$l$, $n$ and $d\geqslant n$ be positive integers. Then all $l$-generated words of length not less than $\Psi(n,d,l)$ either contain $x^d$ or are $n$-divisible. Here $$\Psi(n,d,l)=4^{5+3\log_3 4} l (nd)^{3 \log_3 (nd)+(5+6\log_3 4)}d^2.$$

This theorem after some coarsening and simplification of the estimate implies that for fixed $l$ and $nd \rightarrow\infty$ we have $$\Psi(n,d,l) < 2^{18} l (nd)^{3 \log_3 (nd)+13}d^2
= (nd)^{3(1+o(1))\log_3(nd)},$$

and for fixed $n$ and $l\rightarrow\infty$ we have $$\Psi(n,d,l) < C(n,d)l.$$

Let $l$, $d$ be positive integers, and let an associative $l$-generated algebra $A$ satisfy $x^{d}=0$. Then its nilpotency index is less than $\Psi(d,d,l)$.

Moreover we prove a subexponential estimate which is better for small $n$ and $d$:

\[t2:log\_2\] Let $l$, $n$ and $d\geqslant n$ be positive integers. Then all $l$-generated words of length not less than $\Psi(n,d,l)$ either contain $x^d$ or are $n$-divisible. Here $$\Psi(n,d,l) = 256 l(nd)^{2\log_2 (nd)+10}d^2.$$

For a real number $x$ put $\ulcorner x\urcorner := -[-x].$ Thus we replace non-integer numbers by the closest greater integers.

Proving theorem \[c:main2\] we also prove the following theorem on estimation of essential height:

\[ThThick\] The essential height of an $l$-generated $PI$-algebra with an admissible polynomial identity of degree $n$ over the set of words of length less than 