Bernstein-Bezout Matrix And Mobius Transformations Of Vandermonde Matrices

In recent years the study of Bernstein-Bezoutians mainly involves their computationand structure properties. In this paper, the basic characters of Bernstein-Bezoutians, thezeros of Bernstein polynomials and Mobius transformations of Vandemonde matrices areinvestigated. The paper consists of five chapters.In chapter one, the backgrounds and significances of Bernstein-Bezoutians are simplyintroduced. The main work done in this paper is also listed.From the definition of Bernstein-Bezoutian determined by its generating function inchapter two, we obtain the intertwining relation between Bernstein-Bezoutian andcompanion matrix, generalized Barnett factorization and Vandermonde reduction by meansof pure algebraic methods. Then a new resultant matrix is defined in section three by whichwe can decompose the Bernstein-Bezoutian generated by two Bernstein polynomials.In chapter three we mainly study the regions of the zeros of Bernstein polynomials. Atfirst we compute the traces of matrices ψ(A), ψ(A)~2,ψ(A)~*ψ(A) which determines theregions of the zeros of Bernstein polynomials, then the formulae bounds for the zerosof Bernstein polynomial are obtained.In chapter four the generalized expression is given for confluent Vandermondematrices by Mobius transformation, which develops the Vandermonde reduction ofBezoutian.In chapter five we pose some unsolved questions for further research.