height of the set of not $n$-divisible words over an alphabet of cardinality $l$ relative to the set of words of length less than $n$ does not exceed $\Phi(n,l)$ where $$\Phi(n,l) = E_1 l\cdot n^{E_2+12\log_3 n} ,$$ $E_1 = 4^{21\log_3 4 + 17}, E_2 = 30\log_3 4 + 10.$

This theorem after some coarsening and simplification of the estimate implies that for fixed $l$ and $n \rightarrow\infty$ we have $$\Phi(n,l) < 2^{87} l\cdot n^{12\log_3 n + 48}
= n^{12(1+o(1))\log_3{n}},$$ and for fixed $n$ and $l\rightarrow\infty$ we have $$\Phi(n,l) < C(n)l.$$

The height of an $l$-generated $\PI$-algebra with an admissible polynomial identity of degree $n$ over the set of words of length less than $n$ does not exceed $\Phi(n,l)$.

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

\[t1:log\_2\] The height of the set of not $n$-divisible words over an alphabet of cardinality $l$ relative to the set of words of length less than $n$ does not exceed $\Phi(n,l)$ where $$\Phi(n,l) = 2^{40} l\cdot n^{38+8\log_2 n}.$$

In particular we obtain subexponential estimates for the nilpotency index of $l$-generated nil-algebras of degree $n$ in an arbitrary characteristic.

The second main result of our paper is the following

\[c:main1\] Let 