  We obtain quantitative estimates of local flatness of zero sets of harmonic
polynomials. There are two alternatives: at every point either the zero set
stays uniformly far away from a hyperplane in the Hausdorff distance at all
scales or the zero set becomes locally flat on small scales with arbitrarily
small constant. An application is given to a free boundary problem for harmonic
measure from two sides, where blow-ups of the boundary are zero sets of
harmonic polynomials.
