Factorization of special harmonic polynomials of three variables

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 $m\le 5$ and prove that ${\mathfrak{R}}_6$ and ${\mathfrak{R}}_7$ are finite.