In $N$-dimensional space, we can show by direct calculation that the polynomial
$$
r^{2K+N-2}\nabla_a\nabla_b\nabla_c\cdots \frac{1}{r^{N-2}}
\hspace{1cm}
\text{(with $K$ derivatives)}
$$
is harmonic (annihilated by the Laplacian $\nabla^2$), where $\nabla_n$ is the partial derivative with respect to the $n$th coordinate and $r$ is the distance from the origin.
Is every homogeneous harmonic polynomial of degree $K$ a linear combination of these? If so, how do we prove this? If not, what is a counterexample?
Motive: I'm studying physics, and this seems like a much nicer way to approach the theory of spherical harmonics (just divide this polynomial by $r^K$ to get a spherical harmonic) compared to the typical physics-textbook approach using spherical coordinates, but it's not obvious to me that all spherical harmonics are linear combinations of these for an arbitrary number of dimensions $N$.
Best Answer
This is a theorem of Maxwell and Sylvester, see for example https://arxiv.org/abs/math-ph/0408046 , https://arxiv.org/abs/0805.1904 and https://arxiv.org/abs/astro-ph/0412231 but, it is in 3d only.
In higher dimension, this is the object of a conjecture of Shubin, see page 10 of https://arxiv.org/abs/0704.1174
Great question BTW, related to some interesting and beautiful mathematics.