{"title":"有限域上 Diophantine 元组的显式构造","authors":"Seoyoung Kim, Chi Hoi Yip, Semin Yoo","doi":"10.1007/s11139-024-00888-5","DOIUrl":null,"url":null,"abstract":"<p>A Diophantine <i>m</i>-tuple over a finite field <span>\\({\\mathbb F}_q\\)</span> is a set <span>\\(\\{a_1,\\ldots , a_m\\}\\)</span> of <i>m</i> distinct elements in <span>\\(\\mathbb {F}_{q}^{*}\\)</span> such that <span>\\(a_{i}a_{j}+1\\)</span> is a square in <span>\\({\\mathbb F}_q\\)</span> whenever <span>\\(i\\ne j\\)</span>. In this paper, we study <i>M</i>(<i>q</i>), the maximum size of a Diophantine tuple over <span>\\({\\mathbb F}_q\\)</span>, assuming the characteristic of <span>\\({\\mathbb F}_q\\)</span> is fixed and <span>\\(q \\rightarrow \\infty \\)</span>. By explicit constructions, we improve the lower bound on <i>M</i>(<i>q</i>). In particular, this improves a recent result of Dujella and Kazalicki by a multiplicative factor.</p>","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":"14 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Explicit constructions of Diophantine tuples over finite fields\",\"authors\":\"Seoyoung Kim, Chi Hoi Yip, Semin Yoo\",\"doi\":\"10.1007/s11139-024-00888-5\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>A Diophantine <i>m</i>-tuple over a finite field <span>\\\\({\\\\mathbb F}_q\\\\)</span> is a set <span>\\\\(\\\\{a_1,\\\\ldots , a_m\\\\}\\\\)</span> of <i>m</i> distinct elements in <span>\\\\(\\\\mathbb {F}_{q}^{*}\\\\)</span> such that <span>\\\\(a_{i}a_{j}+1\\\\)</span> is a square in <span>\\\\({\\\\mathbb F}_q\\\\)</span> whenever <span>\\\\(i\\\\ne j\\\\)</span>. In this paper, we study <i>M</i>(<i>q</i>), the maximum size of a Diophantine tuple over <span>\\\\({\\\\mathbb F}_q\\\\)</span>, assuming the characteristic of <span>\\\\({\\\\mathbb F}_q\\\\)</span> is fixed and <span>\\\\(q \\\\rightarrow \\\\infty \\\\)</span>. By explicit constructions, we improve the lower bound on <i>M</i>(<i>q</i>). In particular, this improves a recent result of Dujella and Kazalicki by a multiplicative factor.</p>\",\"PeriodicalId\":501430,\"journal\":{\"name\":\"The Ramanujan Journal\",\"volume\":\"14 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-06-29\",\"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/s11139-024-00888-5\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"The Ramanujan Journal","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s11139-024-00888-5","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Explicit constructions of Diophantine tuples over finite fields
A Diophantine m-tuple over a finite field \({\mathbb F}_q\) is a set \(\{a_1,\ldots , a_m\}\) of m distinct elements in \(\mathbb {F}_{q}^{*}\) such that \(a_{i}a_{j}+1\) is a square in \({\mathbb F}_q\) whenever \(i\ne j\). In this paper, we study M(q), the maximum size of a Diophantine tuple over \({\mathbb F}_q\), assuming the characteristic of \({\mathbb F}_q\) is fixed and \(q \rightarrow \infty \). By explicit constructions, we improve the lower bound on M(q). In particular, this improves a recent result of Dujella and Kazalicki by a multiplicative factor.