Research On Several Classes Of Bezout Matrices

Bezout matrix is a special class of quadratic form, arising from the theory of resultants, and it was introduced to study the root location problems of polynomials as the earliest application. Over the past few decades, the study on Bezoutians has been applied to the fields of symbolic computing, polynomial stability and systems engineering, which plays a significant role in dealing with some related issues therein.In this paper, firstly some research results on Bezoutians achieved by domestic and foreign scholars are summarized, and several basic concepts and properties are recalled, which are useful to the following sections of this paper.Secondly, according to polynomial Taylor expansion, the expression of Bezout matrix of the polynomials under a special basis is given, and the algorithm (namely recursive formulas) of the elements in this Bezoutian form is also obtained.Next, using the fact that a continuous function on a closed interval [0,1] can be uniformly approximated by Bernstein polynomial, this paper introduces the concept on the uniform approximation of Bezoutians. Consequently, some conclusions about them, such as the uniform approximation forms of Barnett type factorization and triangular factorization formulas of classical Bezout matrix, the algorithm of the uniform approximation form and the identical relations on elements of the two classes of uniform approximation forms, are obtained.Finally, the concept called (s,t)-type Bezout matrix is introduced. The algorithms for the elements of two classes of (s,t)-type Bezoutians generated by a pair of polynomials are presented. Being inspired by the form of the algorithms, this paper then introduces a new concept called relevancy to discuss the relation of two classes of (s,t)-type Bezoutians. In addition, the (s,t)-type Bezoutian is generalized to a new form generated by a pair of general functions, and some corresponding properties are obtained.