Details
| Original language | English |
|---|---|
| Pages (from-to) | 4206-4220 |
| Number of pages | 15 |
| Journal | Communications in algebra |
| Volume | 39 |
| Issue number | 11 |
| Early online date | 22 Nov 2011 |
| Publication status | E-pub ahead of print - 22 Nov 2011 |
| Externally published | Yes |
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.
Keywords
- Equisingularity class, Plane branch, Ultrametric distance
ASJC Scopus subject areas
- Mathematics(all)
- Algebra and Number Theory
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In: Communications in algebra, Vol. 39, No. 11, 22.11.2011, p. 4206-4220.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - The Ultrametric Space of Plane Branches
AU - Abío, Ignasi
AU - Alberich-Carramiñana, Maria
AU - González-Alonso, Víctor
PY - 2011/11/22
Y1 - 2011/11/22
N2 - 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.
AB - 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.
KW - Equisingularity class
KW - Plane branch
KW - Ultrametric distance
UR - http://www.scopus.com/inward/record.url?scp=84857926464&partnerID=8YFLogxK
U2 - 10.1080/00927872.2010.521934
DO - 10.1080/00927872.2010.521934
M3 - Article
AN - SCOPUS:84857926464
VL - 39
SP - 4206
EP - 4220
JO - Communications in algebra
JF - Communications in algebra
SN - 0092-7872
IS - 11
ER -