The number of components of complements to level surfaces of partially harmonic polynomials (Q1277490)

The author proves the upper bound \(2m^{n-1}+O(m^{n-2})\) for the number of connected components of the complement in \(\mathbb{R}^n\) to the zero set of a real \(k\)-harmonic polynomial of degree \(m\) in \(n\) variables, where \(1\leq k\leq n\) and \(n\geq 2\) (compare with the fact that for arbitrary real polynomials of degree \(m\) in \(n\) variables the upper bound is \(m^n+O(m^{n-1})\), and the exponent cannot be diminished). A real polynomial \(F\) in \(n\) variables is \(k\)-harmonic if it satisfies the Laplace equation with respect to \(k\) variables: \[ (\partial^2/ \partial x^2_1+\cdots +\partial^2/ \partial x^2_k)F=0. \] The result follows from two observations: (1) The complement in \(\mathbb{R}^n\) to the zero set of a real \(k\)-harmonic polynomial in \(n\), variables does not have bounded components (according to the maximum principle). (2) The number of connected components of the complement in the sphere \(S^{n-1}\) to the zero set of a real polynomial of degree \(m\) in \(n\) variables does not exceed \(2m^{n-1}+ O(m^{n-2})\). The author also proves sharper estimates under the assumptions that the singular set of the zero locus is compact or that the leading homogeneous part of the \(k\)-harmonic polynomial is nondegenerate.