{"title":"Upper bounds for the rank of powers of quadrics","authors":"Cosimo Flavi","doi":"10.1016/j.laa.2024.11.009","DOIUrl":null,"url":null,"abstract":"<div><div>We establish an upper bound for the rank of every power of an arbitrary quadratic form. Specifically, for any <span><math><mi>s</mi><mo>∈</mo><mi>N</mi></math></span>, we prove that the <em>s</em>-th power of a quadratic form of rank <em>n</em> grows as <span><math><msup><mrow><mi>n</mi></mrow><mrow><mi>s</mi></mrow></msup></math></span>. Furthermore, we demonstrate that its rank is subgeneric for all <span><math><mi>n</mi><mo>></mo><msup><mrow><mo>(</mo><mn>2</mn><mi>s</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></math></span>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"707 ","pages":"Pages 49-79"},"PeriodicalIF":1.0000,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Linear Algebra and its Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0024379524004294","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
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Abstract
We establish an upper bound for the rank of every power of an arbitrary quadratic form. Specifically, for any , we prove that the s-th power of a quadratic form of rank n grows as . Furthermore, we demonstrate that its rank is subgeneric for all .
期刊介绍：
Linear Algebra and its Applications publishes articles that contribute new information or new insights to matrix theory and finite dimensional linear algebra in their algebraic, arithmetic, combinatorial, geometric, or numerical aspects. It also publishes articles that give significant applications of matrix theory or linear algebra to other branches of mathematics and to other sciences. Articles that provide new information or perspectives on the historical development of matrix theory and linear algebra are also welcome. Expository articles which can serve as an introduction to a subject for workers in related areas and which bring one to the frontiers of research are encouraged. Reviews of books are published occasionally as are conference reports that provide an historical record of major meetings on matrix theory and linear algebra.
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