Codimensions of products and of intersections of verbally prime T-ideals (Q1279624)

In 1985, A. R. Kemer described the T-prime T-ideals, and subsequently he proved their importance by obtaining the Specht property for associative algebras over a field of characteristic 0 [see \textit{A. R. Kemer}, Ideals of identities of associative algebras, Transl. Math. Monogr. 87 (1991; Zbl 0732.16001)]. The paper under review deals with the asymptotics of the codimension sequences of intersections and of products of T-prime T-ideals. Note that according to Kemer's theory, if \(A\) is a PI algebra over a field \(F\) of characteristic 0 and if \(I\) is the T-ideal of \(A\) then there exist T-prime T-ideals \(J_1,\ldots,J_r\) such that \(J_{i_1}\ldots J_{i_m}\subseteq I\subseteq J_1\cap\ldots\cap J_r\) for every \(1\leq i_k\leq r\). Hence any information about T-ideals of this type is useful for better understanding the nature of PI algebras. The main results of the paper under review are the following. Denote as \(c_n(A)\) the \(n\)-th codimension of the algebra \(A\), and suppose further that \(J_i\), \(i=1,\ldots,m\) are T-prime T-ideals in the free associative algebra \(F(X)\), and \(I=J_1\ldots J_m\). Let \(\alpha_i=\lim_{n\to\infty}\root n\of {c_n(F(X)/J_i)}\), and let \(\alpha=\lim_{n\to\infty}\root n\of {c_n(F(X)/I)}\). Then \(\alpha=\alpha_1+\cdots+\alpha_m\), and in particular \(\alpha\) is an integer. Further, assume that \(c_n(F(X)/J_i)\approx a_in^{e_i}\alpha_i^n\) where \(a_i\), \(e_i\) and \(\alpha_i\) are constants. Then one has \(c_n(F(X)/I)\approx an^e\alpha^n\) where \(e=e_1+\cdots+e_m+m-1\) is a half integer, and \(a=a_1\ldots a_m\alpha_1^{e_1}\ldots\alpha_m^{e_m}(\alpha_1+\cdots+\alpha_m)^{-e}\). Therefore \(a=\pi^{-s/2}\cdot q\) where \(s\) is a non-negative integer and \(q\) is an algebraic number. Furthermore the authors prove that \(c_n(F(X)/J_1\cap\ldots\cap J_m)\approx c_n(F(X)/J_i)\) for some \(1\leq i\leq m\). The authors raise two important questions concerning codimensions and cocharacters. The first is whether for every PI algebra \(A\) the limit \(\lim_{n\to\infty}\root n\of {c_n(A)}\) is an integer. And the second is to investigate the behaviour of the cocharacters instead of codimensions, in the spirit of the results in the paper under consideration.