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.

在本文中，我们研究调和齐次多项式的几何和拓扑性质。基于对单位球面上此类多项式的零级线的研究，我们引入了它们的拓扑类型的概念。我们描述了三次谐波多项式的拓扑类型。

在复值调和多项式的情况下，我们考虑其临界点在球体上那些区域的分布，其中它们的实部和虚部具有恒定的符号。我们证明，当从实数多项式传递到复数多项式时，这些区域的数量会增加，而调和多项式的模数平方的最大值会减少。使用欧拉公式，我们对所考虑的函数的临界点数量做出了某些结论。