{"title":"Partitions into powers of an algebraic number","authors":"Vítězslav Kala, Mikuláš Zindulka","doi":"10.1007/s11139-024-00845-2","DOIUrl":null,"url":null,"abstract":"<p>We study partitions of complex numbers as sums of non-negative powers of a fixed algebraic number <span>\\(\\beta \\)</span>. We prove that if <span>\\( \\beta \\)</span> is real quadratic, then the number of partitions is always finite if and only if some conjugate of <span>\\(\\beta \\)</span> is larger than 1. Further, we show that for <span>\\(\\beta \\)</span> satisfying a certain condition, the partition function attains all non-negative integers as values.</p>","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"The Ramanujan Journal","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s11139-024-00845-2","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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Abstract
We study partitions of complex numbers as sums of non-negative powers of a fixed algebraic number \(\beta \). We prove that if \( \beta \) is real quadratic, then the number of partitions is always finite if and only if some conjugate of \(\beta \) is larger than 1. Further, we show that for \(\beta \) satisfying a certain condition, the partition function attains all non-negative integers as values.
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