{"title":"On Waring numbers of henselian rings","authors":"Tomasz Kowalczyk, Piotr Miska","doi":"10.1112/mtk.12276","DOIUrl":null,"url":null,"abstract":"<p>Let <span></span><math></math> be a positive integer. Let <span></span><math></math> be a henselian local ring with residue field <span></span><math></math> of <span></span><math></math>th level <span></span><math></math>. We give some upper and lower bounds for the <span></span><math></math>th Waring number <span></span><math></math> in terms of <span></span><math></math> and <span></span><math></math>. In large number of cases, we are able to compute <span></span><math></math>. Similar results for the <span></span><math></math>th Waring number of the total ring of fractions of <span></span><math></math> are obtained. We then provide applications. In particular, we compute <span></span><math></math> and <span></span><math></math> for <span></span><math></math> and any prime <span></span><math></math>.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"70 4","pages":""},"PeriodicalIF":0.8000,"publicationDate":"2024-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematika","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/mtk.12276","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Let be a positive integer. Let be a henselian local ring with residue field of th level . We give some upper and lower bounds for the th Waring number in terms of and . In large number of cases, we are able to compute . Similar results for the th Waring number of the total ring of fractions of are obtained. We then provide applications. In particular, we compute and for and any prime .
期刊介绍：
Mathematika publishes both pure and applied mathematical articles and has done so continuously since its founding by Harold Davenport in the 1950s. The traditional emphasis has been towards the purer side of mathematics but applied mathematics and articles addressing both aspects are equally welcome. The journal is published by the London Mathematical Society, on behalf of its owner University College London, and will continue to publish research papers of the highest mathematical quality.
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