Relatively Prime Polynomials and Nonsingular Hankel Matrices
over Finite Fields
# Relatively Prime Polynomials and Nonsingular Hankel Matrices
over Finite Fields

###
Mario Garcia-Armas

#### *
Department of Mathematics, *

University of British Columbia

Vancouver, BC V6T 1Z2, Canada

E-Mail: marioga@math.ubc.ca

### 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

The probability for two monic polynomials of a positive degree *n* with
coefficients in the finite field **F**_{q} to be relatively prime
turns out to be identical with the probability for an *n* x *n*
Hankel matrix over **F**_{q}
to be nonsingular. Motivated by this, we give an explicit map from pairs
of coprime polynomials to nonsingular Hankel matrices that explains this
connection. A basic tool used here is the classical notion of Bezoutian
of two polynomials. Moreover, we give simpler and direct proofs of the
general formulae for the number of *m*-tuples of relatively prime
polynomials over **F**_{q} of given degrees and for the number
of *n* x *n* Hankel matrices over **F**_{q}
of a given rank.

1 | Introduction | 1 |

2 | Preliminaries | 2 |

3 | An Explicit Surjection | 3 |

4 | Relatively Prime Polynomials | 5 |

5 | Hankel matrices over **F**_{q} | 6 |

| References | 10 |

This paper is accepted for publication in the
Journal of Combinatorial Theory, Series A.
[Preprint Version:
arXiv.math/1011.1760v1 (November 2010)]

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