Asymptotics for the Amitsur's Capelli - Type Polynomials and Verbally Prime PI-Algebras

We consider associative PI-algebras over a field of characteristic zero. The main goal of the paper is to prove that the codimensions of a verbally prime algebra [11] are asymptotically equal to the codimensions of the T-ideal generated by some Amitsur\u2019s Capelli-type polynomials E\ua4M;L [1]. In particular we prove that cn(Mk(G)) '' cn(E\ua4k2;k2 ) and cn(Mk;l(G)) '' cn(E\ua4k2+l2;2kl); where G is the Grassmann algebra. These results extend to all verbally prime PI-algebras a theorem of A.Giambruno and M.Zaicev [9] giving the asymptotic equality cn(Mk(F)) '' cn(E\ua4k2;0) between the codimensions of the matrix algebra Mk(F) and the Capelli polynomials