In mathematics, a polynomial whose Laplacian is zero is termed a harmonic polynomial.

The harmonic polynomials form a subspace of the vector space of polynomials over the given field. In fact, they form a graded subspace. For the real field), the harmonic polynomials are important in mathematical physics.

The Laplacian is the sum of second-order partial derivatives with respect to each of the variables, and is an invariant differential operator under the action of the orthogonal group via 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 radial polynomial and a harmonic polynomial. This is equivalent to the statement that the polynomial ring is a free module over the ring of radial polynomials.

Examples

. In order to be harmonic, this polynomial must satisfy.

Note that, as in any vector space, there are other choices of basis for this same space of polynomials.

See also

• Harmonic function

• Spherical harmonics

• Zonal spherical harmonics

• Multilinear polynomial

References

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