Master of Science
There are problems concerning the set of root of a sequence of polynomials. A simple question is to ask if the set of roots lies entirely in real numbers. Many approaches to answering this question are known. The main object of this dissertation is to develop new tools for tackling the above problem. In order to be able to apply the ideas we define a specific numerical sequence, and then we consider the sequence of their minimal polynomials over the rational numbers.
The first step is to find a recursive way of defining the sequence of polynomials by using the so-called Bézout matrices, which are a specific family of matrix polynomials. Having the construction of minimal polynomials as the determinant of some Bézout matrix, we interpret the roots of each polynomial as eigenvalues of the corresponding Bézout matrix. Then by using a symmetric linearization of such matrix polynomial we can talk about the real roots.
Rafiee Sevyeri, Leili, "A Sequence of Symmetric Bézout Matrix Polynomials" (2016). Electronic Thesis and Dissertation Repository. 4083.