convergent solutions of analytic equations

We provide the detailed proof of a sharpened version of the M. Artin Approximation Theorem

Invariants of plane curve singularities and Newton diagrams

We present an intersection-theoretical approach to the invariants of plane curve singularities $\mu$, $\delta$, $r$ related by the Milnor formula $2\delta=\mu+r-1$. Using Newton transformations we give formulae for $\mu$, $\delta$, $r$ which imply planar versions of well-known theorems on nondegenerate singularities