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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