Zeros of 2 by 2 Matrix Polynomials

TL;DR

This paper explores the solutions of matrix polynomial equations over 2x2 matrices, establishing conditions for finite solutions and constructing polynomials with a specified number of solutions.

Contribution

It demonstrates that for any number up to a binomial coefficient, there exists an nth degree matrix polynomial with exactly that many solutions.

Findings

More than ${2n race 2}$ solutions imply infinitely many solutions.

Existence of polynomials with exactly m solutions for all m up to ${2n race 2}$.

Characterization of solution counts for matrix polynomial equations.

Abstract

Consider the $n$th degree polynomial equation, $X^n+A_{n-1}X^{n-1}+...+A_1X+A_0=0$ over the ring of 2 by 2 complex matrices. If this equation has more than ${2n \choose 2}$ solutions, then it has infinitely many solutions. We show here that for any $n,m \in\N$ such that $m\leq{2n \choose 2}$, there exists an $n$th degree polynomial equation with exactly $m$ solutions.