Sbornik: Mathematics
RUS  ENG    JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB  
General information
Latest issue
Forthcoming papers
Archive
Impact factor
Guidelines for authors
License agreement
Submit a manuscript

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Mat. Sb.:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


Sbornik: Mathematics, 2012, Volume 203, Issue 4, Pages 534–553
DOI: https://doi.org/10.1070/SM2012v203n04ABEH004233
(Mi sm7853)
 

This article is cited in 14 scientific papers (total in 14 papers)

Subexponential estimates in Shirshov's theorem on height

A. Ya. Belova, M. I. Kharitonovb

a Moscow Institute of Open Education
b M. V. Lomonosov Moscow State University
References:
Abstract: Suppose that $F_{2,m}$ is a free $2$-generated associative ring with the identity $x^m=0$. In 1993 Zelmanov put the following question: is it true that the nilpotency degree of $F_{2,m}$ has exponential growth?
We give the definitive answer to Zelmanov's question by showing that the nilpotency class of an $l$-generated associative algebra with the identity $x^d=0$ is smaller than $\Psi(d,d,l)$, where
$$ \Psi(n,d,l)=2^{18}l(nd)^{3\log_3(nd)+13}d^2. $$
This result is a consequence of the following fact based on combinatorics of words. Let $l$, $n$ and $d\geqslant n$ be positive integers. Then all words over an alphabet of cardinality $l$ whose length is not less than $\Psi(n,d,l)$ are either $n$-divisible or contain $x^d$; a word $W$ is $n$-divisible if it can be represented in the form $W=W_0W_1\dotsb W_n$ so that $W_1,\dots,W_n$ are placed in lexicographically decreasing order. Our proof uses Dilworth's theorem (according to V. N. Latyshev's idea). We show that the set of not $n$-divisible words over an alphabet of cardinality $l$ has height $h<\Phi(n,l)$ over the set of words of degree $\le n-1$, where
$$ \Phi(n,l)=2^{87}l\cdot n^{12\log_3n+48}. $$

Bibliography: 40 titles.
Keywords: Shirshov's theorem on height, word combinatorics, $n$-divisibility, Dilworth theorem, Burnside-type problems.
Received: 12.02.2011 and 17.10.2011
Bibliographic databases:
Document Type: Article
UDC: 512.552+512.64+519.1
MSC: 16R10, 68R15
Language: English
Original paper language: Russian
Citation: A. Ya. Belov, M. I. Kharitonov, “Subexponential estimates in Shirshov's theorem on height”, Sb. Math., 203:4 (2012), 534–553
Citation in format AMSBIB
\Bibitem{BelKha12}
\by A.~Ya.~Belov, M.~I.~Kharitonov
\paper Subexponential estimates in Shirshov's theorem on height
\jour Sb. Math.
\yr 2012
\vol 203
\issue 4
\pages 534--553
\mathnet{http://mi.mathnet.ru//eng/sm7853}
\crossref{https://doi.org/10.1070/SM2012v203n04ABEH004233}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2976288}
\zmath{https://zbmath.org/?q=an:1254.16015}
\adsnasa{https://adsabs.harvard.edu/cgi-bin/bib_query?2012SbMat.203..534B}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000305396300004}
\elib{https://elibrary.ru/item.asp?id=19066470}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84862633662}
Linking options:
  • https://www.mathnet.ru/eng/sm7853
  • https://doi.org/10.1070/SM2012v203n04ABEH004233
  • https://www.mathnet.ru/eng/sm/v203/i4/p81
  • This publication is cited in the following 14 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Математический сборник Sbornik: Mathematics
    Statistics & downloads:
    Abstract page:702
    Russian version PDF:190
    English version PDF:9
    References:55
    First page:33
     
      Contact us:
     Terms of Use  Registration to the website  Logotypes © Steklov Mathematical Institute RAS, 2024