Date of Award
Bachelor of Science
polynomials, composition, similarity, commute
It is well known that polynomials commute under addition and multiplication. It turns out that certain polynomials also commute under composition. In this paper, we examine polynomials with coefficients in the field of complex numbers that commute under composition (also referred to as “commuting polynomials”). We begin this examination by defining what it means for polynomials to commute under composition. We then introduce sequences of commuting polynomials and observe how the polynomials in these sequences (later defined as chains) along with other commuting polynomials relate to a concept called similarity. These observations allow us to better understand the qualities and characteristics of polynomials that commute under composition. Later, we characterize all chains of polynomials with complex coefficients. We conclude the paper with an exploration of a concept called self similarity and how it relates to commuting polynomials. This exploration reinforces the observations we made about the qualities and characteristics of commuting polynomials and also provides new insights. The examination outlined in this paper provides us with an accessible understanding of polynomials that commute under composition.
Williams, Samuel J., "An Analysis of Polynomials that Commute Under Composition" (2013). Honors Theses. 756.