Newton diagrams and equivalence of plane curve germs. (English) Zbl 1111.32029

García Barroso, Evelia Rosa; Lenarcik, Andrzej; Płoski, Arkadiusz

J. Math. Soc. Japan 59, No. 1, 81-96 (2007).

The authors study an equivalence of plane curve germs which is weaker than Zariski’s equisingularity. Improving M. Lejeune-Jalabert [Thèse d’Etat, Paris VII, Janvier 1973)] and M. Oka [Proc. Symp. Pure Math. 40, Part 2, 259–268 (1983; Zbl 0514.14003)] theorems on the stability of the Newton boundary they show that the set of all Newton diagrams of a germ is an invariant of this equivalence. Then they give a complete description of the set of all Newton diagrams of a plane many-branched singularity in terms of some invariants of branches and their orders of contact. The properties of the ultrametric space of plane curve germs play an important part in the paper.

Reviewer: Andrzej Lenarcik (Kielce)

MSC:

32S55	Milnor fibration; relations with knot theory
14H20	Singularities of curves, local rings

Keywords:

plane curve germ; Newton diagram; equivalence of terms; strong triangle inequality; quasi-branch

Citations:

Zbl 0514.14003