Graduation Year

2021

Document Type

Dissertation

Degree

Ph.D.

Degree Name

Doctor of Philosophy (Ph.D.)

Degree Granting Department

Mathematics and Statistics

Major Professor

Catherine Bénéteau, Ph.D.

Committee Member

Dmitry Khavinson, Ph.D.

Committee Member

Mohamed Elhamdadi, Ph.D.

Committee Member

Seung-Yeop Lee, Ph.D.

Committee Member

Myrto Manolaki, Ph.D.

Keywords

Bezout’s theorem, maximum number of zeros, Green theorem, rational functions, Shwarz function, gravitational lensing

Abstract

In this thesis, we study topics related to harmonic functions, where we are interested in the maximum number of solutions of a harmonic polynomial equation and how it is related to gravitational lensing. In Chapter 2, we study the conditions that we should have on the real or complex coefficients of a polynomial p to get the maximum number of distinct solutions for the equation p(z) − z¯ 2 = 0, where deg p = 2. In Chapter 3, we discuss the lens equation when the lens is an ellipse, a limac¸on, or a Neumann Oval. Also, we give a counterexample to a conjecture by C. Ben´ eteau and N. Hudson in [2]. We also discuss estimates ´ related to the maximum number of solutions for the lens equation for the Neumann Oval.

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