Estimates in Shirshov height theorem

Abstract

In 1993 E. I. Zelmanov asked the following question in Dniester Notebook: "Suppose that $F_{2, m}$ is a 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 show that the nilpotency degree of $l$-generated associative algebra with the identity $x^d=0$ is smaller than $\Psi(d,d,l)$, where $\Psi(n,d,l)=l (nd)^{C \log (nd)}$ and $C$ is a constant. We give the definitive answer to E. I. Zelmanov by this result. It is the consequence of one fact, which is based on combinatorics of words. Let $l, n$ and $d>n$ be positive integers. Then all the words over alphabet of cardinality $l$ which length is greater than $\Psi(n,d,l)$ are either $n$-divided or contain $d$-th power of subword, where a word $W$ is $n$-divided, if it can be represented in the following form $W=W_0 W_1\dots W_n$ such that $W_n \prec W_{n-1}\prec\cdots\prec W_1$. The symbol $\prec$ means lexicographical order here. A. I. Shirshov proved that the set of non $n$-divided words over alphabet of cardinality $l$ has bounded height $h$ over the set $Y$ consisting of all the words of degree $<n$. Original Shirshov's estimation was just recursive, in 1982 double exponent was obtained by A. G. Kolotov and in 1993 A. Ya. Belov obtained exponential estimation. We show, that $h<Phi(n,l)$, where $Phi(n,l) = n^{C \log n} l$ and $C$ is a constant. Our proof uses Latyshev idea of Dilworth theorem application.