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Quaternion Polynomials

Prerequisites

To understand this, you need a knowledge of polynomial equations and some familiarity with complex numbers, as can be had in some North American high-school algebra courses.

Introduction

Traditional polynomials with real numbers were always problematic. The Greeks were first confronted with irrational numbers in their attempts to solve the polynomial equation x² = 2 . Besides casting aspersions, nobody knew what to do with equations such as x² = −1 until the 16th century, when the complex numbers were developed.

Polynomial equations reach their full expression in the algebra of the complex numbers. Every polynomial equation with complex coefficients of degree n has a set of exactly n possibly repeated complex solutions. (This is called the fundamental theorem of algebra).

The behavior of polynomials over a given algebra could not get any nicer than with the algebra of the complex numbers. This point is illustrated Goldilocks-fashion with the quaternion algebra. Where the reals are too small an algebra in which to solve polynomial equations, the quaternions are too large.

Issues with commutativity

There is a question about the meaning of a polynomial, due to the non-commutativity of quaternions,

For example, the most general quadratic polynomial term in the real or complex numbers may be expressed by

However, the same expression among quaternions is different from

and this is again different from

Apparently, the most general quaternion polynomial term of degree n has n + 1 quaternion coefficients. One is also left wondering whether the business of shoving coefficients between all the variable factors was really the best way to proceed.

Roots of polynomial equations

The roots of even the simplest quaternion polynomials are far more complicated than those of complex polynomials. For example, the equation q² = −1 has infinitely many solutions. Writing q as the pair of complex numbers (a, b), and applying the formula for multiplication of quaternions, we have

The second of these equations reduces to

so the problem is reduced to the two non-exclusive cases b = 0 and Re(a) = 0. The case b = 0 is just the case in which q represents a complex number, and results as expected in

The case Re(a) = 0 results in

whose solutions, for any complex number a whose imaginary part is less than 1 in magnitude, consist of all the complex numbers b in a circle with a positive radius. The set of solutions of this simple polynomial equation form a 2-dimensional manifold in the 4-dimensional space of quaternions! They lie in the hyperplane given by Re(a) = 0, and therein form a sphere of radius 1.

However, the same way, the equation q² = 1 is found to have only the solutions ±1.

More generally, the quaternion solutions of q² = z, where z is a complex number, are just the two complex roots of z so long as z is not a negative real number, in which case there are infinitely many solutions.

In the 1940s, Niven and others investigated quaternion polynomials, the coefficients of whose terms were to the left of the power function.

References