Subexponential estimations in Shirshov's height theorem (in English)

TL;DR

This paper establishes subexponential bounds on the nilpotency degree of associative algebras with certain identities, answering Zelmanov's question and advancing the combinatorial understanding of word structures in algebra.

Contribution

It provides a subexponential upper bound on the height of non n-divided words, improving previous exponential estimates and confirming Zelmanov's conjecture.

Findings

The nilpotency degree is smaller than a specific subexponential function Psi(d,d,l).

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

The height of non n-divided words is bounded by Phi(n,l), a subexponential function.

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.