Subexponential estimates in the height theorem and estimates on numbers of periodic parts of small periodsстатья

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1. Полный текст art3A10.10072Fs10958-013-1477-4.pdf 302,6 КБ 15 апреля 2016 [AlexeiBelov]

[1] Belov A. Y., Kharitonov M. I. Subexponential estimates in the height theorem and estimates on numbers of periodic parts of small periods // Journal of Mathematical Sciences. — 2013. — Vol. 193, no. 4. — P. 493–515. Shirshov’s height theorem has proven its strength in many parts of ring theory. The original proof of the theorem can provide bounds for the height. Until now all known upper bounds for the height (as a function in the cardinality l of the generating set of the algebra) are exponential. There are lower bounds known: these are polynomial ones. Consider an alphabet with l letters, and take the set of all words that are not n-divisible. Recall here that a word is n-divisible if one can cut some initial segment of it, and divide the remainder into n consecutive subwords in decreasing order. (The order here is the usual lexicographical order.) Then Shirshov’s Theorem states that if A is a PI algebra in l generators then the set of the non n-divisible words in A is of bounded height with respect to the set of all words of length ≤n−1. The authors give an upper bound Φ(n,l) for the height in this case: Φ(n,l)=E_1l⋅n{E_2+12log3n}; here E_1=421log34+17 and E_2=30log34+10. The reader should note that the bound is subexponential. This upper bound allows the authors to give better upper bounds for the nilpotency index in the Nagata-Higman-Dubnov-Ivanov theorem. Let A be a nil algebra satisfying the identity xd=0. Suppose A is generated by l elements, then its nilpotency index has an upper bound which is also subexponential. The authors give also other applications of these bounds. They study the essential height and its relation to the Gelfand-Kirillov dimension, and also applications to Ramsey theory.Reviewer: Plamen Koshlukov (Campinas). [ DOI ]

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