# Zeros of Complex-Valued Harmonic Functions

## Document Type

Presentation Abstract

## Presentation Date

5-1-2023

## Abstract

BYU has a large number of students involved in mentored research. This past year, we had 362 math majors, and 119 of them were involved in research. Although it is easy to enumerate the benefits to students of doing such research, mentoring these students can be a challenge. In particular, for many of us, our own research is not accessible to undergraduates. In the past 4 years, I have added a new thread to my research program that has led to research projects for 15 undergraduates and 4 MS students. Students can begin with relatively little background, but there are still accessible open problems of interest to the larger community of complex analysts.

Specifically, we study the zeros of complex-valued harmonic polynomials. The Fundamental Theorem of Algebra states that for a polynomial $f$ in one complex variable of degree $n \geq 1$, the number of zeros (counting multiplicity) is exactly $n$. However, if $f=h+\overline{g}$ is the sum of an analytic polynomial and the conjugate of an analytic polynomial, the theorem no longer applies. Strange things can happen; the number of zeros need not equal the degree and the number of zeros can vary with the coefficients.

In this talk, I give an overview of this research, highlighting the contributions of my students. I will also share ideas for making undergraduate research work.

## Recommended Citation

Brooks, Jennifer, "Zeros of Complex-Valued Harmonic Functions" (2023). *Colloquia of the Department of Mathematical Sciences*. 653.

https://scholarworks.umt.edu/mathcolloquia/653

## Additional Details

May 1, 2023 at 3:00 p.m. Math 103