{"title":"Computing Gröbner bases and free resolutions of OI-modules","authors":"Michael Morrow, Uwe Nagel","doi":"10.1016/j.jalgebra.2024.08.009","DOIUrl":null,"url":null,"abstract":"<div><p>Given a sequence of related modules <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> defined over a sequence of related polynomial rings, one may ask how to simultaneously compute a finite Gröbner basis for each <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. Furthermore, one may ask how to simultaneously compute the module of syzygies of each <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. In this paper we address both questions. Working in the setting of OI-modules over a Noetherian polynomial OI-algebra, we provide OI-analogues of Buchberger's Criterion, Buchberger's Algorithm for computing Gröbner bases, and Schreyer's Theorem for computing syzygies. We also establish a stabilization result for Gröbner bases.</p></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Algebra","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0021869324004666","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Given a sequence of related modules defined over a sequence of related polynomial rings, one may ask how to simultaneously compute a finite Gröbner basis for each . Furthermore, one may ask how to simultaneously compute the module of syzygies of each . In this paper we address both questions. Working in the setting of OI-modules over a Noetherian polynomial OI-algebra, we provide OI-analogues of Buchberger's Criterion, Buchberger's Algorithm for computing Gröbner bases, and Schreyer's Theorem for computing syzygies. We also establish a stabilization result for Gröbner bases.
给定一系列定义在一系列相关多项式环上的相关模块 Mn，我们可能会问，如何同时计算每个 Mn 的有限格罗伯纳基。此外，人们还会问如何同时计算每个 Mn 的协同模块。在本文中，我们将解决这两个问题。在诺特多项式 OI 代数上的 OI 模块的背景下，我们提供了布赫伯格准则的 OI 类似方法、计算格罗纳基的布赫伯格算法和计算对称的施雷尔定理。我们还建立了格氏基的稳定结果。
期刊介绍：
The Journal of Algebra is a leading international journal and publishes papers that demonstrate high quality research results in algebra and related computational aspects. Only the very best and most interesting papers are to be considered for publication in the journal. With this in mind, it is important that the contribution offer a substantial result that will have a lasting effect upon the field. The journal also seeks work that presents innovative techniques that offer promising results for future research.
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