{"title":"构建具有无穷多个有理点的对角五元三次方","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":"{\"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}","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
摘要
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。
Construction of diagonal quintic threefolds with infinitely many rational points
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=(B0,B1,B2,B3)B=(B_{0}, B_{1}, B_{2}, B_{3}) of co-prime integers such that for a suitable chosen integer bb (depending on BB), the equation B0X05+B1X15+B2X25+B3X35=bB_{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.
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