Estimations of the particular periodicity in case of the extremal periods in Shirshov's Height theorem (1108.6295v2)

Abstract: Let us recall the well-known Shirshov's Height Theorem. "Let A be a finitely generated algebra of degree d. Then there exists a finite set Y which is the subset of A that A has and an integer h' = h(A) such that A has Shirshov's height h' over set Y. For Y we may take the set of words of length <d. Such Y is called a Shirshov's base of algebra A." Shirshov's original proof was purely combinatorical, but did not provide a reasonable upper estimate for the height. Kolotov obtained an estimate for h(A) < s^s^m (m = deg(A), and s is the number of generators) in 1981. I. Zelmanov asked for an exponential bound in the Dniester Notebook. In 2011 Belov and Kharitonov get the estimate 2^87 m^12 log_3(m+48) l. This result answered Zelmanov question that in fact Shirshov's height has subexponential growth. As a corollary from Amitzur-Levitsky theorem we have the lower estimate of h(A) which is ((m/2)^2 - 1)(s - 1) + 1 . We consider essential height over the sets of small words to obtain exact estimations of Shirshov's height. Define by n-divisible such word w that may be constructed as w = v u_1 u_2...u_n, where for every i from 2 to n u_(i-1) is lexicographically less than u_i. The main results of the paper are the following: Let M be the set of non-n-dividable words with: 1) finite essential height over the words degree 2, then the number of lexicographically comparable words with period 2 is less than (2l-1)(n-1)(n-2)/2; 2) finite essential height over the words degree 3, then the number of lexicographically comparable words with period 3 is less than (2l-1)(n-1)(n-2); 3) finite essential height over the words degree (n-1), then the number of lexicographically comparable words with period (n-1) is less than (l-2)(n-1). The case of the words of length 2 is synthesized to the proof of the exponential estimation in Shirshov's Height theorem.