On some topological characteristics of harmonic polynomials

The paper studies the geometric and topological properties of harmonic homogeneous polynomials. Based on the study of the zero-level lines of polynomials on the unit sphere, the concept of topological type for such polynomials is introduced. Topological types are described for harmonic polynomials up to the third degree inclusive.
In the case of complex-valued harmonic polynomials, the distributions are investigated of their critical points in regions on the sphere in which their real and imaginary parts have constant sign. It is shown 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, conclusions are drawn about the number of critical points of the functions under study.