Subexponential estimates in Shirshov's theorem on height

A. Ya. Belov ; M. I. Kharitonov

Sbornik. Mathematics, Tome 203 (2012) no. 4, pp. 534-553

Résumé

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.

Détail

Keywords: Shirshov's theorem on height, word combinatorics, $n$-divisibility, Dilworth theorem, Burnside-type problems.