On The Greatest Common Divisor With Respect To Some Polynomials

Given a power (monomial) basis, the greatest common divisor (GCD) of polynomials can be achieved by the calculation of resultant matrix, Bezoutian, etc. Based on this technique, the present paper mainly focus on the research of resultant matrix, the Bezoutian and the related problems of GCD in the bases of bilinear transformation function, i.e{αa1(n)(λ)=(1+λ)i(1-λ)n-i,i= 0,1,...,n} as well as its generalized form{βin(λ)= (a+λ)i(b-λ)n-i,i= 0,1,...,n,a≠-b}.The general review of GCD is summarized firstly. Specifically, some gen-eral ways of determining the GCD of polynomials with the given power basis are generalized, which include the classical Euclidean algorithm, and some ma-trix related algorithms, such as resultant matrix, Bezoutian. Then we give an introduction of some GCD methods for polynomials and scaled polynomials with a specified basis, i.e. Bernstein basis.The main contribution of the present paper are introduced in the third and fourth chapters, where some operations including division operation in the basis of bilinear transformation function are given. With the fact that the certain division of polynomials is based on a simple ring isomorphism, the operation work with the bilinear transformation function bases will be transformed to the corresponding work with monomial basis. Then a new way to obtain the GCD of polynomials in this typical basis is developed, which also can be treated as a development of the division operation.In the present work, a novel basis is constructed with the formulation of{βin(λ)= (a+λ)i(b-λ)n-i, i= 0,1,..., n, a≠-b}. Furthermore, with the proposed basis, the corresponding formulations of Bezout matrix, companion matrix, companion resultant matrix as well as their relationships with GCD are given in this paper.