Construction of diagonal quintic threefolds with infinitely many rational points

IF 2.2 2区数学 Q1 MATHEMATICS, APPLIED

Mathematics of Computation Pub Date : 2024-02-07 DOI:10.1090/mcom/3953

Maciej Ulas

{"title":"Construction of diagonal quintic threefolds with infinitely many rational points","authors":"Maciej Ulas","doi":"10.1090/mcom/3953","DOIUrl":null,"url":null,"abstract":"<p>In this note we present a construction of an infinite family of diagonal quintic threefolds defined over <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper Q\"> <mml:semantics> <mml:mrow> <mml:mi mathvariant=\"double-struck\">Q</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\mathbb {Q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> each containing infinitely many rational points. As an application, we prove that there are infinitely many quadruples <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper B equals left-parenthesis upper B 0 comma upper B 1 comma upper B 2 comma upper B 3 right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mi>B</mml:mi> <mml:mo>=</mml:mo> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msub> <mml:mi>B</mml:mi> <mml:mrow> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>B</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>B</mml:mi> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>B</mml:mi> <mml:mrow> <mml:mn>3</mml:mn> </mml:mrow> </mml:msub> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">B=(B_{0}, B_{1}, B_{2}, B_{3})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of co-prime integers such that for a suitable chosen integer <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"b\"> <mml:semantics> <mml:mi>b</mml:mi> <mml:annotation encoding=\"application/x-tex\">b</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (depending on <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper B\"> <mml:semantics> <mml:mi>B</mml:mi> <mml:annotation encoding=\"application/x-tex\">B</mml:annotation> </mml:semantics> </mml:math> </inline-formula>), the equation <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper B 0 upper X 0 Superscript 5 Baseline plus upper B 1 upper X 1 Superscript 5 Baseline plus upper B 2 upper X 2 Superscript 5 Baseline plus upper B 3 upper X 3 Superscript 5 Baseline equals b\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>B</mml:mi> <mml:mrow> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> <mml:msubsup> <mml:mi>X</mml:mi> <mml:mrow> <mml:mn>0</mml:mn> </mml:mrow> <mml:mn>5</mml:mn> </mml:msubsup> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>B</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:msubsup> <mml:mi>X</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> <mml:mn>5</mml:mn> </mml:msubsup> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>B</mml:mi> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> <mml:msubsup> <mml:mi>X</mml:mi> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> <mml:mn>5</mml:mn> </mml:msubsup> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>B</mml:mi> <mml:mrow> <mml:mn>3</mml:mn> </mml:mrow> </mml:msub> <mml:msubsup> <mml:mi>X</mml:mi> <mml:mrow> <mml:mn>3</mml:mn> </mml:mrow> <mml:mrow> <mml:mn>5</mml:mn> </mml:mrow> </mml:msubsup> <mml:mo>=</mml:mo> <mml:mi>b</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">B_{0}X_{0}^5+B_{1}X_{1}^5+B_{2}X_{2}^5+B_{3}X_{3}^{5}=b</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has infinitely many positive integer solutions.</p>","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"147 1","pages":""},"PeriodicalIF":2.2000,"publicationDate":"2024-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematics of Computation","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/mcom/3953","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}

引用次数: 0

Abstract

In this note we present a construction of an infinite family of diagonal quintic threefolds defined over Q \mathbb {Q} each containing infinitely many rational points. As an application, we prove that there are infinitely many quadruples B = ( B 0 , B 1 , B 2 , B 3 ) B=(B_{0}, B_{1}, B_{2}, B_{3}) of co-prime integers such that for a suitable chosen integer b b (depending on B B ), the equation B 0 X 0 5 + B 1 X 1 5 + B 2 X 2 5 + B 3 X 3 5 = b B_{0}X_{0}^5+B_{1}X_{1}^5+B_{2}X_{2}^5+B_{3}X_{3}^{5}=b has infinitely many positive integer solutions.

查看原文本刊更多论文

构建具有无穷多个有理点的对角五元三次方

在本注释中，我们提出了一个定义在 Q \mathbb {Q} 上的对角五元三次方的无穷族的构造，每个对角五元三次方都包含无穷多个有理点。作为应用，我们证明存在无穷多个四元数 B = ( B 0 , B 1 , B 2 , B 3 ) B=(B_{0}, B_{1}, B_{2}, B_{3})，对于一个合适的选定整数 b b (取决于 B B )、方程 B 0 X 0 5 + B 1 X 1 5 + B 2 X 2 5 + B 3 X 3 5 = b B_{0}X_{0}^5+B_{1}X_{1}^5+B_{2}X_{2}^5+B_{3}X_{3}^{5}=b 有无穷多个正整数解。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Mathematics of Computation 数学-应用数学

CiteScore

3.90

自引率

5.00%

发文量

审稿时长

7.0 months

期刊介绍： All articles submitted to this journal are peer-reviewed. The AMS has a single blind peer-review process in which the reviewers know who the authors of the manuscript are, but the authors do not have access to the information on who the peer reviewers are. This journal is devoted to research articles of the highest quality in computational mathematics. Areas covered include numerical analysis, computational discrete mathematics, including number theory, algebra and combinatorics, and related fields such as stochastic numerical methods. Articles must be of significant computational interest and contain original and substantial mathematical analysis or development of computational methodology.