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Mat. Sb. (N.S.), 1985, Volume 128(170), Number 1(9), Pages 124–132 (Mi msb2021)  

This article is cited in 8 scientific papers (total in 8 papers)

An independence theorem and its consequences

V. A. Ufnarovskii


Abstract: The following theorem is proved: Let $A_1,…,A_d$ be linear operators in a vector space $V$, $v\in V$, and let the word $C=A_{k_1}A_{k_2}…A_{k_n}$ be maximal in the right lexicographical order among all words of length $n$ satisfying the condition $Cv\ne0$. If all the operators corresponding to the subwords of $C$ are nilpotent, then the vectors $v$, $A_{k_n}v$, $A_{k_{n-1}}A_{k_n}v,…,A_{k_1}A_{k_2}\cdots A_{k_n}v$ are independent.
As a corollary, a proof is presented of Shestakov's conjecture about the number of nil-conditions necessary for a subalgebra of a matrix algebra to be nilpotent.
Bibliography: 5 titles.

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English version:
Mathematics of the USSR-Sbornik, 1987, 56:1, 121–129

Bibliographic databases:

UDC: 512.55
MSC: Primary 16A22; Secondary 16A30, 15A03
Received: 27.06.1984

Citation: V. A. Ufnarovskii, “An independence theorem and its consequences”, Mat. Sb. (N.S.), 128(170):1(9) (1985), 124–132; Math. USSR-Sb., 56:1 (1987), 121–129

Citation in format AMSBIB
\Bibitem{Ufn85}
\by V.~A.~Ufnarovskii
\paper An independence theorem and its consequences
\jour Mat. Sb. (N.S.)
\yr 1985
\vol 128(170)
\issue 1(9)
\pages 124--132
\mathnet{http://mi.mathnet.ru/msb2021}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=805699}
\zmath{https://zbmath.org/?q=an:0605.15002|0598.15002}
\transl
\jour Math. USSR-Sb.
\yr 1987
\vol 56
\issue 1
\pages 121--129
\crossref{https://doi.org/10.1070/SM1987v056n01ABEH003027}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. A. Ya. Belov, “On a Shirshov basis of relatively free algebras of complexity $n$”, Math. USSR-Sb., 63:2 (1989), 363–374  mathnet  crossref  mathscinet  zmath
    2. N. A. Koreshkov, D. Yu. Kharitonov, “On the nilpotency of Engel algebras”, Russian Math. (Iz. VUZ), 45:11 (2001), 15–18  mathnet  mathscinet  zmath  elib
    3. Drensky V., “Polynomial identity rings - Part A - Combinatorial aspects in PI-rings”, Polynomial Identity Rings, Advanced Courses in Mathematics Crm Barcelona, 2004, 1  isi
    4. A. Ya. Belov, “The Kurosh problem, height theorem, nilpotency of the radical, and algebraicity identity”, J. Math. Sci., 154:2 (2008), 125–142  mathnet  crossref  mathscinet  zmath  elib  elib
    5. A. Ya. Belov, “Burnside-type problems, theorems on height, and independence”, J. Math. Sci., 156:2 (2009), 219–260  mathnet  crossref  mathscinet  zmath  elib  elib
    6. A. Ya. Belov, M. I. Kharitonov, “Subexponential estimates in Shirshov's theorem on height”, Sb. Math., 203:4 (2012), 534–553  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    7. A. Ya. Belov, M. I. Kharitonov, “Subexponential estimates in the height theorem and estimates on numbers of periodic parts of small periods”, J. Math. Sci., 193:4 (2013), 493–515  mathnet  crossref
    8. M. I. Kharitonov, “Otsenki, svyazannye s teoremoi Shirshova o vysote”, Chebyshevskii sb., 15:4 (2014), 55–123  mathnet
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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