  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.
