ATTILA BÉRCZES, YANN BUGEAUD, KÁLMÁN GYŐRY, JORGE MELLO, ALINA OSTAFE, MIN SHA
{"title":"Multiplicative dependence of rational values modulo approximate finitely generated groups","authors":"ATTILA BÉRCZES, YANN BUGEAUD, KÁLMÁN GYŐRY, JORGE MELLO, ALINA OSTAFE, MIN SHA","doi":"10.1017/s0305004124000173","DOIUrl":null,"url":null,"abstract":"<p>In this paper, we establish some finiteness results about the multiplicative dependence of rational values modulo sets which are ‘close’ (with respect to the Weil height) to division groups of finitely generated multiplicative groups of a number field <span>K</span>. For example, we show that under some conditions on rational functions <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240918084446080-0836:S0305004124000173:S0305004124000173_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$f_1, \\ldots, f_n\\in K(X)$</span></span></img></span></span>, there are only finitely many elements <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240918084446080-0836:S0305004124000173:S0305004124000173_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$\\alpha \\in K$</span></span></img></span></span> such that <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240918084446080-0836:S0305004124000173:S0305004124000173_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$f_1(\\alpha),\\ldots,f_n(\\alpha)$</span></span></img></span></span> are multiplicatively dependent modulo such sets.</p>","PeriodicalId":18320,"journal":{"name":"Mathematical Proceedings of the Cambridge Philosophical Society","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Proceedings of the Cambridge Philosophical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1017/s0305004124000173","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper, we establish some finiteness results about the multiplicative dependence of rational values modulo sets which are ‘close’ (with respect to the Weil height) to division groups of finitely generated multiplicative groups of a number field K. For example, we show that under some conditions on rational functions $f_1, \ldots, f_n\in K(X)$, there are only finitely many elements $\alpha \in K$ such that $f_1(\alpha),\ldots,f_n(\alpha)$ are multiplicatively dependent modulo such sets.
期刊介绍:
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