{"title":"论具有周期性连续分数展开 $$sqrt{boldsymbol{f}}$ 的多项式序列 $$\\boldsymbol{f}$","authors":"G. V. Fedorov","doi":"10.3103/s002713222470013x","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>For each <span>\\(n\\geqslant 3\\)</span>, three nonequivalent polynomials <span>\\(f\\in\\mathbb{Q}[x]\\)</span> of degree <span>\\(n\\)</span> were previously constructed for which <span>\\(\\sqrt{f}\\)</span> has a periodic continued fraction expansion in the field <span>\\(\\mathbb{Q}((x))\\)</span>. In this paper, for each <span>\\(n\\geqslant 5\\)</span>, two new polynomials <span>\\(f\\in K[x]\\)</span> of degree <span>\\(n\\)</span> are found, defined over the field <span>\\(K\\)</span>, <span>\\([K:\\mathbb{Q}]=[(n-1)/2]\\)</span>, for which <span>\\(\\sqrt{f}\\)</span> has a periodic continued fraction expansion in the field <span>\\(K((x))\\)</span>.</p>","PeriodicalId":42963,"journal":{"name":"Moscow University Mathematics Bulletin","volume":null,"pages":null},"PeriodicalIF":0.2000,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the Sequences of Polynomials $$\\\\boldsymbol{f}$$ with a Periodic Continued Fraction Expansion $$\\\\sqrt{\\\\boldsymbol{f}}$$\",\"authors\":\"G. V. Fedorov\",\"doi\":\"10.3103/s002713222470013x\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<h3 data-test=\\\"abstract-sub-heading\\\">Abstract</h3><p>For each <span>\\\\(n\\\\geqslant 3\\\\)</span>, three nonequivalent polynomials <span>\\\\(f\\\\in\\\\mathbb{Q}[x]\\\\)</span> of degree <span>\\\\(n\\\\)</span> were previously constructed for which <span>\\\\(\\\\sqrt{f}\\\\)</span> has a periodic continued fraction expansion in the field <span>\\\\(\\\\mathbb{Q}((x))\\\\)</span>. In this paper, for each <span>\\\\(n\\\\geqslant 5\\\\)</span>, two new polynomials <span>\\\\(f\\\\in K[x]\\\\)</span> of degree <span>\\\\(n\\\\)</span> are found, defined over the field <span>\\\\(K\\\\)</span>, <span>\\\\([K:\\\\mathbb{Q}]=[(n-1)/2]\\\\)</span>, for which <span>\\\\(\\\\sqrt{f}\\\\)</span> has a periodic continued fraction expansion in the field <span>\\\\(K((x))\\\\)</span>.</p>\",\"PeriodicalId\":42963,\"journal\":{\"name\":\"Moscow University Mathematics Bulletin\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.2000,\"publicationDate\":\"2024-07-29\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Moscow University Mathematics Bulletin\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.3103/s002713222470013x\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Moscow University Mathematics Bulletin","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.3103/s002713222470013x","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
AbstractFor each \(n\geqslant 3\), three nonequivalent polynomials \(f\in\mathbb{Q}[x]\) of degree \(n\) previously been constructed for which \(\sqrt{f}\) has a periodic continued fraction expansion in the field \(\mathbb{Q}((x))\).在本文中,对于每一个 \(ngeqslant 5\), 都找到了两个新的度(n)的多项式 \(f\in K[x]\), 定义在 \(K\) 场上,\([K:\mathbb{Q}]=[(n-1)/2]\),对于这些多项式,\(\sqrt{f}\) 在 \(K((x))\) 场中有一个周期性的连续分数展开。
On the Sequences of Polynomials $$\boldsymbol{f}$$ with a Periodic Continued Fraction Expansion $$\sqrt{\boldsymbol{f}}$$
Abstract
For each \(n\geqslant 3\), three nonequivalent polynomials \(f\in\mathbb{Q}[x]\) of degree \(n\) were previously constructed for which \(\sqrt{f}\) has a periodic continued fraction expansion in the field \(\mathbb{Q}((x))\). In this paper, for each \(n\geqslant 5\), two new polynomials \(f\in K[x]\) of degree \(n\) are found, defined over the field \(K\), \([K:\mathbb{Q}]=[(n-1)/2]\), for which \(\sqrt{f}\) has a periodic continued fraction expansion in the field \(K((x))\).
期刊介绍:
Moscow University Mathematics Bulletin is the journal of scientific publications reflecting the most important areas of mathematical studies at Lomonosov Moscow State University. The journal covers research in theory of functions, functional analysis, algebra, geometry, topology, ordinary and partial differential equations, probability theory, stochastic processes, mathematical statistics, optimal control, number theory, mathematical logic, theory of algorithms, discrete mathematics and computational mathematics.