{"title":"Maximal Generating Degrees of Powers of Homogeneous Ideals","authors":"Le Tuan Hoa","doi":"10.1007/s40306-021-00469-4","DOIUrl":null,"url":null,"abstract":"<div><p>The degree excess function <i>𝜖</i>(<i>I</i>; <i>n</i>) is the difference between the maximal generating degree <i>d</i>(<i>I</i><sup><i>n</i></sup>) of the n-th power of a homogeneous ideal <i>I</i> of a polynomial ring and <i>p</i>(<i>I</i>)<i>n</i>, where <i>p</i>(<i>I</i>) is the leading coefficient of the asymptotically linear function <i>d</i>(<i>I</i><sup><i>n</i></sup>). It is shown that any non-increasing numerical function can be realized as a degree excess function, and there is a monomial ideal <i>I</i> whose <i>𝜖</i>(<i>I</i>; <i>n</i>) has exactly a given number of local maxima. In the case of monomial ideals, an upper bound on <i>𝜖</i>(<i>I</i>; <i>n</i>) is provided. As an application, it is shown that in the worst case, the so-called stability index of the Castelnuovo-Mumford regularity of a monomial ideal <i>I</i> must be at least an exponential function of the number of variables.</p></div>","PeriodicalId":45527,"journal":{"name":"Acta Mathematica Vietnamica","volume":null,"pages":null},"PeriodicalIF":0.3000,"publicationDate":"2022-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Mathematica Vietnamica","FirstCategoryId":"1085","ListUrlMain":"https://link.springer.com/article/10.1007/s40306-021-00469-4","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 1
Abstract
The degree excess function 𝜖(I; n) is the difference between the maximal generating degree d(In) of the n-th power of a homogeneous ideal I of a polynomial ring and p(I)n, where p(I) is the leading coefficient of the asymptotically linear function d(In). It is shown that any non-increasing numerical function can be realized as a degree excess function, and there is a monomial ideal I whose 𝜖(I; n) has exactly a given number of local maxima. In the case of monomial ideals, an upper bound on 𝜖(I; n) is provided. As an application, it is shown that in the worst case, the so-called stability index of the Castelnuovo-Mumford regularity of a monomial ideal I must be at least an exponential function of the number of variables.
期刊介绍:
Acta Mathematica Vietnamica is a peer-reviewed mathematical journal. The journal publishes original papers of high quality in all branches of Mathematics with strong focus on Algebraic Geometry and Commutative Algebra, Algebraic Topology, Complex Analysis, Dynamical Systems, Optimization and Partial Differential Equations.