Asymptotics of the Codimensions $c_n$ in the Algebra $F^{(7)}$

The paper studies the additive structure of the algebra $F^{(7)}$, i.e., a relatively free associative countably generated algebra with the identity $[x_1,\dots,x_7]=0$ over an infinite field of characteristic $\ne 2,3$. First, the space of proper multilinear polynomials in this algebra is investigated. As an application, estimates for the codimensions $c_n=\dim F_n^{(7)}$ are obtained, where $F_n^{(7)}$ stands for the subspace of multilinear polynomials of degree $n$ in the algebra $F^{(7)}$.