The trace codimensions give a quantitative description of the identities satisfied by an algebra with trace. Here we study the asymptotic behaviour of the sequence of trace codimensions cntr(A) and of pure trace codimensions cnptr(A) of a finite-dimensional algebra A over a field of characteristic zero. We find an upper and lower bound of both codimensions and as a consequence we get that the limits limn→∞cntr(A)n and limn→∞cnptr(A)n always exist and are integers. This result gives a positive answer to a conjecture of Amitsur in this setting. Finally we characterize the varieties of algebras whose exponential growth is bounded by 2.
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