Exact asymptotic behaviour of the codimensions of some P.I. algebras (Q677455)

Let \(K\) be a field, and let \(A\) be a PI algebra over \(K\). One of the important numerical characteristics of \(A\) is the sequence of its codimensions \(c_n=c_n(A)\). It is well-known that the growth of \(c_n\) does not exceed the exponential one: \(c_n\leq\alpha^n\) for some constant \(\alpha\). Another well-known result states that if the growth of \(c_n\) is polynomial i.e., if \(c_n\approx qn^k\) asymptotically, and if the characteristic of \(k\) is 0 then \(q\) is rational. The present paper gives upper and lower bounds for that rational number \(q\). Namely, it is shown that when \(1\in A\) then \((k!)^{-1}\leq q\leq(2!)^{-1}-(3!)^{-1}+\cdots+(-1)^k(k!)^{-1}\). If \(A\) is not unitary the authors construct examples proving that \(q\) can take every positive (rational) value.