A symmetry property of some harmonic algebraic curves

TL;DR

This paper reveals a surprising symmetry property of harmonic algebraic curves formed by roots of polynomials on the unit circle, showing they create regular polygons when certain argument conditions are met.

Contribution

It introduces a novel symmetry property of harmonic algebraic curves associated with polynomials with roots on the unit circle.

Findings

Points on the unit circle where Arg(P(z))=θ form a regular n-gon.

The symmetry holds for roots of polynomials with roots on the unit circle.

The property links polynomial roots to geometric regular polygons.

Abstract

The aim of this note is to give a surprising symmetry property of some harmonic algebraic curves: when all the roots $z_i$ of a complex polynomial $P$ lie on the unit circle $\U$, the points of $\U$ different from the $z_i$, and such that $\Arg(P(z))=\theta$, form a regular $n$-gon, where $n$ is the degree of $P$.