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Let $F$ be a field of characteristic zero and $W$ be an associative affine$F$-algebra satisfying a polynomial identity (PI). The codimension sequenceassociated 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 analgebraic interpretation of the polynomial part (the constant $t$) by means ofKemer'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 aconjecture of Giambruno.

The Polynomial Part of the Codimension Growth of Affine PI Algebras