The enumeration of isomorphism classes of modules of a
polynomial algebra in several variables over a finite field is the same as
the classifi cation of commuting tuples of matrices over a finite field up
to simultaneous similarity. Let C_{n,k}(q) denote the number of
isomorphism classes of n-dimensional Fq[x1,...,xk]-modules. The generating
function in k of the C_{n,k}(q) is a rational function. The computation of
this was done explicitly for n <= 4. I shall give a summary of my recently
published work on this study of the C_{n,k}(q)s for n <= 2.