Zero-sets of quaternionic and octonionic analytic functions with central coefficients

Abstract

We prove that the zero set of any quaternionic (or octonionic) analytic function f with central (that is, real) coefficients is the disjoint union of codimension two spheres in R⁴ or R⁸ (respectively) and certain purely real points. In particular, for polynomials with real coefficients, the complete root-set is geometrically characterisable from the lay-out of the roots in the complex plane. The root-set becomes the union of a finite number of codimension 2 Euclidean spheres together with a finite number of real points. We also find the preimages f^-1 for any quaternion (or octonion) A. We demonstrate that this surprising phenomenon of complete spheres being part of the solution set is very markedly a special 'real' phenomenon. For example, the quaternionic or octonionic Nth roots of any non-real quaternion (respectively octonion) turn out to be precisely N distinct points. All this allows us to do some interesting topology for self-maps of spheres.