Zeros of regular functions and polynomials of a quaternionic variable

The fundamental elements of a new theory of regular functions of a quaternionic variable have been recently developed, following an idea of Cullen. In this paper we present a detailed study of the structure of the zero set of Cullen-regular functions. We prove that the zero sets of the functions under investigation consist of isolated points or isolated 2-spheres, in the 4-dimensional real space of quaternions. Moreover, the zeros of a regular function can be factored by means of a non-standard product. The Fundamental Theorem of Algebra for quaternions, and the approach here adopted lead, in particular, to a deeper insight of the geometric and algebraic properties of the zero sets of polynomials with quaternionic coefficients.