Codimensions of generalized polynomial identities

It is proved that for every finite-dimensional associative algebra $A$ over a field of characteristic zero there are numbers $C\in\mathbb Q_+$ and $t\in\mathbb Z_+$ such that $gc_n(A)\sim Cn^td^n$ as $n\to\infty$, where $d=PI\exp(A)\in\mathbb Z_+$. Thus, Amitsur's and Regev's conjectures hold for the codimensions $gc_n(A)$ of the generalized polynomial identities.
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