Topological Degree of Some Mappings

1. In previous papers [2; 3], the problem of proving an existence theorem for a certain functional equation was reduced to that of computing the topological degree of a mapping in Euclidean w-space defined by homogeneous polynomials or infinite series. The complex case of the latter problem was solved in [4]. Since the problem is analogous to that of studying the roots of a polynomial equation, we would expect the real case to be more complicated. Here we obtain a result that is an analogue of the theorem that a real polynomial equation of odd degree has at least one real root. Also we describe the solution for the case w = 2 if the mapping is defined by homogeneous polynomials.