On extensions of theorems in the geometry of polynomials

Abstract

The geometry of polynomials started with the famous Gauss-Lucas theorem at the beginning of the 19th century. Many giants in mathematics such as Laguerre, Legendre, Chebyshev, Polya and Szego also contributed to theories that have become fundamental tools in multiple areas in applied mathematics and computer science.

This thesis extends several pivotal results in the developments of the geometry of polynomials. Chapter 2 is about the extension of the basic binary relation on two polynomials, apolarity, to different degree cases, and therefore our extension reduces the additional condition on the multiplicities of zeros in a theorem about the canonical representation of apolar pair given in the paper by J. P. Brennan, J. V. Chipalkatti, and R. M. Fossum. The second part of Chapter 2 provides several novel and original examples of stability preservers for all circular domains and the unit disk domain.

The next topic of interest is the location of zeros of polynomials under transformations. By considering the change of basis of polynomials from monomials to monic orthogonal polynomial system, the corresponding analogy of a classical result about the positivity of determinants of Vandermonde matrix is now found and proved with aid of a result by Obrechkoff on the rule of sign. Another transformation is the Wronskian of a polynomial, regarding it as a binary transformation of polynomials, we succeeded in refining a result of Dilcher to construct a class of polynomials with zeros in the open left half unit disk from one special type of real transcendental entire functions of significant interest in the field.

Finally Chapter 4 gives our original studies to the characteristic polynomials of Jacobi matrices. We start this chapter by giving the explicit Jacobi matrices with concyclic relations on the roots of the complexification of characteristic polynomials, as well as several analyses with different tools on the distribution of zeros of the complexification of a consecutive pair of Laguerre polynomials. Lastly, the thesis ends with pointing out our interests on a new class of Jacobi matrices with a unique property called equideterminant.