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abstract: |
    In 1993 E. I. Zelmanov has put the following question in Dniester Notebook:[*Suppose that $F_{2, m}$ is a free 2-generated associative ring with the identity $x^m=0.$ Is it true that the nilpotency degree of $F_{2, m}$ has exponential growth?*]{}

    We give the definitive answer to E. I. Zelmanov’s question 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 $l$, $n$ и $d \geqslant n$ 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 $d$-th power of a subword; a word $W$ is [*$n$-divisible*]{} if it can be represented in the form $W=W_0W_1\cdots W_n$ such that $W_1,W_2,\dots,W_n$ are placed in lexicographically decreasing order. Our proof uses Dilworth 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 $\leqslant (n-1)$, where $$\Phi(n,l) = 2^{87} l\cdot n^{12\log_3 n + 48}.$$
author:
- 'Alexei 