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

In this paper k-harmonic polynomials in R-n, i.e., polynomials satisfying the Laplace equation with respect to k variables: (partial derivative(2)/partial derivative x(1)(2) + ... + partial derivative(2)/partial derivative x(k)(2))F = 0 are considered; here 1 less than or equal to k less than or equal to n and n greater than or equal to 2. 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.