Publication

The ultrametric space of plane branches

Journal Article (2011)

Journal

Communications in Algebra

Pages

4206-4220

Volume

39

Number

11

Doc link

http://dx.doi.org/10.1080/00927872.2010.521934

File

Download the digital copy of the doc pdf document

Authors

Abstract

We study properties of the space of irreducible germs of plane curves (branches), seen as an ultrametric space. We provide various geometrical methods to measure the distance between two branches and to compare distances between branches, in terms of topological invariants of the singularity which comprises some of the branches. We show that, in spite of being very close to the notion of intersection multiplicity between two germs, this notion of distance behaves very differently. For instance, any value in [0, 1] ∩ Q is attained as the distance between a fixed branch and some other branch, in contrast with the fact that the semigroup of the fixed branch has gaps. We also present results that lead to interpret this distance as a sort of geometric distance between the topological equivalence or equisingularity classes of branches.

Categories

robot kinematics.

Author keywords

equisingularity class, plane branch, ultrametric distance

Scientific reference

I. Abío, M. Alberich-Carramiñana and V. González-Alonso. The ultrametric space of plane branches. Communications in Algebra, 39(11): 4206-4220, 2011.