Remarks on Wilmshurst's theorem
Complex Variables
2013-08-30 v1 Algebraic Geometry
Abstract
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 two. Examples are provided to show that this conclusion cannot be improved.
Cite
@article{arxiv.1308.6474,
title = {Remarks on Wilmshurst's theorem},
author = {Seung-Yeop Lee and Antonio Lerario and Erik Lundberg},
journal= {arXiv preprint arXiv:1308.6474},
year = {2013}
}
Comments
14 pages, 3 figures