On a quaternionic Picard theorem

Proceedings of the American Mathematical Society, Series B Pub Date : 2020-06-27 DOI:10.1090/bproc/54

C. Bisi, J. Winkelmann

{"title":"On a quaternionic Picard theorem","authors":"C. Bisi, J. Winkelmann","doi":"10.1090/bproc/54","DOIUrl":null,"url":null,"abstract":"<p>The classical theorem of Picard states that a non-constant holomorphic function <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"f colon double-struck upper C right-arrow double-struck upper C\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>f</mml:mi>\n <mml:mo>:</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">C</mml:mi>\n </mml:mrow>\n <mml:mo stretchy=\"false\">→</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">C</mml:mi>\n </mml:mrow>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">f:\\mathbb {C}\\to \\mathbb {C}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> can avoid at most one value.</p>\n\n<p>We investigate how many values a non-constant slice regular function of a quaternionic variable <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"f colon double-struck upper H right-arrow double-struck upper H\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>f</mml:mi>\n <mml:mo>:</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">H</mml:mi>\n </mml:mrow>\n <mml:mo stretchy=\"false\">→</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">H</mml:mi>\n </mml:mrow>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">f:\\mathbb {H}\\to \\mathbb {H}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> may avoid.</p>","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"58 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2020-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"8","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the American Mathematical Society, Series B","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/bproc/54","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}