Zeros of Harmonic Functions whose Caustic is a Non-Singular Image of an Epicycloid

TL;DR

This paper studies the zeros of a family of harmonic functions whose critical curves map to non-singular images of epicycloids, providing a detailed zero-counting theorem using the harmonic Argument Principle.

Contribution

It introduces a new class of harmonic functions with critical curves as non-singular epicycloid images and derives a zero-counting theorem for them.

Findings

Zero-counting theorem for harmonic functions with epicycloid critical images

Analysis of the critical curve's geometry and its impact on zeros

Extension of the harmonic Argument Principle to this new setting

Abstract

Recent researchers have investigated how the zeros of certain families of complex harmonic functions change with a single parameter. Many leverage the well-behaved images of the critical curve and the harmonic analogue of the Argument Principle to prove zero-counting theorems. In this paper, we investigate the zeros of a family of harmonic functions for which the image of its critical curve is a non-singular linear image of an epicycloid. By analyzing this curve and using the harmonic analogue of the Argument Principle, we obtain a detailed zero-counting theorem for our family.