Geodesic
Factorization of special harmonic polynomials of three variables
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 17 (2020), pp. 1299-1312.

Voir la notice de l'article provenant de la source Math-Net.Ru

We consider homogeneous 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 homogeneous rotation invariant polynomials $p$ of degree $m$ such that $p$ is reducible over the rationals and $(x+yi)^{n}p$ is harmonic for some $n\in\mathbb{N}$. We describe $\mathfrak{R}_{m}$ for $m\leq5$ and prove that $\mathfrak{R}_{6}$ and $\mathfrak{R}_{7}$ are finite.
Keywords: Legendre functions, harmonic polynomials, factorization. 
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V. M. Gichev. Factorization of special harmonic polynomials of three variables. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 17 (2020), pp. 1299-1312. http://geodesic.mathdoc.fr/item/SEMR_2020_17_a138/

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