Simultaneous Similarity Classes of Commuting matrices over a finite field

Abstract

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 c n,k (q) denote the number of isomorphism classes of n-dimensional F q [x 1 , . . . , x k ]-modules. The generating function k c n,k (q)t k is a rational function. We compute this function for n ≤ 4. We find that its coefficients are polynomial functions in q with non-negative integer coefficients.