On Some Topological Characteristics of Harmonic Polynomials

Abstract

In this paper, we study geometric and topological properties of harmonic homogeneous polynomials. Based on the study of zero-level lines of such polynomials on the unit sphere, we introduce the notion of their topological type. We describe topological types of harmonic polynomials up to the third degree inclusive.

In the case of complex-valued harmonic polynomials, we consider distributions of their critical points in those regions on the sphere, where their real and imaginary parts have constant signs. We demonstrate that when passing from real to complex polynomials, the number of such regions increases and the maximal values of the square of the modulus of the harmonic polynomial decrease. Using the Euler formula, we make certain conclusions about the number of critical points of functions under consideration.