On the jacobian Newton polygon, branches and dandelions
Streszczenie
Let $f\in\mathbb{C}\{X,Y\}$ be a convergent series and let $f=0$ be a germ of an isolated plane curve singularity. To describe the singularity we apply the Eggers–Płoski tree which is graphically equivalent to the Eggers tree. Instead of the contact exponent between branches (as in the original construction) we use Płoski’s logarithmic distance between them. An advantage of this approach is that the tree can be constructed without fixing any coordinate system. We call a germ a dandelion if all branches go through all the black vertices of the tree of $f$. Then these balls form a single chain. We consider the jacobian Newton polygon $\mathcal{N}_{\mathbf J}(f)$ introduced by Bernard Teissier. We say that the jacobian Newton polygon determines the equisingularity class of $f=0$ among all singularities if for any germ $g=0$ the equality $\mathcal{N}_{\mathbf J}(f)=\mathcal{N}_{\mathbf J}(g)$ implies the equisingularity of the germs $f=0$ and $g=0$. Evelia García Barroso and Janusz Gwoździewicz proved that every branch $f=0$ satisfies this property. In this way they obtained a new criterion of irreducibility. We present an analogous result for dandelions.
