EXACT ASYMPTOTIC-BEHAVIOR OF THE CODIMENSIONS OF SOME PI ALGEBRAS

Let c(n)(A) denote the codimensions of a P.I. algebra A, and assume c( n)(A) has a polynomial growth: c(n)(A)similar or equal to/n-->infinity qn(k). Then, necessarily, q is an element of Q [D3]. If 1 is an eleme nt of A, we show that 1/K! less than or equal to q less than or equal to 1/2! - 1/3! + - ... + (-1)(k)/k! approximate to 1/e, where e = 2.71 .... In the non-unitary case, for any 0 < q is an element of Q, we con struct A, with a suitable k, such that c(n)(A)similar or equal to/n--> infinity qn(k).