Subexponential estimates in Shirshov's theorem on height

A. Ya Belov, M. I. Kharitonov

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

Suppose that F2,m is a free 2-generated associative ring with the identity xm = 0. In 1993 Zelmanov put the following question: is it true that the nilpotency degree of F2,m has exponential growth? We give the defnitive answer to Zelmanov's question by showing that the nilpotency class of an l-generated associative algebra with the identity xd = 0 is smaller than Ψ(d,d,l), where (equation) This result is a consequence of the following fact based on combinatorics of words. Let l, n and d ≥ n be positive integers. Then all words over an alphabet of cardinality l whose length is not less than Ψ(n,d,l) are either n-divisible or contain xd; a word W is n-divisible if it can be represented in the form W = W 0W1 · · · Wn so that W1,., Wn 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 < Φ(n, l) over the set of words of degree ≤ n-1, where 87 12 logc, n+48 Φ(n, l) = 287l · n12log 3n+48 Bibliography: 40 titles.

Original languageEnglish
Pages (from-to)534-553
Number of pages20
JournalSbornik Mathematics
Volume203
Issue number4
DOIs
StatePublished - 2012
Externally publishedYes

Keywords

  • Burnside-type problems
  • Dilworth theorem
  • Shirshov theorem on height
  • Word combinatorics
  • n-divisibility

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