Physical Sciences and Mathematics Commons^™

An Analysis Of Polynomials That Commute Under Composition, Samuel J. Williams

Honors Theses

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 …

Factorization Of Polynomials And Real Analytic Function, Radoslaw L. Stefanski

Honors Theses

In this project, we address the question: When can a polynomial p(x, y) of two variables be factored as p(x, y) = f(x)g(y), where f and g are polynomials of one variable. We answer this question, using linear algebra, and create a Mathematica program which carries out this factorization. For example,

3+3x-5x^3+y+xy-5/3x^3y+y^2+xy^2-5/3x^3y^2 = (1+x-5/3x^3)(3+y+y^2)

We then generalize this concept and ask: When can p(x,y) can be written as

p(x,y) = f1(x)g2(y)+f2(x)g2(y)+...+fr(x)gr(y)

where fj,gj are polynomials. This can certainly be done (for large enough r). Which is the minimum such r? Again, we have a Mathematica program which carries out this …