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Article: Rationality of meromorphic functions between real algebraic sets in the plane

TitleRationality of meromorphic functions between real algebraic sets in the plane
Authors
Ng, Tuen-WaiYao, Xiao
Keywordsmeromorphic function
Real algebraic curve
Schwarz function
Issue Date15-Sep-2022
PublisherAmerican Mathematical Society
Citation
Proceedings of the American Mathematical Society, 2023, v. 151, n. 2, p. 623-631 How to Cite?
DOI: http://dx.doi.org/10.1090/proc/16109
Abstract

This paper considers the following problem:
   Under what conditions must a holomorphic mapping f:Cm→Cn sending a real algebraic set A⊂Cm onto another real algebraic set A′⊂Cn be algebraic?
   This is the general problem of the famous ones studied by Poincaré (1883), Tanaka (1962), Alexander (1974), Forstnerič (1989) and Webster (1977). The paper investigates the mentioned-problem in the case m=n=1, i.e., one variable meromorphic functions mapping a planar real algebraic set A to another real algebraic set in the complex plane. By using the theory of Schwarz reflection functions, they show that for certain A, these meromorphic functions must be rational. In particular, when A is the standard unit circle, they obtain a one-dimensional analog of Poincaré, Tanaka and Alexander's rationality results for a (2m−1)-dimensional sphere in Cm when m≥2.


Persistent Identifierhttp://hdl.handle.net/10722/331308
ISSN
0002-9939
2021 Impact Factor: 0.971
2020 SCImago Journal Rankings: 0.968

 

DC FieldValueLanguage
dc.contributor.authorNg, Tuen-Wai-
dc.contributor.authorYao, Xiao-
dc.date.accessioned2023-09-21T06:54:34Z-
dc.date.available2023-09-21T06:54:34Z-
dc.date.issued2022-09-15-
dc.identifier.citationProceedings of the American Mathematical Society, 2023, v. 151, n. 2, p. 623-631-
dc.identifier.issn0002-9939-
dc.identifier.urihttp://hdl.handle.net/10722/331308-
dc.description.abstract<p>This paper considers the following problem:<br>   Under what conditions must a holomorphic mapping f:Cm→Cn sending a real algebraic set A⊂Cm onto another real algebraic set A′⊂Cn be algebraic?<br>   This is the general problem of the famous ones studied by Poincaré (1883), Tanaka (1962), Alexander (1974), Forstnerič (1989) and Webster (1977). The paper investigates the mentioned-problem in the case m=n=1, i.e., one variable meromorphic functions mapping a planar real algebraic set A to another real algebraic set in the complex plane. By using the theory of Schwarz reflection functions, they show that for certain A, these meromorphic functions must be rational. In particular, when A is the standard unit circle, they obtain a one-dimensional analog of Poincaré, Tanaka and Alexander's rationality results for a (2m−1)-dimensional sphere in Cm when m≥2.</p>-
dc.languageeng-
dc.publisherAmerican Mathematical Society-
dc.relation.ispartofProceedings of the American Mathematical Society-
dc.rightsThis work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.-
dc.subjectmeromorphic function-
dc.subjectReal algebraic curve-
dc.subjectSchwarz function-
dc.titleRationality of meromorphic functions between real algebraic sets in the plane-
dc.typeArticle-
dc.identifier.doi10.1090/proc/16109-
dc.identifier.scopuseid_2-s2.0-85144812337-
dc.identifier.volume151-
dc.identifier.issue2-
dc.identifier.spage623-
dc.identifier.epage631-
dc.identifier.eissn1088-6826-
dc.identifier.issnl0002-9939-

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