Harmonic polynomial

Remarks

Example: $p(x, y, z) = x^2 + y^2 - 2z^2$ is harmonic, since $\Delta p = 2 + 2 - 4 = 0$. Non-example: $p(x, y) = x^2 + y^2$ is not harmonic, since $\Delta p = 4$. The Laplacian maps a homogeneous polynomial of degree $k$ to one of degree $k - 2$, so a polynomial is harmonic if and only if each of its homogeneous parts is harmonic. The homogeneous harmonic polynomials of degree $k$ in $n$ variables form a vector space $\mathcal{H}_k(\R^n)$ of dimension $\binom{n+k-1}{k} - \binom{n+k-3}{k-2}$, where the second term is $0$ for $k < 2$. For $n = 2$ and $k \ge 1$ this dimension is $2$, with basis $\operatorname{Re}(x + iy)^k$ and $\operatorname{Im}(x + iy)^k$; for $n = 3$ it is $2k + 1$. The restrictions of the elements of $\mathcal{H}_k(\R^n)$ to the unit sphere are the spherical harmonics of degree $k$.