Searching for the maximal valence of harmonic polynomials: a new example

TL;DR

This paper establishes a new lower bound for the maximum number of zeros of harmonic polynomials with specific degree constraints, advancing understanding in polynomial zero distribution.

Contribution

It introduces a novel example that improves the known lower bounds for zeros of harmonic polynomials with degrees n and n-2.

Findings

New lower bound for zeros of harmonic polynomials

Explicit example demonstrating the bound

Enhanced understanding of harmonic polynomial zeros

Abstract

We find a new lower bound for the maximal number of zeros to harmonic polynomials, $p(z)+\overline{q(z)}$, when ${\rm deg}\, p = n$ and ${\rm deg}\, q = n-2$.