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.

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