Block companion Singer cycles, Primitive recursive vector sequences, and Coprime polynomial pairs over finite fields

Block companion Singer cycles, Primitive recursive vector sequences, and Coprime polynomial pairs over finite fields


Sudhir R. Ghorpade

Department of Mathematics
Indian Institute of Technology Bombay
Powai, Mumbai 400076, India

E-Mail: srg@math.iitb.ac.in

and

Samrith Ram

Department of Mathematics
Indian Institute of Technology Bombay
Powai, Mumbai 400076, India

E-Mail: samrith@mgmail.com


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.


1 Introduction 1
2 The characteristic map 3
3 Relatively prime polynomials 4
4 The case m=2 5
5 Splitting subspaces 7
6 Asymptotic formula 9
7 Application to Toeplitz matrices 11
References 12


This paper is accepted for publication in the Finite Fields and Their Applications . [Preprint Version: arXiv.math/1102.5335v1 (February 2011)]

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