Sharp bounds for the valence of certain harmonic polynomials

D. Khavinson and G. Swiatek proved that harmonic polynomials p(z)+q(z), where p is holomorphic, q is antiholomorphic, and deg p = n > 1 = deg q, can have at most 3n-2 complex zeros. We show that this bound is sharp for all n by proving a conjecture of Sarason and Crofoot about the existence of holomorphic polynomials p which map all critical points to their complex conjugates. We also count the number of equivalence classes of real polynomials solving this system of equations. The methods employed rely on Thurston’s characterization of post-critically finite rational maps, in particular the results on polynomials by S. Levy and A. Poirier.