Zeros of harmonic polynomials, critical lemniscates and caustics

TL;DR

This paper improves bounds on the maximum zeros of complex harmonic polynomials and explores how the curvature of critical lemniscates influences caustic geometry and zero distribution.

Contribution

It significantly sharpens existing estimates on zeros of harmonic polynomials and analyzes the geometric relationship between lemniscates and caustics.

Findings

Improved bounds on the number of zeros of harmonic polynomials

Established connection between lemniscate curvature and caustic geometry

Analyzed the impact of lemniscate curvature on zero distribution

Abstract

In this paper we sharpen significantly several known estimates on the maximal number of zeros of complex harmonic polynomials. We also study the relation between the curvature of critical lemniscates and its impact on geometry of caustics and the number of zeros of harmonic polynomials.