Research / Harmonic Mappings
The valence of harmonic polynomials: Wilmshurst showed that if a polynomial planar harmonic mapping has bounded zero set, then the number of zeros is at most the square of the degree (Bezout's theorem can be applied without any genericity assumption). He went on to conjecture that $$ N \leq m(m-1) + 3n-2$$ Seung-Yeop Lee, Antonio Lerario, and I have a preprint investigating this theorem and Wilmshurst's conjecture that Bezout's bound can be refined in terms of the separate analytic and anti-analytic degrees. We provide counterexamples to the specific bound conjectured by Wilmshurst, and we conjecture an upper bound that is linear in the analytic degree when the anti-analytic degree is fixed. We also initiate a discussion of Wilmshurst's theorem in higher dimensions.
Random harmonic polynomials: Given the high level of variability in the number of zeros, it is interesting to ask about the valence of a random harmonic polynomial. W. Li and A. Wei posed this problem, applied the Kac-Rice formula, and performed asymptotic analysis when the polynomials p and q are sampled independently from the complex Kostlan ensemble. A. Lerario and I, and more recently A. Thomack and Z. Tyree have studied asymptotics for other models.