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 Fq to be relatively prime
turns out to be identical with the probability for an n x n
Hankel matrix over Fq
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 Fq of given degrees and for the number
of n x n Hankel matrices over Fq
of a given rank.
| 1 | Introduction | 1 |
| 2 | Preliminaries | 2 |
| 3 | An Explicit Surjection | 3 |
| 4 | Relatively Prime Polynomials | 5 |
| 5 | Hankel matrices over Fq | 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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