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

A. Ya. Belov ; M. I. Kharitonov

Fundamentalʹnaâ i prikladnaâ matematika, Tome 17 (2012) no. 5, pp. 21-54.

Résumé

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 $W=W_0W_1\cdots W_n$, where $W_1\prec W_2\prec\dots\prec W_n$. If an affine algebra $A$ satisfies a polynomial identity of degree $n$, then $A$ is spanned by non $n$-divisible words of generators $a_1\prec\dots\prec a_l$. 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 $\leq n-1$. We show that $h\Phi(n,l)$, where $\Phi(n,l)=2^{87}l\cdot n^{12\log_3n+48}$. Let $l,n$, and $d\geq n$ be positive integers. Then all words over an alphabet of cardinality $l$ whose length is greater than $\Psi(n,d,l)$ are either $n$-divisible or contain the $d$th power of a subword, where $\Psi(n,d,l)=2^{18}l(nd)^{3\log_3(nd)+13}d^2$. 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 $x^m=0$. 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 $x^d=0$ is smaller than $\Psi(d,d,l)$. 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.

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