Субэкспоненциальные оценки в теореме Ширшова о высоте

What is it about?

Let $F_{2,m}$ be a free 2-generated associative ring with the identity $x^m=0$. In 1993, E. Zelʹmanov asked a question about subexponentiality for the growth of the class of nilpotency in $m$ of the ring $F_{2,m}$. In the paper under review the authors answer the above question, establishing that for an $l$-generated associative algebra with the identity $x^d=0$ its nilpotency class is less than $\Psi(d,d,l)$ for $\Psi(n,d,l)=2^{18}l(nd)^{3\log_3(nd)+13}d^2$. This result is a corollary of the following fact concerning the combinatorics of words: Let $l$, $ n$ and $d\geq n$ be natural numbers. Then all words over an $l$-lettered alphabet of length not less than $\Psi(n,d,l)$ either include $x^d$ or are $n$-decomposable, where the word $W$ is called $n$-decomposable if it could be expressed as $W=W_0\cdots W_n$ where the subwords $W_1,\dotsc ,W_n$ are lexicographically diminishing. The proof of this combinatorial fact uses Dilworth's theorem. The authors show that the set of all non-$n$-decomposable words over an $l$-lettered alphabet is of height $h<\Phi(n,l)$ over words of degree at most $n-1$ for $\Phi(n,l)=2^{87}\ln^{12\log_3n+48}$. Reviewed by Tsetska Grigorova Rashkova

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