Simultaneous similarity classes of commuting matrices over a finite field

This paper concerns the enumeration of isomorphism classes of modules of a polynomial algebra in several variables over a finite field. This is the same as the classification of commuting tuples of matrices over a finite field up to simultaneous similarity. Let cn,k(q)cn,k(q) denote the number of isomorphism classes of n  -dimensional Fq[x1,…,xk]Fq[x1,…,xk]-modules. The generating function ∑kcn,k(q)tk∑kcn,k(q)tk is a rational function. We compute this function for n≤4n≤4. We find that its coefficients are polynomial functions in q with non-negative integer coefficients.