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
Authors
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Abío, Ignasi
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Alberich Carramiñana, Maria
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González, Víctor
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.
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