Towards an efficient Meat-Axe algorithm using f-cyclic matrices: The density of uncyclic matrices in M(n,q)

[Truncated abstract] An element X in the algebra M(n,F) of all n×n matrices over a field F is said to be f-cyclic if the underlying vector space considered as an F[X]-module has at least one cyclic primary component. These are the matrices considered to be “good” in the Holt–Rees version of Norton's irreducibility test in the Meat-axe algorithm. We prove that, for any finite field Fq, the proportion of matrices in M(n,Fq) that are “not good” decays exponentially to zero as the dimension n approaches infinity...