  Let $F$ be a field of characteristic zero and $W$ be an associative affine
$F$-algebra satisfying a polynomial identity (PI). The codimension sequence
associated to $W$, $c_n(W)$, is known to be of the form $\Theta (c n^t d^n)$,
where $d$ is the well known (PI) exponent of $W$. In this paper we establish an
algebraic interpretation of the polynomial part (the constant $t$) by means of
Kemer's theory. In particular, we show that in case $W$ is a basic algebra,
then $t = \frac{d-q}{2} + s$, where $q$ is the number of simple component in
$W/J(W)$ and $s+1$ is the nilpotency degree of $J(W)$. Thus proving a
conjecture of Giambruno.
