{"title":"欧几里得域上多变量多项式环中的强格罗布纳基和线性代数","authors":"Erhard Aichinger","doi":"10.1016/j.exmath.2024.125627","DOIUrl":null,"url":null,"abstract":"<div><div>We provide a self-contained introduction to Gröbner bases of submodules of <span><math><mrow><mi>R</mi><msup><mrow><mrow><mo>[</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>]</mo></mrow></mrow><mrow><mi>k</mi></mrow></msup></mrow></math></span>, where <span><math><mi>R</mi></math></span> is a Euclidean domain, and explain how to use these bases to solve linear systems over <span><math><mrow><mi>R</mi><mrow><mo>[</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>]</mo></mrow></mrow></math></span>.</div></div>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Strong Gröbner bases and linear algebra in multivariate polynomial rings over Euclidean domains\",\"authors\":\"Erhard Aichinger\",\"doi\":\"10.1016/j.exmath.2024.125627\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>We provide a self-contained introduction to Gröbner bases of submodules of <span><math><mrow><mi>R</mi><msup><mrow><mrow><mo>[</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>]</mo></mrow></mrow><mrow><mi>k</mi></mrow></msup></mrow></math></span>, where <span><math><mi>R</mi></math></span> is a Euclidean domain, and explain how to use these bases to solve linear systems over <span><math><mrow><mi>R</mi><mrow><mo>[</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>]</mo></mrow></mrow></math></span>.</div></div>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-10-30\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S072308692400094X\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S072308692400094X","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
我们对 R[x1,...,xn]k 的子模的格洛布纳基（其中 R 是欧几里得域）进行了完整的介绍，并解释了如何使用这些基来求解 R[x1,...,xn] 上的线性系统。
Strong Gröbner bases and linear algebra in multivariate polynomial rings over Euclidean domains
We provide a self-contained introduction to Gröbner bases of submodules of , where is a Euclidean domain, and explain how to use these bases to solve linear systems over .
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