{"title":"二元格氏基的 m-adic 算法","authors":"Éric Schost , Catherine St-Pierre","doi":"10.1016/j.jsc.2024.102389","DOIUrl":null,"url":null,"abstract":"<div><div>Let <span><math><mi>A</mi></math></span> be a domain, with <span><math><mi>m</mi><mo>⊆</mo><mi>A</mi></math></span> a maximal ideal, and let <span><math><mi>F</mi><mo>⊆</mo><mi>A</mi><mo>[</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>]</mo></math></span> be any finite generating set of an ideal with finitely many roots (in an algebraic closure of the fraction field <span><math><mi>K</mi></math></span> of <span><math><mi>A</mi></math></span>). We present a randomized <span><math><mi>m</mi></math></span>-adic algorithm to recover the lexicographic Gröbner basis <span><math><mi>G</mi></math></span> of <span><math><mo>〈</mo><mi>F</mi><mo>〉</mo><mo>⊆</mo><mi>K</mi><mo>[</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>]</mo></math></span>, or of its primary component at the origin. We observe that previous results of Lazard's that use Hermite normal forms to compute Gröbner bases for ideals with two generators can be generalized to a generating set <span><math><mi>F</mi></math></span> of cardinality greater than two. We use this result to bound the size of the coefficients of <span><math><mi>G</mi></math></span>, and to control the probability of choosing a <em>good</em> maximal ideal <span><math><mi>m</mi><mo>⊆</mo><mi>A</mi></math></span>. We give a complete cost analysis over number fields (<span><math><mi>K</mi><mo>=</mo><mi>Q</mi><mo>(</mo><mi>α</mi><mo>)</mo></math></span>) and function fields (<figure><img></figure>), and we obtain a complexity that is less than cubic in terms of the dimension of <span><math><mi>K</mi><mo>/</mo><mo>〈</mo><mi>G</mi><mo>〉</mo></math></span> and softly linear in the size of its coefficients.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2024-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"An m-adic algorithm for bivariate Gröbner bases\",\"authors\":\"Éric Schost , Catherine St-Pierre\",\"doi\":\"10.1016/j.jsc.2024.102389\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>Let <span><math><mi>A</mi></math></span> be a domain, with <span><math><mi>m</mi><mo>⊆</mo><mi>A</mi></math></span> a maximal ideal, and let <span><math><mi>F</mi><mo>⊆</mo><mi>A</mi><mo>[</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>]</mo></math></span> be any finite generating set of an ideal with finitely many roots (in an algebraic closure of the fraction field <span><math><mi>K</mi></math></span> of <span><math><mi>A</mi></math></span>). We present a randomized <span><math><mi>m</mi></math></span>-adic algorithm to recover the lexicographic Gröbner basis <span><math><mi>G</mi></math></span> of <span><math><mo>〈</mo><mi>F</mi><mo>〉</mo><mo>⊆</mo><mi>K</mi><mo>[</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>]</mo></math></span>, or of its primary component at the origin. We observe that previous results of Lazard's that use Hermite normal forms to compute Gröbner bases for ideals with two generators can be generalized to a generating set <span><math><mi>F</mi></math></span> of cardinality greater than two. We use this result to bound the size of the coefficients of <span><math><mi>G</mi></math></span>, and to control the probability of choosing a <em>good</em> maximal ideal <span><math><mi>m</mi><mo>⊆</mo><mi>A</mi></math></span>. We give a complete cost analysis over number fields (<span><math><mi>K</mi><mo>=</mo><mi>Q</mi><mo>(</mo><mi>α</mi><mo>)</mo></math></span>) and function fields (<figure><img></figure>), and we obtain a complexity that is less than cubic in terms of the dimension of <span><math><mi>K</mi><mo>/</mo><mo>〈</mo><mi>G</mi><mo>〉</mo></math></span> and softly linear in the size of its coefficients.</div></div>\",\"PeriodicalId\":50031,\"journal\":{\"name\":\"Journal of Symbolic Computation\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2024-09-26\",\"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/S0747717124000932\",\"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/S0747717124000932","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
设 A 是一个域,m⊆A 是一个最大理想,设 F⊆A[x,y]是具有有限多个根(在 A 的分数域 K 的代数闭包中)的理想的任意有限生成集。我们提出了一种随机 m-adic 算法来恢复〈F〉⊆K[x,y]的词典格罗伯纳基 G 或其在原点的主成分。我们注意到,拉扎德之前利用赫米特正则表达式计算有两个生成子的理想的格罗伯纳基的结果,可以推广到心数大于两个的生成集 F。我们利用这一结果来约束 G 的系数大小,并控制选择一个好的最大理想 m⊆A 的概率。我们对数域(K=Q(α))和函数域()进行了完整的代价分析,得到的复杂度小于 K/〈G〉维数的立方,与其系数的大小呈软线性关系。
Let be a domain, with a maximal ideal, and let be any finite generating set of an ideal with finitely many roots (in an algebraic closure of the fraction field of ). We present a randomized -adic algorithm to recover the lexicographic Gröbner basis of , or of its primary component at the origin. We observe that previous results of Lazard's that use Hermite normal forms to compute Gröbner bases for ideals with two generators can be generalized to a generating set of cardinality greater than two. We use this result to bound the size of the coefficients of , and to control the probability of choosing a good maximal ideal . We give a complete cost analysis over number fields () and function fields (), and we obtain a complexity that is less than cubic in terms of the dimension of and softly linear in the size of its coefficients.
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.