S 单位外部积的边界

IF 0.9 1区数学 Q2 MATHEMATICS

Algebra & Number Theory Pub Date : 2024-09-19 DOI:10.2140/ant.2024.18.1589

Shabnam Akhtari, Jeffrey D. Vaaler

{"title":"S 单位外部积的边界","authors":"Shabnam Akhtari, Jeffrey D. Vaaler","doi":"10.2140/ant.2024.18.1589","DOIUrl":null,"url":null,"abstract":"<p>We generalize an inequality for the determinant of a real matrix proved by A. Schinzel, to more general exterior products of vectors in Euclidean space. We apply this inequality to the logarithmic embedding of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>S</mi></math>-units contained in a number field <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>k</mi></math>. This leads to a bound for the exterior product of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>S</mi></math>-units expressed as a product of heights. Using a volume formula of P. McMullen we show that our inequality is sharp up to a constant that depends only on the rank of the <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>S</mi></math>-unit group but not on the field <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>k</mi></math>. Our inequality is related to a conjecture of F. Rodriguez Villegas. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"64 1","pages":""},"PeriodicalIF":0.9000,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A bound for the exterior product of S-units\",\"authors\":\"Shabnam Akhtari, Jeffrey D. Vaaler\",\"doi\":\"10.2140/ant.2024.18.1589\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We generalize an inequality for the determinant of a real matrix proved by A. Schinzel, to more general exterior products of vectors in Euclidean space. We apply this inequality to the logarithmic embedding of <math display=\\\"inline\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><mi>S</mi></math>-units contained in a number field <math display=\\\"inline\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><mi>k</mi></math>. This leads to a bound for the exterior product of <math display=\\\"inline\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><mi>S</mi></math>-units expressed as a product of heights. Using a volume formula of P. McMullen we show that our inequality is sharp up to a constant that depends only on the rank of the <math display=\\\"inline\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><mi>S</mi></math>-unit group but not on the field <math display=\\\"inline\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><mi>k</mi></math>. Our inequality is related to a conjecture of F. Rodriguez Villegas. </p>\",\"PeriodicalId\":50828,\"journal\":{\"name\":\"Algebra & Number Theory\",\"volume\":\"64 1\",\"pages\":\"\"},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2024-09-19\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Algebra & Number Theory\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.2140/ant.2024.18.1589\",\"RegionNum\":1,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Algebra & Number Theory","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.2140/ant.2024.18.1589","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

摘要

我们将 A. Schinzel 证明的实矩阵行列式不等式推广到欧几里得空间中更一般的向量外部积。我们将这一不等式应用于包含在数域 k 中的 S 单位的对数嵌入，从而得出以高的乘积表示的 S 单位外部乘积的约束。利用麦克马伦（P. McMullen）的一个体积公式，我们证明了我们的不等式在一个常数以内都是尖锐的，这个常数只取决于 S 单位群的秩，而不取决于域 k。我们的不等式与罗德里格斯-比列加斯（F. Rodriguez Villegas）的一个猜想有关。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文本刊更多论文

A bound for the exterior product of S-units

We generalize an inequality for the determinant of a real matrix proved by A. Schinzel, to more general exterior products of vectors in Euclidean space. We apply this inequality to the logarithmic embedding of $S$ -units contained in a number field $k$ . This leads to a bound for the exterior product of $S$ -units expressed as a product of heights. Using a volume formula of P. McMullen we show that our inequality is sharp up to a constant that depends only on the rank of the $S$ -unit group but not on the field $k$ . Our inequality is related to a conjecture of F. Rodriguez Villegas.

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

来源期刊

Algebra & Number Theory MATHEMATICS-

CiteScore

1.80

自引率

7.70%

发文量

审稿时长

6-12 weeks

期刊介绍： ANT’s inclusive definition of algebra and number theory allows it to print research covering a wide range of subtopics, including algebraic and arithmetic geometry. ANT publishes high-quality articles of interest to a broad readership, at a level surpassing all but the top four or five mathematics journals. It exists in both print and electronic forms. The policies of ANT are set by the editorial board — a group of working mathematicians — rather than by a profit-oriented company, so they will remain friendly to mathematicians'' interests. In particular, they will promote broad dissemination, easy electronic access, and permissive use of content to the greatest extent compatible with survival of the journal. All electronic content becomes free and open access 5 years after publication.