Enestr¨om-Kakeya Type Results for Complex and Quaternionic Polynomials

Author

Matthew Gladin, East Tennessee State UniversityFollow

Degree Name

MS (Master of Science)

Program

Mathematical Sciences

Date of Award

5-2023

Committee Chair or Co-Chairs

Robert Gardner

Committee Members

Anant Godbole, Rodney Keaton

Abstract

The well known Eneström-Kakeya Theorem states that: for P(z)=∑_i=0ⁿ a_i zⁱ, a polynomial of degree n with real coefficients satisfying 0 ≤ a₀ ≤ a₁ ≤ ⋯≤ a_n, all zeros of P(z) lie in |z|≤1 in the complex plane. In this thesis, we will find inner and outer bounds in which the zeros of complex and quaternionic polynomials lie. We will do this by imposing restrictions on the real and imaginary parts, and on the moduli, of the complex and quaternionic coefficients. We also apply similar restrictions on complex polynomials with complex coefficients to give a bound on the number of zeros in a disk centered at the origin. For each result, we will consider lacunary polynomials, that is polynomials of the form P(z)=a₀+∑_i=mⁿ a_i zⁱ, as well as a new class of polynomials P(z)=a₀+∑_i=m^m' a_i zⁱ + a_n zⁿ.

Document Type

Thesis - unrestricted

Recommended Citation

Gladin, Matthew, "Enestr¨om-Kakeya Type Results for Complex and Quaternionic Polynomials" (2023). Electronic Theses and Dissertations. Paper 4167. https://dc.etsu.edu/etd/4167

Copyright

Copyright by the authors.