{"title":"嵌入变量、x阶和解的唯一性或有限性","authors":"E. Ballico","doi":"10.1007/s13370-023-01133-w","DOIUrl":null,"url":null,"abstract":"<div><p>Let <span>\\(X\\subset \\mathbb {P}^r\\)</span> be an integral and non-degenerate variety. For any <span>\\(q\\in \\mathbb {P}^r\\)</span> its <i>X</i>-rank <span>\\(r_X(q)\\)</span> is the minimal cardinality of a finite subset of <i>X</i> whose linear span contains <i>q</i>. The solution set <span>\\(\\mathcal {S}(X,q)\\)</span> of <span>\\(q\\in \\mathbb {P}^r\\)</span> is the set of all <span>\\(S\\subset X\\)</span> such that <span>\\(\\#S=r_X(q)\\)</span> and <i>S</i> spans <i>q</i>. We prove that if <span>\\(X\\ne \\mathbb {P}^r\\)</span> there is at least one <i>q</i> with <span>\\(\\#\\mathcal {S}(X,q)>1\\)</span> and that for almost all pairs (<i>X</i>, <i>q</i>) we have <span>\\(\\dim \\mathcal {S}(X,q)>0\\)</span>.</p></div>","PeriodicalId":46107,"journal":{"name":"Afrika Matematika","volume":"34 4","pages":""},"PeriodicalIF":0.9000,"publicationDate":"2023-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Embedded varieties, X-ranks and uniqueness or finiteness of the solutions\",\"authors\":\"E. Ballico\",\"doi\":\"10.1007/s13370-023-01133-w\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Let <span>\\\\(X\\\\subset \\\\mathbb {P}^r\\\\)</span> be an integral and non-degenerate variety. For any <span>\\\\(q\\\\in \\\\mathbb {P}^r\\\\)</span> its <i>X</i>-rank <span>\\\\(r_X(q)\\\\)</span> is the minimal cardinality of a finite subset of <i>X</i> whose linear span contains <i>q</i>. The solution set <span>\\\\(\\\\mathcal {S}(X,q)\\\\)</span> of <span>\\\\(q\\\\in \\\\mathbb {P}^r\\\\)</span> is the set of all <span>\\\\(S\\\\subset X\\\\)</span> such that <span>\\\\(\\\\#S=r_X(q)\\\\)</span> and <i>S</i> spans <i>q</i>. We prove that if <span>\\\\(X\\\\ne \\\\mathbb {P}^r\\\\)</span> there is at least one <i>q</i> with <span>\\\\(\\\\#\\\\mathcal {S}(X,q)>1\\\\)</span> and that for almost all pairs (<i>X</i>, <i>q</i>) we have <span>\\\\(\\\\dim \\\\mathcal {S}(X,q)>0\\\\)</span>.</p></div>\",\"PeriodicalId\":46107,\"journal\":{\"name\":\"Afrika Matematika\",\"volume\":\"34 4\",\"pages\":\"\"},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2023-11-14\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Afrika Matematika\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s13370-023-01133-w\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Afrika Matematika","FirstCategoryId":"1085","ListUrlMain":"https://link.springer.com/article/10.1007/s13370-023-01133-w","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
Embedded varieties, X-ranks and uniqueness or finiteness of the solutions
Let \(X\subset \mathbb {P}^r\) be an integral and non-degenerate variety. For any \(q\in \mathbb {P}^r\) its X-rank \(r_X(q)\) is the minimal cardinality of a finite subset of X whose linear span contains q. The solution set \(\mathcal {S}(X,q)\) of \(q\in \mathbb {P}^r\) is the set of all \(S\subset X\) such that \(\#S=r_X(q)\) and S spans q. We prove that if \(X\ne \mathbb {P}^r\) there is at least one q with \(\#\mathcal {S}(X,q)>1\) and that for almost all pairs (X, q) we have \(\dim \mathcal {S}(X,q)>0\).