FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2011/2012, VOLUME 17, NUMBER 5, PAGES 21-54

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

A. Ya. Belov

M. I. Kharitonov

Abstract

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The paper is devoted to subexponential estimates in Shirshov's height
theorem.
A word $W$ is *$n$-divisible* if it can
be represented in the form $W\; =\; W$_{0}W_{1}...W_{n},
where
$\$\; W\_1\backslash prec\; W\_2\backslash prec\; \backslash dots\; \backslash prec\; W\_n\; \$$.
If an affine algebra $A$ satisfies
a polynomial identity of degree $n$, then $A$ is spanned by non
$n$-divisible
words of generators $\$\; a\_1\backslash prec\; \backslash dots\; \backslash prec\; a\_l\; \$$.
A. I. Shirshov proved that the set of non $n$-divisible words over an
alphabet of cardinality $l$ has bounded
height $h$
over the set $Y$ consisting of all words
of degree $$£ n
- 1.
We show that $h\; <$F(n,l), where
$$F(n,l) =
2^{87} l n^{12 log3 n + 48}.

Let $l$,
$n$, and
$d$³
n be positive integers.
Then all words over an alphabet of cardinality $l$ whose length is greater
than $$Y(n,d,l) are either
$n$-divisible or
contain the $d$th
power of a subword, where $$Y(n,d,l) =
2^{18} l (nd)^{3 log3(nd) + 13} d^{2}.

In 1993, E. I. Zelmanov asked the following question in the
Dniester Notebook: Suppose that $F$_{2,m} is
a $2$-generated associative
ring with the identity $xm=\; 0$.
Is it true that the nilpotency degree of $F$_{2,m} has
exponential growth? We give the definitive answer to
E. I. Zelmanov by this result.
We show that the nilpotency degree of the $l$-generated associative
algebra with the identity $xd=\; 0$ is smaller
than $$Y(d,d,l).
This implies subexponential estimates on the nilpotency index of
nil-algebras of arbitrary characteristic.
Shirshov's original estimate was just recursive, in 1982 double
exponent was obtained, and an exponential estimate was obtained in
1992.

Our proof uses Latyshev's idea of an application of the Dilworth
theorem.
We think that Shirshov's height theorem is deeply connected to
problems of modern combinatorics.
In particular, this theorem is related to the Ramsey theory.
We obtain lower and upper estimates of the number of periods of length
$2$, $3$, $n$-
1 in some non $n$-divisible word.
These estimates differ only by a constant.

Location: http://mech.math.msu.su/~fpm/eng/k1112/k115/k11502h.htm

Last modified: October 18, 2012