{"title":"6 × 6 矩阵的帕斯猜想证明","authors":"M.A. Khrystik , A.M. Maksaev","doi":"10.1016/j.laa.2024.10.019","DOIUrl":null,"url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>F</mi><mo>)</mo></math></span> be the algebra of <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> matrices over a field <span><math><mi>F</mi></math></span> and let <span><math><mi>S</mi></math></span> be its generating set (as an <span><math><mi>F</mi></math></span>-algebra). The length of <span><math><mi>S</mi></math></span> is the smallest number <em>k</em> such that <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>F</mi><mo>)</mo></math></span> equals the <span><math><mi>F</mi></math></span>-linear span of all products of the length at most <em>k</em> of matrices from <span><math><mi>S</mi></math></span>. The length of <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>F</mi><mo>)</mo></math></span>, denoted by <span><math><mi>l</mi><mo>(</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>F</mi><mo>)</mo><mo>)</mo></math></span>, is defined to be the maximal length of any of its generating sets. In 1984, Paz conjectured that <span><math><mi>l</mi><mo>(</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>F</mi><mo>)</mo><mo>)</mo><mo>=</mo><mn>2</mn><mi>n</mi><mo>−</mo><mn>2</mn></math></span>, for any field <span><math><mi>F</mi></math></span>. This conjecture has been verified only for <span><math><mi>n</mi><mo>⩽</mo><mn>5</mn></math></span>. In this paper, we prove Paz's conjecture for <span><math><mi>n</mi><mo>=</mo><mn>6</mn></math></span>, meaning that <span><math><mi>l</mi><mo>(</mo><msub><mrow><mi>M</mi></mrow><mrow><mn>6</mn></mrow></msub><mo>(</mo><mi>F</mi><mo>)</mo><mo>)</mo><mo>=</mo><mn>10</mn></math></span>. We also prove that <span><math><mn>12</mn><mo>⩽</mo><mi>l</mi><mo>(</mo><msub><mrow><mi>M</mi></mrow><mrow><mn>7</mn></mrow></msub><mo>(</mo><mi>F</mi><mo>)</mo><mo>)</mo><mo>⩽</mo><mn>13</mn></math></span>.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"704 ","pages":"Pages 249-269"},"PeriodicalIF":1.0000,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A proof of the Paz conjecture for 6 × 6 matrices\",\"authors\":\"M.A. Khrystik , A.M. Maksaev\",\"doi\":\"10.1016/j.laa.2024.10.019\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>Let <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>F</mi><mo>)</mo></math></span> be the algebra of <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> matrices over a field <span><math><mi>F</mi></math></span> and let <span><math><mi>S</mi></math></span> be its generating set (as an <span><math><mi>F</mi></math></span>-algebra). The length of <span><math><mi>S</mi></math></span> is the smallest number <em>k</em> such that <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>F</mi><mo>)</mo></math></span> equals the <span><math><mi>F</mi></math></span>-linear span of all products of the length at most <em>k</em> of matrices from <span><math><mi>S</mi></math></span>. The length of <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>F</mi><mo>)</mo></math></span>, denoted by <span><math><mi>l</mi><mo>(</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>F</mi><mo>)</mo><mo>)</mo></math></span>, is defined to be the maximal length of any of its generating sets. In 1984, Paz conjectured that <span><math><mi>l</mi><mo>(</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>F</mi><mo>)</mo><mo>)</mo><mo>=</mo><mn>2</mn><mi>n</mi><mo>−</mo><mn>2</mn></math></span>, for any field <span><math><mi>F</mi></math></span>. This conjecture has been verified only for <span><math><mi>n</mi><mo>⩽</mo><mn>5</mn></math></span>. In this paper, we prove Paz's conjecture for <span><math><mi>n</mi><mo>=</mo><mn>6</mn></math></span>, meaning that <span><math><mi>l</mi><mo>(</mo><msub><mrow><mi>M</mi></mrow><mrow><mn>6</mn></mrow></msub><mo>(</mo><mi>F</mi><mo>)</mo><mo>)</mo><mo>=</mo><mn>10</mn></math></span>. We also prove that <span><math><mn>12</mn><mo>⩽</mo><mi>l</mi><mo>(</mo><msub><mrow><mi>M</mi></mrow><mrow><mn>7</mn></mrow></msub><mo>(</mo><mi>F</mi><mo>)</mo><mo>)</mo><mo>⩽</mo><mn>13</mn></math></span>.</div></div>\",\"PeriodicalId\":18043,\"journal\":{\"name\":\"Linear Algebra and its Applications\",\"volume\":\"704 \",\"pages\":\"Pages 249-269\"},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2024-10-21\",\"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/S0024379524003975\",\"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":"Linear Algebra and its Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0024379524003975","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
设 Mn(F) 是一个域 F 上 n×n 矩阵的代数，设 S 是它的生成集（作为一个 F 代数）。S 的长度是最小数 k，使得 Mn(F) 等于来自 S 的矩阵的所有长度至多为 k 的乘积的 F 线性跨度。Mn(F) 的长度用 l(Mn(F)) 表示，定义为其任何一个生成集的最大长度。1984 年，帕兹猜想，对于任意域 F，l(Mn(F))=2n-2。在本文中，我们证明了 n=6 时帕斯的猜想，即 l(M6(F))=10。我们还证明了 12⩽l(M7(F))⩽13。
Let be the algebra of matrices over a field and let be its generating set (as an -algebra). The length of is the smallest number k such that equals the -linear span of all products of the length at most k of matrices from . The length of , denoted by , is defined to be the maximal length of any of its generating sets. In 1984, Paz conjectured that , for any field . This conjecture has been verified only for . In this paper, we prove Paz's conjecture for , meaning that . We also prove that .
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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