Simultaneous Similarity Classes of Commuting matrices over a finite field

TL;DR

This paper classifies and counts isomorphism classes of modules over polynomial algebras in several variables over finite fields, linking them to simultaneous similarity classes of commuting matrices and providing explicit generating functions.

Contribution

It computes the generating functions for isomorphism classes of modules for small dimensions and shows these are rational functions with polynomial coefficients in q.

Findings

Generated functions are rational with polynomial coefficients in q.

Explicit enumeration for n ≤ 4.

Coefficients are non-negative integers.

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 $\mathbb{F}_q[x_1,\dotsc,x_k]$-modules. The generating function $\sum_k c_{n,k}(q)t^k$ is a rational function. We compute this function for $n\leq 4$. We find that its coefficients are polynomial functions in $q$ with non-negative integer coefficients.