Greatest common divisor of several polynomials

Abstract

Given a polynomial a(λ) with degree n, and polynomials b1(λ), …, bm(λ) of degree not greater than n – 1, then the degree k of the greatest common divisor of the polynomials is equal to the rank defect of the matrix R = [b1(A), b2(A), …, bm(A)], where A is a suitable companion matrix of a(λ). Furthermore, it is shown that if the first k rows of R are expressed as linear combinations of the remaining n – k rows (which are linearly independent) then the greatest common divisor is given by the coefficients of row k + 1 in these expressions. A simple expression is derived for R and a permutation of the columns of this matrix establishes a direct connexion with controllability of a constant linear control system. Finally, when m = 1 a relationship between the corresponding R and Sylvester's matrix is exhibited.