We demonstrate counterexamples toWilmshurst's conjecture on the valence of harmonic polynomials in the plane, and we conjecture a bound that is linear in the analytic degree for each fixed anti-analytic degree. Then, we initiate a discussion of Wilmshurt's theorem in more than two dimensions, showing that if the zero set of a polynomial harmonic field is bounded, then it must have codimension at least 2. Examples are provided to show that this conclusion cannot be improved.
