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  "body": "\n## Summary\n\nA harmonic polynomial is a polynomial in $n$ real variables whose Laplacian is the zero polynomial.\nHarmonic polynomials are the simplest [[harmonic functions|0DEB0JMN0W]] on all of $\\R^n$.\nThe homogeneous ones of degree $k$ form a finite-dimensional space, and their restrictions to the unit sphere are the [[spherical harmonics|3F32Y21HP4]].\nEvery harmonic function on a ball is a locally uniform limit of harmonic polynomials, so they play the role that monomials play for power series.\n\n## Explanations\n\n### L1 human\n\nTake a polynomial such as $x^2 - y^2$ and add up its second derivatives in each variable: $2 - 2 = 0$.\nWhen that sum is zero, the polynomial is called harmonic.\nIn the plane, the real and imaginary parts of $(x + iy)^k$ give two harmonic polynomials of each degree $k \\ge 1$, for example $x^2 - y^2$ and $2xy$.\nThe polynomial $x^2 + y^2$ is not harmonic, since the sum of its second derivatives is $4$.\n",
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