Relatively Prime Polynomials and Nonsingular Hankel Matrices over Finite Fields
Combinatorics
2011-02-10 v1
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
Cite
@article{arxiv.1011.1760,
title = {Relatively Prime Polynomials and Nonsingular Hankel Matrices over Finite Fields},
author = {Mario Garcia Armas and Sudhir R. Ghorpade and Samrith Ram},
journal= {arXiv preprint arXiv:1011.1760},
year = {2011}
}
Comments
10 pages; to appear in the Journal of Combinatorial Theory, Series A