The number of components of complements to level surfaces of partially harmonic polynomials

In this paper $k$-harmonic polynomials in $\mathbb R^n$ i.e. polynomials satisfying the Laplace equation with respect to $k$ variables: $(\partial^2/\partial x_1^2+\dots+\partial^2/\partial x_k^2)F=0$ are considered; here $1\le k\le n$, $n\ge2$. For a polynomial $F$ (of degree $m$) of this type, it is proved that the number of components of the complements of its level sets does not exceed $2m^{n-1}+O(m^{n-2})$. Under the assumptions that the singular set of the level surface is compact or that the leading homogeneous part of the $k$-harmonic polynomial $F$ is nondegenerate, sharper estimates are also established.