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.