Relatively prime polynomials and nonsingular Hankel matrices over finite fields

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. (C) 2010 Elsevier Inc. All rights reserved.