Counting Similarity Classes of Tuples of Commuting Matrices Over a Finite Field[HBNI Th119]

Abstract

This thesis concerns the problem of enumerating isomorphism classes of n dimensional modules over polynomial algebras F q [x 1 , . . . , x k ], in k variables over a finite field of order q. This problem is the same as the classification of k-tuples of commuting n × n matrices over a finite field up to simultaneous similarity. Let c(n, k, q) denote the number of isomorphism classes of n-dimensional Fq [x 1 , . . . , x k]-modules. In this thesis, we analyse the asymptotic behaviour of c(n, k, q) as a function of k, keeping n and q fixed. We also give an explicit formula of c(n, k, q) for n ≤ 4. This thesis is divided into four chapters. In the first chapter, we state our main results about c(n, k, q). In the second chapter, we discuss the preliminaries required for understanding the thesis. In the third chapter, we discuss the asymptotic behaviour of c(n, k, q) as a function of k, for a fixed n and q. In the fourth chapter, we explicitly calculate c(n, k, q) for n = 2, 3, 4, and any k ≥ 1.