On the Complexity Functions for $T$-Ideals of Associative Algebras

Let $c_n(\mathbf V)$ be the sequence of codimension growth for a variety $\mathbf V$ of associative algebras. We study the complexity function $\mathscr C(\mathbf V,z)=\sum_{n=0}^\infty c_n(\mathbf V)z^n/n!$, which is the exponential generating function for the sequence of codimensions. Earlier, the complexity functions were used to study varieties of Lie algebras. The objective of the note is to start the systematic investigation of complexity functions in the associative case. These functions turn out to be a useful tool to study the growth of varieties over a field of arbitrary characteristic. In the present note, the Schreier formula for the complexity functions of one-sided ideals of a free associative algebra is found. This formula is applied to the study of products of $T$-ideals. An exact formula is obtained for the complexity function of the variety $\mathbf U_c$ of associative algebras generated by the algebra of upper triangular matrices, and it is proved that the function $c_n(\mathbf U_c)$ is a quasi-polynomial. The complexity functions for proper identities are investigated. The results for the complexity functions are applied to study the asymptotics of codimension growth. Analogies between the complexity functions of varieties and the Hilbert–Poincaré series of finitely generated algebras are traced.