Remarks on Wilmshurst's theorem

TL;DR

This paper presents counterexamples to Wilmshurst's conjecture on harmonic polynomial valence, proposes a new linear bound, and explores higher-dimensional generalizations of harmonic polynomial zero sets.

Contribution

It provides the first counterexamples to Wilmshurst's conjecture and extends the discussion of harmonic polynomial zero sets to higher dimensions.

Findings

Counterexamples to Wilmshurst's conjecture on valence.

Proposed linear bound in analytic degree.

Higher-dimensional zero set codimension results.

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.