{"title":"Hilbert Series of Generic Ideals in Products of Projective Spaces","authors":"R. Fröberg","doi":"10.1080/10586458.2021.1925999","DOIUrl":null,"url":null,"abstract":"Abstract If , k a field, is a standard graded algebra, then the Hilbert series of R is the formal power series . It is known already since Macaulay which power series are Hilbert series of graded algebras. A much harder question is which series are Hilbert series if we fix the number of generators of I and their degrees, say for ideals , . In some sense “most” ideals with fixed degrees of their generators have the same Hilbert series. There is a conjecture for the Hilbert series of those “generic” ideals, see below. In this article we make a conjecture, and prove it in some cases, in the case of generic ideals of fixed degrees in the coordinate ring of , which might be easier to prove.","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/10586458.2021.1925999","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1080/10586458.2021.1925999","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
Abstract If , k a field, is a standard graded algebra, then the Hilbert series of R is the formal power series . It is known already since Macaulay which power series are Hilbert series of graded algebras. A much harder question is which series are Hilbert series if we fix the number of generators of I and their degrees, say for ideals , . In some sense “most” ideals with fixed degrees of their generators have the same Hilbert series. There is a conjecture for the Hilbert series of those “generic” ideals, see below. In this article we make a conjecture, and prove it in some cases, in the case of generic ideals of fixed degrees in the coordinate ring of , which might be easier to prove.