Consider $U_n = \mathbb{C}[x_1,...,x_n]$ - the vector space of all complex polynomials over with $n$ variables $x_1,...,x_n$ ($n \ge 2$). A polynomial $h \in U_n$ is homogeneous of degree $d \in \mathbb{N}$ when $h(\lambda x_1, ..., \lambda x_n) = \lambda^d h(x_1, ..., x_n) \; \forall \lambda \in \mathbb{C}$. It's known that the set of all homogeneous polynomials of degree $d$ in $U_n$ is a subspace of $U_n$, now called $V_{n,d}$. We have $\dim V_{n,d} = \binom{n+d-1}{n-1}$.
A polynomial $h \in V_{n,d}$ is harmonic when its Laplacian $\Delta h = 0 \in V_{n,d-2}$. Then, the set of all harmonic polynomials in $V_{n,d}$ is a subspace of $V_{n,d}$, now called $H_{n,d}$.
To exclude some trivial cases, we only consider $n \ge 2$ and $d \ge 2$.
Problem: Given $n, d \in \mathbb{N}$, describe an algorithm to systemically find a basis of $H_{n,d} \subset V_{n,d} \subset U_n$.
So far, I only know the fact that $\dim H_{3,d} = 2d+1$ for all $d$. I assume there is a formula for $\dim H_{n,d}$ w.r.t $n,d$ (I guess it's $\binom{n+d-1}{n-1} - \binom{n+d-3}{n-1}$). Regardless, besides the trivial cases, I have not achieved any progress in this problem.