{"title":"The Collatz map analogue in polynomial rings and in completions","authors":"","doi":"10.1016/j.disc.2024.114273","DOIUrl":null,"url":null,"abstract":"<div><div>We study an analogue of the Collatz map in the polynomial ring <span><math><mi>R</mi><mo>[</mo><mi>x</mi><mo>]</mo></math></span>, where <em>R</em> is an arbitrary commutative ring. We prove that if <em>R</em> is of positive characteristic, then every polynomial in <span><math><mi>R</mi><mo>[</mo><mi>x</mi><mo>]</mo></math></span> is eventually periodic with respect to this map. This extends previous works of the authors and of Hicks, Mullen, Yucas and Zavislak, who studied the Collatz map on <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>[</mo><mi>x</mi><mo>]</mo></math></span> and <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>[</mo><mi>x</mi><mo>]</mo></math></span>, respectively. We also consider the Collatz map on the ring of formal power series <span><math><mi>R</mi><mo>[</mo><mo>[</mo><mi>x</mi><mo>]</mo><mo>]</mo></math></span> when <em>R</em> is finite: we characterize the eventually periodic series in this ring, and give formulas for the number of cycles induced by the Collatz map, of any given length. We provide similar formulas for the original Collatz map defined on the ring <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> of 2-adic integers, extending previous results of Lagarias.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2024-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0012365X24004047","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
We study an analogue of the Collatz map in the polynomial ring , where R is an arbitrary commutative ring. We prove that if R is of positive characteristic, then every polynomial in is eventually periodic with respect to this map. This extends previous works of the authors and of Hicks, Mullen, Yucas and Zavislak, who studied the Collatz map on and , respectively. We also consider the Collatz map on the ring of formal power series when R is finite: we characterize the eventually periodic series in this ring, and give formulas for the number of cycles induced by the Collatz map, of any given length. We provide similar formulas for the original Collatz map defined on the ring of 2-adic integers, extending previous results of Lagarias.
期刊介绍:
Discrete Mathematics provides a common forum for significant research in many areas of discrete mathematics and combinatorics. Among the fields covered by Discrete Mathematics are graph and hypergraph theory, enumeration, coding theory, block designs, the combinatorics of partially ordered sets, extremal set theory, matroid theory, algebraic combinatorics, discrete geometry, matrices, and discrete probability theory.
Items in the journal include research articles (Contributions or Notes, depending on length) and survey/expository articles (Perspectives). Efforts are made to process the submission of Notes (short articles) quickly. The Perspectives section features expository articles accessible to a broad audience that cast new light or present unifying points of view on well-known or insufficiently-known topics.