Central polynomials of algebras and their growth (Q824444)

A central polynomial for an algebra \(A\) is a polynomial \(f(x_1,\ldots, x_n)\) such that \(f(a_1,\ldots,a_n)\) lies in the center of \(A\) for all \(a_1,\ldots,a_n\in A\). This paper studies two codimension sequences for central polynomials analoguous to the ordinary codimension sequence. Let \(P_n\) be the space of multilinear degree \(n\) polynomials in \(x_1,\ldots,x_n\). Then let \(I_n\) be the space of polynomial identities of \(A\) contained in \(P_n\) and let \(I_n\subset \Delta_n\) be the space of central polynomials. The three codimensions are \(c_n=\dim (P_n/I_n)\), the ordinary codimension; \(c_n^Z=\dim(P_n/J_n)\), the central codimension; and \(\delta_n=\dim(\Delta_n/I_n)\), the proper central codimension. Note that \(c_n^Z+\delta_n=c_n\). These codimensions were defined by \textit{A. Regev} [Commun. Algebra 44, No. 10, 4411--4421 (2016; Zbl 1354.16023)] and studied by the authors in [Proc. Am. Math. Soc. 147, No. 3, 909--919 (2019; Zbl 1409.16018)]. All three codimension sequences have an integral exponential rate of growth, namely for each one the limit of the \(n\)-th root of the \(n\)-th codimension converges to an integer, and the ordinary and central codimensions have the same exponential rates of growth. The current paper describes the results of these papers with some interesting examples and some new results. These include algebras with arbitrarily large gaps between the exponential rates of growth of \(c_n(A)\) and \(\delta_n(A)\) and a non-associative algebra in which all three exponential rates of growth are non-integral. For the entire collection see [Zbl 1461.16003].