Zeros of 2 by 2 Matrix Polynomials

Abstract

Consider the $n$th degree polynomial equation, $X^n+A_{n-1}{X}^{n-1}+...+A_{1}X+A_0=0$ over the ring of 2 by 2 complex matrices. If this equation has more than $\left(\genfrac{}{}{0ex}{}{2n}{2}\right)$ solutions, then it has infinitely many solutions. We show here that for any $n,m\in \mathbb{N}$ such that $m\le \left(\genfrac{}{}{0ex}{}{2n}{2}\right)$, there exists an $n$th degree polynomial equation with exactly $m$ solutions.

14 Pages. Research done at an REU at Rutgers Universtiy in 2006 under Robert Wilson