Paweł Pielasa , Matouš Šafránek , Anatoli Shatsila
{"title":"一般子分支的精确值","authors":"Paweł Pielasa , Matouš Šafránek , Anatoli Shatsila","doi":"10.1016/j.aim.2025.110234","DOIUrl":null,"url":null,"abstract":"<div><div>In this article we prove the subrank of a generic tensor in <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>n</mi><mo>,</mo><mi>n</mi></mrow></msup></math></span> to be <span><math><mi>Q</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mo>⌊</mo><msqrt><mrow><mn>3</mn><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msqrt><mo>⌋</mo></math></span> by providing a lower bound to the known upper bound. More generally, we find the generic subrank of tensors of all orders and dimensions. This answers two open questions posed in <span><span>[5]</span></span>. Finally, we compute dimensions of varieties of tensors of subrank at least <em>r</em>.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"469 ","pages":"Article 110234"},"PeriodicalIF":1.5000,"publicationDate":"2025-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Exact values of generic subrank\",\"authors\":\"Paweł Pielasa , Matouš Šafránek , Anatoli Shatsila\",\"doi\":\"10.1016/j.aim.2025.110234\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>In this article we prove the subrank of a generic tensor in <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>n</mi><mo>,</mo><mi>n</mi></mrow></msup></math></span> to be <span><math><mi>Q</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mo>⌊</mo><msqrt><mrow><mn>3</mn><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msqrt><mo>⌋</mo></math></span> by providing a lower bound to the known upper bound. More generally, we find the generic subrank of tensors of all orders and dimensions. This answers two open questions posed in <span><span>[5]</span></span>. Finally, we compute dimensions of varieties of tensors of subrank at least <em>r</em>.</div></div>\",\"PeriodicalId\":50860,\"journal\":{\"name\":\"Advances in Mathematics\",\"volume\":\"469 \",\"pages\":\"Article 110234\"},\"PeriodicalIF\":1.5000,\"publicationDate\":\"2025-04-02\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Advances in Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S000187082500132X\",\"RegionNum\":1,\"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":"Advances in Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S000187082500132X","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
In this article we prove the subrank of a generic tensor in to be by providing a lower bound to the known upper bound. More generally, we find the generic subrank of tensors of all orders and dimensions. This answers two open questions posed in [5]. Finally, we compute dimensions of varieties of tensors of subrank at least r.
期刊介绍:
Emphasizing contributions that represent significant advances in all areas of pure mathematics, Advances in Mathematics provides research mathematicians with an effective medium for communicating important recent developments in their areas of specialization to colleagues and to scientists in related disciplines.