A topological version of the argument principle and Rouche’s theorem

Abstract

In this paper, the topological nature of the generalization of the classical argument principle well-known in multidimensional complex analysis is discussed. The topological approach offered here ensures some topological results on the structure of pole and zero sets of holomorphic maps of bounded domains in complex manifolds. Some connections with integral representations of holomorphic functions are studied and a geometric interpretation of the Martinelli-Bochner complex-valued, differential-form realization is given.