Remarks on Wilmshurst's theorem S. LeeAntonio LerarioErik Lundberg 30C55harmonic polynomialvalencedegree theory We demonstrate counterexamples to Wilmshurst'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. Indiana University Mathematics Journal 2015 text pdf 10.1512/iumj.2015.64.5526 10.1512/iumj.2015.64.5526 en Indiana Univ. Math. J. 64 (2015) 1153 - 1167 state-of-the-art mathematics http://iumj.org/access/