Factorization of special harmonic polynomials of three variables

TL;DR

This paper studies special harmonic polynomials in three variables, focusing on their factorization properties and classifying reducible cases up to degree 7, revealing finiteness in higher degrees.

Contribution

It characterizes reducible harmonic polynomials of degrees up to 5 and proves finiteness of reducible cases for degrees 6 and 7.

Findings

Classified reducible harmonic polynomials for degrees ≤ 5.

Proved finiteness of reducible polynomials for degrees 6 and 7.

Abstract

We consider harmonic polynomials of real variables $x,y,z$ that are eigenfunctions of the rotations about the axis $z$. They have the form $(x\pm yi)^{n}p(x,y,z)$, where $p$ is a rotation invariant polynomial. Let ${\mathfrak R}_{m}$ be the family of the polynomials $p$ of degree $m$ which are reducible over the rationals. We describe ${\mathfrak R}_{m}$ for $m\leq5$ and prove that ${\mathfrak R}_{6}$ and ${\mathfrak R}_{7}$ are finite.