Generalized polynomial Bézoutian with respect to a Jacobson chain basis over an arbitrary field (Q969002)

The aim of this paper is to use the polynomial model approach and the theory of operator representation to investigate the generalized polynomial Bézoutian with respect to a Jacobson chain basis over an arbitrary field. For that purpose a bilinear form using a matrix-vector product is introduced and it is shown that the underlying Bézoutian is the matrix representation of an operator relative to a pair of dual bases. Then, the Barnett-type formula for factorization and an intertwining relation are derived. Also, the diagonal reduction of the Bézoutian via a generalized confluent Vandermonde matrix is investigated. For a pair of given polynomials, the form of the diagonal reduction is unique and does not depend on the choice of the Jacobson chain basis.