升级Castelnuovo-Mumford规则和Gröbner基地

IF 1.1 4区数学 Q4 COMPUTER SCIENCE, THEORY & METHODS

Journal of Symbolic Computation Pub Date : 2025-08-19 DOI:10.1016/j.jsc.2025.102487

Matías Bender , Laurent Busé , Carles Checa , Elias Tsigaridas

{"title":"升级Castelnuovo-Mumford规则和Gröbner基地","authors":"Matías Bender , Laurent Busé , Carles Checa , Elias Tsigaridas","doi":"10.1016/j.jsc.2025.102487","DOIUrl":null,"url":null,"abstract":"<div><div>We study the relation between the bigraded Castelnuovo-Mumford regularity of a bihomogeneous ideal <em>I</em> in the coordinate ring of the product of two projective spaces and the bidegrees of a Gröbner basis of <em>I</em> with respect to the degree reverse lexicographical monomial order in generic coordinates. For the single-graded case, Bayer and Stillman unraveled all aspects of this relationship forty years ago and these results led to complexity estimates for computations with Gröbner bases. We build on this work to introduce a bounding region of the bidegrees of minimal generators of bihomogeneous Gröbner bases for <em>I</em>. We also use this region to certify the presence of some minimal generators close to its boundary. Finally, we show that, up to a certain shift, this region is related to the bigraded Castelnuovo-Mumford regularity of <em>I</em>.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"133 ","pages":"Article 102487"},"PeriodicalIF":1.1000,"publicationDate":"2025-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Bigraded Castelnuovo-Mumford regularity and Gröbner bases\",\"authors\":\"Matías Bender , Laurent Busé , Carles Checa , Elias Tsigaridas\",\"doi\":\"10.1016/j.jsc.2025.102487\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>We study the relation between the bigraded Castelnuovo-Mumford regularity of a bihomogeneous ideal <em>I</em> in the coordinate ring of the product of two projective spaces and the bidegrees of a Gröbner basis of <em>I</em> with respect to the degree reverse lexicographical monomial order in generic coordinates. For the single-graded case, Bayer and Stillman unraveled all aspects of this relationship forty years ago and these results led to complexity estimates for computations with Gröbner bases. We build on this work to introduce a bounding region of the bidegrees of minimal generators of bihomogeneous Gröbner bases for <em>I</em>. We also use this region to certify the presence of some minimal generators close to its boundary. Finally, we show that, up to a certain shift, this region is related to the bigraded Castelnuovo-Mumford regularity of <em>I</em>.</div></div>\",\"PeriodicalId\":50031,\"journal\":{\"name\":\"Journal of Symbolic Computation\",\"volume\":\"133 \",\"pages\":\"Article 102487\"},\"PeriodicalIF\":1.1000,\"publicationDate\":\"2025-08-19\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Symbolic Computation\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0747717125000690\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"COMPUTER SCIENCE, THEORY & METHODS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Symbolic Computation","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0747717125000690","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}

引用次数: 0

摘要

研究了两个射影空间积的坐标环上的双齐次理想I的重阶Castelnuovo-Mumford正则性与I的Gröbner基的双阶在一般坐标下关于逆字典单阶阶的关系。对于单一等级的情况，Bayer和Stillman在40年前就揭示了这种关系的所有方面，这些结果导致了Gröbner基计算的复杂性估计。在此基础上，我们引入了i的双齐次Gröbner基的最小发生器的度的边界区域，并利用该区域证明了在其边界附近存在一些最小发生器。最后，我们发现，在一定的位移范围内，该区域与I的渐变Castelnuovo-Mumford正则性有关。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文本刊更多论文

Bigraded Castelnuovo-Mumford regularity and Gröbner bases

We study the relation between the bigraded Castelnuovo-Mumford regularity of a bihomogeneous ideal I in the coordinate ring of the product of two projective spaces and the bidegrees of a Gröbner basis of I with respect to the degree reverse lexicographical monomial order in generic coordinates. For the single-graded case, Bayer and Stillman unraveled all aspects of this relationship forty years ago and these results led to complexity estimates for computations with Gröbner bases. We build on this work to introduce a bounding region of the bidegrees of minimal generators of bihomogeneous Gröbner bases for I. We also use this region to certify the presence of some minimal generators close to its boundary. Finally, we show that, up to a certain shift, this region is related to the bigraded Castelnuovo-Mumford regularity of I.

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助

来源期刊

Journal of Symbolic Computation 工程技术-计算机：理论方法

CiteScore

2.10

自引率

14.30%

发文量

审稿时长

142 days

期刊介绍： An international journal, the Journal of Symbolic Computation, founded by Bruno Buchberger in 1985, is directed to mathematicians and computer scientists who have a particular interest in symbolic computation. The journal provides a forum for research in the algorithmic treatment of all types of symbolic objects: objects in formal languages (terms, formulas, programs); algebraic objects (elements in basic number domains, polynomials, residue classes, etc.); and geometrical objects. It is the explicit goal of the journal to promote the integration of symbolic computation by establishing one common avenue of communication for researchers working in the different subareas. It is also important that the algorithmic achievements of these areas should be made available to the human problem-solver in integrated software systems for symbolic computation. To help this integration, the journal publishes invited tutorial surveys as well as Applications Letters and System Descriptions.