The Ultrametric Space of Plane Branches

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 c...

Celý popis

Uložené v:

Podrobná bibliografia
Vydané v:	Communications in algebra Ročník 39; číslo 11; s. 4206 - 4220
Hlavní autori:	Abío, Ignasi, Alberich-Carramiñana, Maria, González-Alonso, Víctor
Médium:	Journal Article Publikácia
Jazyk:	English
Vydavateľské údaje:	Taylor & Francis Group 01.11.2011
Predmet:	14 Algebraic geometry 14H Curves 32 Several complex variables and analytic spaces 32S Singularities Classificació AMS Corbes planes Curves, Plane Equisingularity class Matemàtiques i estadística Plane branch Primary 14H20 Secondary 32S05, 14C17 Singularitats (Matemàtica) Ultrametric distance Àrees temàtiques de la UPC
ISSN:	0092-7872, 1532-4125
On-line prístup:	Získať plný text
Tagy:	Pridať tag Žiadne tagy, Buďte prvý, kto otaguje tento záznam!

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.
AbstractList	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. 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] ∩ 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. Peer Reviewed
Author	Alberich-Carramiñana, Maria González-Alonso, Víctor Abío, Ignasi
Author_xml	– sequence: 1 givenname: Ignasi surname: Abío fullname: Abío, Ignasi organization: Departament de Matemàtica Aplicada I , Universitat Politècnica de Catalunya – sequence: 2 givenname: Maria surname: Alberich-Carramiñana fullname: Alberich-Carramiñana, Maria email: maria.alberich@upc.edu organization: Institut de Robòtica i Informàtica Industrial (CSIC-UPC) – sequence: 3 givenname: Víctor surname: González-Alonso fullname: González-Alonso, Víctor organization: Departament de Matemàtica Aplicada I , Universitat Politècnica de Catalunya
BookMark	eNqFkE1LAzEQhoNUsK3-Aw9787Q6k4_djRfR4hcUFGzPIc0mNLLdlGRF-u_dpRXBgx6GYYb3eQ_PhIza0FpCzhEuESq4ApC0rEp6SaF_CYqS8SMyRsFozpGKERkPkXzInJBJSu8AKMqKjsnFYm2zZdNFvbFd9CZ722pjs-Cy10a3NruLujVrm07JsdNNsmeHPSXLh_vF7Cmfvzw-z27nuWFSdDnjrIDa0VoYW1K0IAWvUKOUyFaoCwH1yomC17qS1lHmTO0AV1DICkxRczYluO816cOoaI2NRncqaP9zDEOhpAorwRn2DD8wMaQUrVPb6Dc67hSCGgSpb0FqEKT2gnrs-hdmfKc7H9rehm_-g2_2sG9diBv9GWJTq07vmhDd4Mwnxf5s-AJQBH3X
CitedBy_id	crossref_primary_10_1016_j_aim_2015_10_011 crossref_primary_10_1142_S0219498821500328
Cites_doi	10.1007/BF01418370 10.1017/CBO9780511617560 10.1007/b100262 10.1017/CBO9780511569326 10.4310/AJM.2007.v11.n3.a3
ContentType	Journal Article Publication
Contributor	Universitat Politècnica de Catalunya. EGSA - Equacions Diferencials, Geometria, Sistemes Dinàmics i de Control, i Aplicacions Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada I
Contributor_xml	– sequence: 1 fullname: Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada I – sequence: 2 fullname: Universitat Politècnica de Catalunya. EGSA - Equacions Diferencials, Geometria, Sistemes Dinàmics i de Control, i Aplicacions
Copyright	Copyright Taylor & Francis Group, LLC 2011 Attribution-NonCommercial-NoDerivs 3.0 Spain info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by-nc-nd/3.0/es
Copyright_xml	– notice: Copyright Taylor & Francis Group, LLC 2011 – notice: Attribution-NonCommercial-NoDerivs 3.0 Spain info:eu-repo/semantics/openAccess <a href="http://creativecommons.org/licenses/by-nc-nd/3.0/es/">http://creativecommons.org/licenses/by-nc-nd/3.0/es/</a>
DBID	AAYXX CITATION XX2
DOI	10.1080/00927872.2010.521934
DatabaseName	CrossRef Recercat
DatabaseTitle	CrossRef
DatabaseTitleList
DeliveryMethod	fulltext_linktorsrc
Discipline	Mathematics
EISSN	1532-4125
EndPage	4220
ExternalDocumentID	oai_recercat_cat_2072_185431 10_1080_00927872_2010_521934 521934
GroupedDBID	-DZ -~X .7F .QJ 0BK 0R~ 29F 2DF 30N 4.4 5GY 5VS AAENE AAGDL AAHIA AAJMT AALDU AAMIU AAPUL AAQRR ABCCY ABFIM ABHAV ABJNI ABLIJ ABPAQ ABPEM ABTAI ABXUL ABXYU ACGEJ ACGFS ACTCW ACTIO ADCVX ADGTB ADXPE AEISY AENEX AEOZL AEPSL AEYOC AFKVX AFRVT AGCQS AGDLA AGMYJ AHDZW AIJEM AIYEW AJWEG AKBVH AKOOK ALMA_UNASSIGNED_HOLDINGS ALQZU AMVHM AQRUH AQTUD AVBZW AWYRJ BLEHA CAG CCCUG CE4 COF CS3 DGEBU DKSSO DU5 EBS EJD E~A E~B GTTXZ H13 HF~ HZ~ H~P IPNFZ J.P KYCEM M4Z NA5 NY~ O9- P2P PQQKQ RIG RNANH ROSJB RTWRZ S-T SNACF TASJS TBQAZ TDBHL TEJ TFL TFT TFW TN5 TOXWX TTHFI TUROJ TWF UPT UT5 UU3 WH7 XSW ZGOLN ~S~ 07G 1TA 6TJ AAGZG AAIKQ AAKBW AAYXX ACAGQ ACGEE AEUMN AGLEN AGROQ AHMOU ALCKM AMEWO AMXXU BCCOT BPLKW C06 CITATION CRFIH DMQIW DWIFK H~9 IVXBP LJTGL MVM NUSFT OHT QCRFL TAQ TFMCV UB9 UU8 V3K V4Q ZY4 XX2
ID	FETCH-LOGICAL-c395t-34360df2d5ce721e095481a19913b1a650dbf564da89ef23fcdf01b06980c6d43
IEDL.DBID	TFW
ISICitedReferencesCount	3
ISICitedReferencesURI	http://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=Summon&SrcAuth=ProQuest&DestLinkType=CitingArticles&DestApp=WOS_CPL&KeyUT=000299732500022&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D
ISSN	0092-7872
IngestDate	Fri Nov 07 13:47:31 EST 2025 Sat Nov 29 02:27:42 EST 2025 Tue Nov 18 22:21:07 EST 2025 Mon Oct 20 23:40:28 EDT 2025
IsDoiOpenAccess	true
IsOpenAccess	true
IsPeerReviewed	true
IsScholarly	true
Issue	11
Language	English
LinkModel	DirectLink
MergedId	FETCHMERGED-LOGICAL-c395t-34360df2d5ce721e095481a19913b1a650dbf564da89ef23fcdf01b06980c6d43
OpenAccessLink	https://recercat.cat/handle/2072/185431
PageCount	15
ParticipantIDs	crossref_primary_10_1080_00927872_2010_521934 informaworld_taylorfrancis_310_1080_00927872_2010_521934 csuc_recercat_oai_recercat_cat_2072_185431 crossref_citationtrail_10_1080_00927872_2010_521934
PublicationCentury	2000
PublicationDate	2011-11-00
PublicationDateYYYYMMDD	2011-11-01
PublicationDate_xml	– month: 11 year: 2011 text: 2011-11-00
PublicationDecade	2010
PublicationTitle	Communications in algebra
PublicationYear	2011
Publisher	Taylor & Francis Group
Publisher_xml	– name: Taylor & Francis Group
References	Favre C. (CIT0004) 2004 Casas-Alvero E. (CIT0001) 2000 Merle M. (CIT0006) 1977; 41 Casas-Alvero E. (CIT0002) 2007; 11 Perron O. (CIT0003) 1954 Wall C. T. C. (CIT0007) 2004 Płoski A. (CIT0005) 1985; 33
References_xml	– volume: 33 start-page: 601 year: 1985 ident: CIT0005 publication-title: Bulletin of the Polish Academy of Sciences Mathematics. – volume-title: Die Lehre von den Kettenbrüchen. Bd I. Elementare Kettenbrüche year: 1954 ident: CIT0003 – volume: 41 start-page: 103 year: 1977 ident: CIT0006 publication-title: Invent. Math. doi: 10.1007/BF01418370 – volume-title: Singular Points of Plane Curves year: 2004 ident: CIT0007 doi: 10.1017/CBO9780511617560 – volume-title: The Valuative Tree year: 2004 ident: CIT0004 doi: 10.1007/b100262 – volume-title: Singularities of Plane Curves year: 2000 ident: CIT0001 doi: 10.1017/CBO9780511569326 – volume: 11 start-page: 373 year: 2007 ident: CIT0002 publication-title: Asian J. Math. doi: 10.4310/AJM.2007.v11.n3.a3
SSID	ssj0015782
Score	1.9253411
Snippet	We study properties of the space of irreducible germs of plane curves (branches), seen as an ultrametric space. We provide various geometrical methods to...
SourceID	csuc crossref informaworld
SourceType	Open Access Repository Enrichment Source Index Database Publisher
StartPage	4206
SubjectTerms	14 Algebraic geometry 14H Curves 32 Several complex variables and analytic spaces 32S Singularities Classificació AMS Corbes planes Curves, Plane Equisingularity class Matemàtiques i estadística Plane branch Primary 14H20 Secondary 32S05, 14C17 Singularitats (Matemàtica) Ultrametric distance Àrees temàtiques de la UPC
Title	The Ultrametric Space of Plane Branches
URI	https://www.tandfonline.com/doi/abs/10.1080/00927872.2010.521934 https://recercat.cat/handle/2072/185431
Volume	39
WOSCitedRecordID	wos000299732500022&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D
hasFullText	1
inHoldings	1
isFullTextHit
isPrint
journalDatabaseRights	– providerCode: PRVAWR databaseName: Taylor & Francis Journals customDbUrl: eissn: 1532-4125 dateEnd: 99991231 omitProxy: false ssIdentifier: ssj0015782 issn: 0092-7872 databaseCode: TFW dateStart: 19740101 isFulltext: true titleUrlDefault: https://www.tandfonline.com providerName: Taylor & Francis
link	http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwpV1LS8RADA6yeNCDb3F90YMgCMVpp8-jiosHXQR3dW9Dm5lBQVbZVn-_SWe3rIIKeii00LQlkzZf0uQLwJGQYY6YST9MTelHhbV-VmQ57UUxJiVa3aQu7q_Tfj8bjfLbuS5-LqvkGNo6oojmW80vd1FWs4q4U-YJIjsLXWEW-Z9cMiEoAXs28UHvof2NwFztjoaSYCRJzHrnvrnIJ9_UweoNvxCYzrme3ur_H3oNVqaw0ztzdrIOC2a8Acs3LWdrtQnHZDHe8Lnmai2m7ffuKJw23ov1eLCR8c55BMejqbZg2LscXFz50zEKPso8rn0ZyURoG-oYDcV7RjDHW1BwzZMsg4Igmi5tnESa1sjYUFrUVgSlSPJMYKIjuQ2d8cvY7IAnBRKcLDQi4ag40OTKjDWRzlKT6jSIuiBnClQ45RjnURfPKmipSJ0aFKtBOTV0wW-lXh3Hxi_nn_DaKHIJZoJFrZgiuz3gLRQkQUiE0FEXsvkVVHWTCLFuaomSP91n9--ie7DUJJ-bpsV96NSTN3MAi_heP1WTw8ZIPwAuTt9g
linkProvider	Taylor & Francis
linkToHtml	http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwrV1LS8NAEB60CurBt1ifOQiCENxk8zyqWCq2RbDV3pZkHyhIlSb19zuTpKEKKoiHQAKZJMxOMt9Ovv0G4IRxN5Yy4rYb6tT2EmPsKIli3PN8GaTSqKJ08dAJe71oOIzvKjZhVtEqaQ5tSqGI4ltNLzcVo6eUuHMSCsJAc0tmFiagmHvzsOBjqiVWX7_1WP9IILX2UogSgSSaTFfPfXOVT9mpIbOJ_CJhOpN8Wmv_8NjrsFohT-uiDJUNmNOjTVjp1rKt2RacYtBYg5ecCFuk3G_d44xaW6_Got5G2rqkLhxPOtuGQeu6f9W2q04KtuSxn9vc4wFTxlW-1Djl04xk3pyEaE88dRJEaSo1fuApHCZtXG6kMsxJWRBHTAbK4zvQGL2O9C5YnElElImSEqGU7yjMZtpoT0WhDlXoeE3gUw8KWcmMU7eLF-HUaqSlGwS5QZRuaIJdW72VMhu_nH9GgyMwK-ixTHJBKtn1AW0uQwsEIwiQmhDNDqHIi1qIKRuXCP7Tffb-bnoMS-1-tyM6N73bfVguatHFGsYDaOTjiT6ERfmeP2fjoyJiPwBa1OOB
linkToPdf	http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwpV1ZS8NAEB60iuiDt1jPPAiCENxkN9ejV1GspWCrvi3JHiiUtjSpv9_ZbBuqoII-BBLIJGF2kvlmMvMNwAmhfiJETF0_UpnLUq3dOI0T3GOBCDOhZZm6eGpGrVb88pK0Z7r4TVmliaG1JYoov9Xm5R5KPa2IOzc8QWhnvi3MQv-TUDYPCyU3Flp0p_Fc_UcwZO2WhxJxJIpMm-e-ucon51QT-Vh8YTCd8T2Ntf8_9TqsTnCnc2ENZQPmVH8TVh4q0tZ8C07RZJxurzDlWoa333nEeFo5A-2YyUbKuTQzOF5Vvg3dxk3n6tadzFFwBU2CwqWMhkRqXwZCYcCniCF581JT9EQzL0WMJjMdhEziIintUy2kJl5GwiQmIpSM7kCtP-irXXAoEYgnUykEAqnAk-jLlFZMxpGKZOSxOtCpArmYkIybWRc97lVcpFYN3KiBWzXUwa2khpZk45fzz8zacPQJaiTSghuO7OrAbD5BCYQiCI_qEM-uIC_KTIi2Y0s4_ek-e38XPYal9nWDN-9a9_uwXCaiywbGA6gVo7E6hEXxXrzlo6PSXj8AjdjiJQ
openUrl	ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=The+Ultrametric+Space+of+Plane+Branches&rft.jtitle=Communications+in+algebra&rft.au=Ab%C3%ADo%2C+Ignasi&rft.au=Alberich-Carrami%C3%B1ana%2C+Maria&rft.au=Gonz%C3%A1lez-Alonso%2C+V%C3%ADctor&rft.date=2011-11-01&rft.pub=Taylor+%26+Francis+Group&rft.issn=0092-7872&rft.eissn=1532-4125&rft.volume=39&rft.issue=11&rft.spage=4206&rft.epage=4220&rft_id=info:doi/10.1080%2F00927872.2010.521934&rft.externalDocID=521934
thumbnail_l	http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0092-7872&client=summon
thumbnail_m	http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0092-7872&client=summon
thumbnail_s	http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0092-7872&client=summon