Block Companion Singer Cycles, Primitive Recursive Vector Sequences, and Coprime Polynomial Pairs over Finite Fields

TL;DR

This paper explores the enumeration of special nonsingular matrices over finite fields, connecting polynomial coprimality, block companion matrices, and applications to Toeplitz matrices, with some conjectures and asymptotic results.

Contribution

It proves a special case of a conjecture on block companion matrices and links it to polynomial coprimality and splitting subspaces over finite fields.

Findings

Proved a special case of the conjecture using coprimality probabilities.

Established an asymptotic version of the conjectural enumeration formula.

Discussed applications to counting nonsingular Toeplitz matrices.

Abstract

We discuss a conjecture concerning the enumeration of nonsingular matrices over a finite field that are block companion and whose order is the maximum possible in the corresponding general linear group. A special case is proved using some recent results on the probability that a pair of polynomials with coefficients in a finite field is coprime. Connection with an older problem of Niederreiter about the number of splitting subspaces of a given dimension are outlined and an asymptotic version of the conjectural formula is established. Some applications to the enumeration of nonsingular Toeplitz matrices of a given size over a finite field are also discussed.