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)]

### Download the full paper as:

Back to the List of Publications

Back to the Sudhir Ghorpade's Home Page