Sharp bounds for the valence of certain harmonic polynomials

In Khavinson and Swiatek, (2002) it was proved that harmonic polynomials z - <(p(z))over bar>, where p is a holomorphic polynomial of degree n > 1, 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 certain extremal polynomials p. We also count the number of equivalence classes of these polynomials.