{"title":"Topology and geometry of the general composition of formal power series - towards Fréchet-Lie group-like formalism","authors":"Dawid Bugajewski","doi":"arxiv-2409.09853","DOIUrl":null,"url":null,"abstract":"In this article, we study the properties of the autonomous superposition\noperator on the space of formal power series, including those with nonzero\nconstant term. We prove its continuity and smoothness with respect to the\ntopology of pointwise convergence and a natural Fr\\'echet manifold structure. A\nnecessary and sufficient condition for the left composition inverse of a formal\npower series to exist is provided. We also present some properties of the\nFr\\'echet-Lie group structures on the set of nonunit formal power series.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"8 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Rings and Algebras","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.09853","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 this article, we study the properties of the autonomous superposition
operator on the space of formal power series, including those with nonzero
constant term. We prove its continuity and smoothness with respect to the
topology of pointwise convergence and a natural Fr\'echet manifold structure. A
necessary and sufficient condition for the left composition inverse of a formal
power series to exist is provided. We also present some properties of the
Fr\'echet-Lie group structures on the set of nonunit formal power series.
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