关于平面三元分环多项式的评述

Pub Date : 2021-12-23 DOI:10.3336/gm.56.2.03

Bin Zhang

{"title":"关于平面三元分环多项式的评述","authors":"Bin Zhang","doi":"10.3336/gm.56.2.03","DOIUrl":null,"url":null,"abstract":"Let \\(\\Phi_n(x)\\) be the \\(n\\)-th cyclotomic polynomial. In this paper, for odd primes \\(p\\lt q \\lt r\\) with \\(q\\equiv \\pm1\\pmod p\\) and \\(8r\\equiv \\pm1\\pmod {pq}\\), we prove that the coefficients of \\(\\Phi_{pqr}(x)\\) do not exceed \\(1\\) in modulus if and only if (i) \\(p=3\\), \\(q\\geq 19\\) and \\(q\\equiv 1\\pmod 3\\) or (ii) \\(p=7\\), \\(q\\geq83\\) and \\(q\\equiv -1\\pmod 7\\).","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"A remark on flat ternary cyclotomic polynomials\",\"authors\":\"Bin Zhang\",\"doi\":\"10.3336/gm.56.2.03\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let \\\\(\\\\Phi_n(x)\\\\) be the \\\\(n\\\\)-th cyclotomic polynomial. In this paper, for odd primes \\\\(p\\\\lt q \\\\lt r\\\\) with \\\\(q\\\\equiv \\\\pm1\\\\pmod p\\\\) and \\\\(8r\\\\equiv \\\\pm1\\\\pmod {pq}\\\\), we prove that the coefficients of \\\\(\\\\Phi_{pqr}(x)\\\\) do not exceed \\\\(1\\\\) in modulus if and only if (i) \\\\(p=3\\\\), \\\\(q\\\\geq 19\\\\) and \\\\(q\\\\equiv 1\\\\pmod 3\\\\) or (ii) \\\\(p=7\\\\), \\\\(q\\\\geq83\\\\) and \\\\(q\\\\equiv -1\\\\pmod 7\\\\).\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2021-12-23\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.3336/gm.56.2.03\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.3336/gm.56.2.03","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}

引用次数: 1

摘要

设\(\Phi_n(x)\)为\(n\) -第一个分圈多项式。对于\(q\equiv \pm1\pmod p\)和\(8r\equiv \pm1\pmod {pq}\)的奇素数\(p\lt q \lt r\)，证明了\(\Phi_{pqr}(x)\)的系数模不超过\(1\)当且仅当(i) \(p=3\)、\(q\geq 19\)和\(q\equiv 1\pmod 3\)或(ii) \(p=7\)、\(q\geq83\)和\(q\equiv -1\pmod 7\)。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文

A remark on flat ternary cyclotomic polynomials

Let \(\Phi_n(x)\) be the \(n\)-th cyclotomic polynomial. In this paper, for odd primes \(p\lt q \lt r\) with \(q\equiv \pm1\pmod p\) and \(8r\equiv \pm1\pmod {pq}\), we prove that the coefficients of \(\Phi_{pqr}(x)\) do not exceed \(1\) in modulus if and only if (i) \(p=3\), \(q\geq 19\) and \(q\equiv 1\pmod 3\) or (ii) \(p=7\), \(q\geq83\) and \(q\equiv -1\pmod 7\).

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助