Relatively Prime Polynomials and Nonsingular Hankel Matrices over Finite Fields

TL;DR

This paper explores the connection between coprime polynomials and nonsingular Hankel matrices over finite fields, providing explicit mappings and simplified counting formulas.

Contribution

It establishes a direct link between coprime polynomials and Hankel matrix nonsingularity, with explicit mappings and simplified enumeration methods.

Findings

Probability of coprime polynomials equals probability of nonsingular Hankel matrices

Explicit map from coprime polynomials to Hankel matrices

Simplified formulas for counting coprime polynomials and Hankel matrices

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