The Image of Critical Circle and Zero-free Curve for Quadrinomials

TL;DR

This paper investigates the zero distribution of a family of harmonic quadrinomials, revealing that the image of a critical circle forms a hypocycloid and identifying a zero-free curve for quadratic quadrinomials.

Contribution

It characterizes the image of critical circles as hypocycloids and identifies zero-free regions for a two-parameter family of quadratic quadrinomials.

Findings

The image of a critical circle is a hypocycloid.

A zero-free curve for quadratic quadrinomials is determined.

The zero distribution depends on the parameters of the family.

Abstract

The location of the zeros of a two-parameter family of complex-valued harmonic quadrinomials depends on the parameters. In this paper, we determine and demonstrate that the image of some critical circle under these two-parameter family of complex-valued harmonic quadrinomial is a hypocycloid. We also determine a zero-free curve for two-parameter family of quadratic qudrinomial.