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Article: Birational morphisms of the plane

TitleBirational morphisms of the plane
Authors
KeywordsAffine Plane
Birational Morphisms
Peak Reduction
Issue Date2004
PublisherAmerican Mathematical Society. The Journal's web site is located at http://www.ams.org/proc
Citation
Proceedings Of The American Mathematical Society, 2004, v. 132 n. 9, p. 2511-2515 How to Cite?
AbstractLet A 2 be the affine plane over a field K of characteristic 0. Birational morphisms of A 2 are mappings A 2 → A 2 given by polynomial mappings φ of the polynomial algebra K[x, y] such that for the quotient fields, one has K(φ(x), φ(y)) = K(x, y). Polynomial automorphisms are obvious examples of such mappings. Another obvious example is the mapping τ x given by x → x, y → xy. For a while, it was an open question whether every birational morphism is a product of polynomial automorphisms and copies of τ x. This question was answered in the negative by P. Russell (in an informal communication). In this paper, we give a simple combinatorial solution of the same problem. More importantly, our method yields an algorithm for deciding whether a given birational morphism can be factored that way.
Persistent Identifierhttp://hdl.handle.net/10722/156212
ISSN
2015 Impact Factor: 0.7
2015 SCImago Journal Rankings: 1.082
ISI Accession Number ID
References

 

DC FieldValueLanguage
dc.contributor.authorShpilrain, Ven_US
dc.contributor.authorYu, JTen_US
dc.date.accessioned2012-08-08T08:40:51Z-
dc.date.available2012-08-08T08:40:51Z-
dc.date.issued2004en_US
dc.identifier.citationProceedings Of The American Mathematical Society, 2004, v. 132 n. 9, p. 2511-2515en_US
dc.identifier.issn0002-9939en_US
dc.identifier.urihttp://hdl.handle.net/10722/156212-
dc.description.abstractLet A 2 be the affine plane over a field K of characteristic 0. Birational morphisms of A 2 are mappings A 2 → A 2 given by polynomial mappings φ of the polynomial algebra K[x, y] such that for the quotient fields, one has K(φ(x), φ(y)) = K(x, y). Polynomial automorphisms are obvious examples of such mappings. Another obvious example is the mapping τ x given by x → x, y → xy. For a while, it was an open question whether every birational morphism is a product of polynomial automorphisms and copies of τ x. This question was answered in the negative by P. Russell (in an informal communication). In this paper, we give a simple combinatorial solution of the same problem. More importantly, our method yields an algorithm for deciding whether a given birational morphism can be factored that way.en_US
dc.languageengen_US
dc.publisherAmerican Mathematical Society. The Journal's web site is located at http://www.ams.org/procen_US
dc.relation.ispartofProceedings of the American Mathematical Societyen_US
dc.rightsCreative Commons: Attribution 3.0 Hong Kong License-
dc.rightsFirst published in [Proceedings of the American Mathematical Society] in [2004, v. 132 n. 9], published by the American Mathematical Society-
dc.subjectAffine Planeen_US
dc.subjectBirational Morphismsen_US
dc.subjectPeak Reductionen_US
dc.titleBirational morphisms of the planeen_US
dc.typeArticleen_US
dc.identifier.emailYu, JT:yujt@hku.hken_US
dc.identifier.authorityYu, JT=rp00834en_US
dc.description.naturepublished_or_final_versionen_US
dc.identifier.doi10.1090/S0002-9939-04-07490-8en_US
dc.identifier.scopuseid_2-s2.0-4344713057en_US
dc.relation.referenceshttp://www.scopus.com/mlt/select.url?eid=2-s2.0-4344713057&selection=ref&src=s&origin=recordpageen_US
dc.identifier.volume132en_US
dc.identifier.issue9en_US
dc.identifier.spage2511en_US
dc.identifier.epage2515en_US
dc.identifier.isiWOS:000222122200003-
dc.publisher.placeUnited Statesen_US
dc.identifier.scopusauthoridShpilrain, V=6603904879en_US
dc.identifier.scopusauthoridYu, JT=7405530208en_US

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