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 Mat. Sb., 2012, Volume 203, Number 4, Pages 81–102 (Mi msb7853)

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

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 $d\ge n$ 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 $W_1,…,W_n$ 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.
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DOI: https://doi.org/10.4213/sm7853

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English version:
Sbornik: Mathematics, 2012, 203:4, 534–553

Bibliographic databases:

UDC: 512.552+512.64+519.1
MSC: 16R10, 68R15

Citation: A. Ya. Belov, M. I. Kharitonov, “Subexponential estimates in Shirshov's theorem on height”, Mat. Sb., 203:4 (2012), 81–102; Sb. Math., 203:4 (2012), 534–553

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb7853
• https://doi.org/10.4213/sm7853
• http://mi.mathnet.ru/eng/msb/v203/i4/p81

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This publication is cited in the following articles:
1. 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
2. Kharitonov M.I., “Dvustoronnie otsenki suschestvennoi vysoty v teoreme Shirshova o vysote”, Vestn. Mosk. un-ta. Ser. 1. Matem., mekh., 2012, no. 2, 20–24
3. Lopatin A.A., “On the nilpotency degree of the algebra with identity $x^n=0$”, J. Algebra, 371 (2012), 350–366
4. Lopatin A.A., Shestakov I.P., “Associative nil-algebras over finite fields”, Internat. J. Algebra Comput., 23:8 (2013), 1881–1894
5. M. I. Kharitonov, “Piecewise periodicity structure estimates in Shirshov's height theorem”, Moscow University Mathematics Bulletin, 68:1 (2013), 26–31
6. M. I. Kharitonov, “Otsenki, svyazannye s teoremoi Shirshova o vysote”, Chebyshevskii sb., 15:4 (2014), 55–123
7. A. Kanel-Belov, Y. Karasik, L. Rowen, Computational aspects of polynomial identities, v. 1, Monographs and Research Notes in Mathematics, 16, Kemer's Theorems, Second edition, CRC Press, Boca Raton, FL, 2016, 407 pp.
8. Domokos M., “Polynomial Bound For the Nilpotency Index of Finitely Generated Nil Algebras”, Algebr. Number Theory, 12:5 (2018), 1233–1242
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