{"title":"多项式环和完备中的科拉茨图类似物","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":"{\"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}","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
摘要
我们研究了多项式环 R[x] 中的科拉茨映射,其中 R 是任意交换环。我们证明,如果 R 是正特征,那么 R[x] 中的每个多项式最终都是关于这个映射的周期性多项式。这扩展了作者以及希克斯、马伦、尤卡斯和扎维斯拉克之前的工作,他们分别研究了 Fp[x] 和 F2[x] 上的科拉茨映射。我们还考虑了当 R 有限时形式幂级数环 R[[x]] 上的科拉茨映射:我们描述了该环中最终周期数列的特征,并给出了任何给定长度的科拉茨映射诱导的循环数公式。我们为定义在二阶整数环 Z2 上的原始科拉茨映射提供了类似的公式,扩展了拉加里亚斯以前的结果。
The Collatz map analogue in polynomial rings and in completions
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.