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SUMMARY Mikhail Igorevich Kharitonov
The research is devoted to sub exp onential estimations in Shirshov's Height theorem. A word

W

is

n

-divisible, if it can b e represented in the following form:

W = W0 W1 · · · Wn
is spanned by non

such that

W1

an ane algebra

W A

2

···

Wn a
1

, where

is comparison in lexicographical sense. If

satises p olynomial identity of degree

n

then

A

n n

-divisible words of generators

···

a

l . A. I. Shirshov proved that the set of non

-divisible words over alphab et of cardinality

l

has b ounded height

h

over the set

Y

consisting of all the words of degree We show, that

n - 1.
12 log3 n+36 log3 log3 n+91

h < (n, l),

where

(n, l) = 296 l · n
Let

.

l, n

Х

d

n

b e p ositive integers. Then all the words over alphab et of cardinality

l

which length is greater than

(n, d, l)

are either

n-divisible

or contain

d

-th p ower of

subword, where

(n, d, l) = 227 l(nd)3

log3 (nd)+9 log3 log3 (nd)+36

. = 0.
Is it

In 1993 E. I. Zelmanov asked the following question in Dniester Noteb o ok: m Supp ose that F2,m is a 2-generated asso ciative ring with the identity x true, that the nilp otency degree of nilp otency degree of than

F

2,m has exp onential growth?

"

We give the denitive answer to E. I. Zelmanov by this result. We show that the l-generated associative algebra with the identity xd = 0 is smaller

(d, d, l).

This imply sub exp onential estimations on the nilp otency index of nil-

algebras of an arbitrary characteristics. Original Shirshov's estimation was just recursive, in 1982 double exp onent was obtained, an exp onential estimation was obtained in 1992. Our pro of uses Latyshev idea of Dilworth's theorem application. We think that Shirshov's height theorem is deeply connected to problems of mo dern combinatorics. In particular this theorem is related to the Ramsey theory. Ab ove all we obtain lower and upp er estimates of the numb er of p erio ds of length

2, 3, (n - 1)

in some non

n-divisible

word. These estimates are dier only by a constant.

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