{"title":"Homogeneous Khovanskii bases and MUVAK bases","authors":"Johannes Schmitt","doi":"arxiv-2409.01146","DOIUrl":null,"url":null,"abstract":"In 2019, Kaveh and Manon introduced Khovanskii bases as a special\n'Gr\\\"obner-like' generating system of an algebra. We extend their work by\nconsidering an arbitrary grading on the algebra and propose a definition for a\n'homogeneous Khovanskii basis' that respects this grading. We generalize\nKhovanskii bases further by taking multiple valuations into account (MUVAK\nbases). We give algorithms in both cases. MUVAK bases appear in the computation of the Cox ring of a minimal model of a\nquotient singularity. Our algorithm is an improvement of an algorithm by\nYamagishi in this situation.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":"8 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Commutative Algebra","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.01146","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
In 2019, Kaveh and Manon introduced Khovanskii bases as a special
'Gr\"obner-like' generating system of an algebra. We extend their work by
considering an arbitrary grading on the algebra and propose a definition for a
'homogeneous Khovanskii basis' that respects this grading. We generalize
Khovanskii bases further by taking multiple valuations into account (MUVAK
bases). We give algorithms in both cases. MUVAK bases appear in the computation of the Cox ring of a minimal model of a
quotient singularity. Our algorithm is an improvement of an algorithm by
Yamagishi in this situation.
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