finite-dimensional. This theorem implies

Let $A$ be a finitely generated $\PI$-algebra. Then $$\GK(A)<\infty.$$

$\GK(A)$ is [*the Gelfand — Kirillov dimension of the algebra $A$*]{}, that is, $$\GK(A)=\lim_{n\to\infty}\ln V_A(n)/\ln(n)$$ where $V_A(n)$ is [*the growth function of $A$*]{}, the dimension of the vector space generated by the words of degree not greater than $n$ in the generators of $A$.

Indeed, it suffices to observe that the number of solutions for the inequality $k_1 |v_1|+\cdots+k_h|v_h|\leqslant n$ with $h\leqslant H$ exceeds $N^{H}$, so that $\GK(A)\leqslant \Ht(A)$.

The number $m=\deg(A)$ will mean [*the degree of the algebra*]{}, or the minimal degree of an identity valid in it. The number $n=\Pid(A)$ is [*the complexity*]{} of $A$, or the maximal $k$ such that ${\Bbb M}_k$, the algebra of matrices of size $k$, belongs to the variety $\Var(A)$ generated by $A$.

Instead of the notion of [*height*]{}, it is more suitable to use the close notion of [*essential height*]{}.

An algebra $A$ has [*essential height $h=H_{Ess}(A)$*]{} over a finite set $Y$ called [*an $s$-basis for $A$*]{} if there exists a finite set $D\subset A$ such that $A$ is linearly representable by elements of the form $t_1\cdot\ldots\cdot t_l$, where $l\leqslant
2h+1$, $\forall i (t_i\!\in\! D \vee t_i=y_i^{k_i};y_i\in Y)$ and 