Title: Remarks on Wilmshurst's theorem

Authors: Seung-Yeop Lee, Antonio Lerario and Erik Lundberg

Issue: Volume 64 (2015), Issue 4, 1153-1167


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.