{"title":"Multiplicity of powers of squarefree monomial ideals","authors":"Phan Thi Thuy, Thanh Vu","doi":"10.1007/s00013-025-02116-y","DOIUrl":null,"url":null,"abstract":"<div><p>Let <i>I</i> be an arbitrary nonzero squarefree monomial ideal of dimension <i>d</i> in a polynomial ring <span>\\(S = \\textrm{k}[x_1,\\ldots ,x_n]\\)</span>. Let <span>\\(\\mu \\)</span> be the number of associated primes of <i>S</i>/<i>I</i> of dimension <i>d</i>. We prove that the multiplicity of powers of <i>I</i> is given by </p><div><div><span>$$\\begin{aligned} e_0(S/I^s) = \\mu \\left( {\\begin{array}{c}n-d+s-1\\\\ s-1\\end{array}}\\right) \\end{aligned}$$</span></div></div><p>for all <span>\\(s \\ge 1\\)</span>. Consequently, we compute the multiplicity of all powers of path ideals of cycles.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"125 1","pages":"9 - 15"},"PeriodicalIF":0.5000,"publicationDate":"2025-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Archiv der Mathematik","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00013-025-02116-y","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Let I be an arbitrary nonzero squarefree monomial ideal of dimension d in a polynomial ring \(S = \textrm{k}[x_1,\ldots ,x_n]\). Let \(\mu \) be the number of associated primes of S/I of dimension d. We prove that the multiplicity of powers of I is given by
期刊介绍:
Archiv der Mathematik (AdM) publishes short high quality research papers in every area of mathematics which are not overly technical in nature and addressed to a broad readership.