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  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $p \\in \\R[x_1, \\dots, x_n]$ be a [[polynomial|1G0B73WPX8]] with real coefficients.\nThe polynomial $p$ is *harmonic* if its [[Laplacian|0E0B6RA8K0]] is the zero polynomial:\n$$\\Delta p = \\frac{\\partial^2 p}{\\partial x_1^2} + \\dots + \\frac{\\partial^2 p}{\\partial x_n^2} = 0.$$\n\n## Remarks\n\nExample: $p(x, y, z) = x^2 + y^2 - 2z^2$ is harmonic, since $\\Delta p = 2 + 2 - 4 = 0$.\nNon-example: $p(x, y) = x^2 + y^2$ is not harmonic, since $\\Delta p = 4$.\nThe Laplacian maps a [[homogeneous polynomial|2C3PAQ7H50]] 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.\nThe 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 $k \u003c 2$.\nFor $n = 2$ 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 $2k + 1$.\nThe restrictions of the elements of $\\mathcal{H}_k(\\R^n)$ to the unit sphere are the [[spherical harmonics|3F32Y21HP4]] of degree $k$.\n",
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