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

摘要

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