Subexponential estimations in Shirshov's height theorem

TL;DR

This paper establishes subexponential bounds on Shirshov's height theorem, providing a definitive answer to Zelmanov's question about the growth of nilpotency degree in certain associative algebras, using combinatorics of words.

Contribution

It introduces subexponential estimations for Shirshov's height, improving previous exponential bounds and resolving Zelmanov's question with a combinatorial approach.

Findings

Nilpotency degree is smaller than Psi(d,d,l) for associative algebras with x^d=0.

Words longer than Psi(n,d,l) are either n-divided or contain a d-th power of a subword.

Height h over Y is bounded by Phi(n,l), refining previous exponential bounds.

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)=2^{18} l (nd)^{3 log_3 (nd)+13}d^2. 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...W_n such that W_1 >' W_2>'...>'W_n. The symbol >' 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) = 2^{87} n^{12 log_3 n + 48} l. Our proof uses Latyshev idea of Dilworth theorem application.