Explicit expressions for Bezoutians

AbstractThe following commutator identity is proved:[u(S∗), v(S)] = [v1(S∗), u1(S)]. Here S is the n by n matrix of the truncated shift operator S = (Γi,i+1), i = 0, 1,…, n − 1, and u, v are two polynomials of degree not exceeding n. The reciprocal polynomial f;1 of a polynomial f; of degree ⩽n is defined by f1(z) = znf(1z). The commutator identity is closely related to some properties of the Bezoutian matrix of a pair of polynomials; it is used to obtain the Bezoutian matrix in the form of a simple expression in terms of S and S∗. To demonstrate the advantage of this expression, we show how it can be used to obtain simple proofs of some interesting corollaries