Harmonic polynomial

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In mathematics, in abstract algebra, a multivariate polynomial over a field whose Laplacian is zero is termed a harmonic polynomial.

The harmonic polynomials form a vector subspace of the vector space of polynomials over the field. In fact, they form a graded subspace.

The Laplacian is the sum of second partials with respect to all the variables, and is an invariant differential operator under the action of the orthogonal group viz the group of rotations.

The standard separation of variables theorem states that every multivariate polynomial over a field can be decomposed as a finite sum of products of a radical polynomial and a harmonic polynomial. This is equivalent to the statement that the polynomial ring is a free module over the ring of radical polynomials.


  • Lie Group Representations of Polynomial Rings by Bertram Kostant published in the American Journal of Mathematics Vol 85 No 3 (July 1963)