Exponential Growth for Codimensions of Some p.i. Algebras

By the Giambruno–Zaicev theorem for associative p.i. algebras, the exponential rate of growth of the codimensions of such a p.i. algebra is always a positive integer. Here we calculate that integer for various generic p.i. algebras which are given by a single identity. These include Capelli-type identities and the various powers of the standard polynomials.