Zeros of Harmonic Polynomials and Related Applications

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) z2 = 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. B en 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.