{"title":"立方三折与非退化赫米提三折交点的最大点数","authors":"Mrinmoy Datta , Subrata Manna","doi":"10.1016/j.ffa.2024.102462","DOIUrl":null,"url":null,"abstract":"<div><p>It was conjectured by Edoukou in 2008 that a non-degenerate Hermitian threefold in <span><math><msup><mrow><mi>P</mi></mrow><mrow><mn>4</mn></mrow></msup><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub><mo>)</mo></math></span> has at most <span><math><mi>d</mi><mo>(</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>5</mn></mrow></msup><mo>+</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mo>+</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>+</mo><mn>1</mn></math></span> points in common with a threefold of degree <em>d</em> defined over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub></math></span>. He proved the conjecture for <span><math><mi>d</mi><mo>=</mo><mn>2</mn></math></span>. In this paper, we show that the conjecture is true for <span><math><mi>d</mi><mo>=</mo><mn>3</mn></math></span> and <span><math><mi>q</mi><mo>≥</mo><mn>7</mn></math></span>.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"98 ","pages":"Article 102462"},"PeriodicalIF":1.2000,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Maximum number of points on an intersection of a cubic threefold and a non-degenerate Hermitian threefold\",\"authors\":\"Mrinmoy Datta , Subrata Manna\",\"doi\":\"10.1016/j.ffa.2024.102462\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>It was conjectured by Edoukou in 2008 that a non-degenerate Hermitian threefold in <span><math><msup><mrow><mi>P</mi></mrow><mrow><mn>4</mn></mrow></msup><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub><mo>)</mo></math></span> has at most <span><math><mi>d</mi><mo>(</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>5</mn></mrow></msup><mo>+</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mo>+</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>+</mo><mn>1</mn></math></span> points in common with a threefold of degree <em>d</em> defined over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub></math></span>. He proved the conjecture for <span><math><mi>d</mi><mo>=</mo><mn>2</mn></math></span>. In this paper, we show that the conjecture is true for <span><math><mi>d</mi><mo>=</mo><mn>3</mn></math></span> and <span><math><mi>q</mi><mo>≥</mo><mn>7</mn></math></span>.</p></div>\",\"PeriodicalId\":50446,\"journal\":{\"name\":\"Finite Fields and Their Applications\",\"volume\":\"98 \",\"pages\":\"Article 102462\"},\"PeriodicalIF\":1.2000,\"publicationDate\":\"2024-06-25\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Finite Fields and Their Applications\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S1071579724001011\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Finite Fields and Their Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S1071579724001011","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
Maximum number of points on an intersection of a cubic threefold and a non-degenerate Hermitian threefold
It was conjectured by Edoukou in 2008 that a non-degenerate Hermitian threefold in has at most points in common with a threefold of degree d defined over . He proved the conjecture for . In this paper, we show that the conjecture is true for and .
期刊介绍：
Finite Fields and Their Applications is a peer-reviewed technical journal publishing papers in finite field theory as well as in applications of finite fields. As a result of applications in a wide variety of areas, finite fields are increasingly important in several areas of mathematics, including linear and abstract algebra, number theory and algebraic geometry, as well as in computer science, statistics, information theory, and engineering.
For cohesion, and because so many applications rely on various theoretical properties of finite fields, it is essential that there be a core of high-quality papers on theoretical aspects. In addition, since much of the vitality of the area comes from computational problems, the journal publishes papers on computational aspects of finite fields as well as on algorithms and complexity of finite field-related methods.
The journal also publishes papers in various applications including, but not limited to, algebraic coding theory, cryptology, combinatorial design theory, pseudorandom number generation, and linear recurring sequences. There are other areas of application to be included, but the important point is that finite fields play a nontrivial role in the theory, application, or algorithm.
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