Roots and Derivatives in the Complex Plane

Abstract

In this paper, we explore the relationships between roots and derivatives in the complex plane. Unfortunately, there exists no one-size-fits all solution like Rolle’s Theorem which describes the relationship between solutions and roots of derivatives, but there are some very interesting patterns that emerge, particularly with complex polynomials. With the help of a very powerful piece of computer software called “Maxima”, we explore these patterns in depth, and hope to start building an intuition for the relationship between roots and derivatives in the complex plane. Many of the patterns which emerge are left unproved; This will hopefully inspire future research into the subject. We also delve into research done by French Mathematician, Jean Dieudonn´e, who explored these concepts in far greater generality.